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		<title>free range statistics - R</title>
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				<title>Reproducing and adapting the UN Population Projections by @ellis2013nz</title>
					<description>&lt;p&gt;I set out to see if it was possible to reproduce the UN’s &lt;a href=&quot;https://population.un.org/wpp/&quot;&gt;2022 Revision of World Population Prospects&lt;/a&gt; for a given country by cohort component projection from the fertility, mortality and immigration rates and population starting point published as part of their projection. The motivation is to make small changes to some of those parameters - for example by substituting in a recent census result for the population and a given year - and “re-run” the projections to see the impact of changes, or to get a more up-to-date version with data that wasn’t available to the UN at the time of their projection.&lt;/p&gt;

&lt;p&gt;It turns out this wasn’t too hard (one morning’s work for the modelling, then a few hours of write-up), particularly in cases where migration is small in the projection period. I was able to reproduce almost exactly population, birth and death totals to 2100 for Vanuatu, and demonstrate the impact of updating the 2020 year for their recent census population totals and fertility rates, and getting a slightly lower projection as a result.&lt;/p&gt;

&lt;p&gt;Here’s my reproduction of Vanuatu’s population projection from 2020 to 2100, using just the 2020 population totals and the forecast fertility, mortality and migration rates. As you can see it’s basically identical to the UN totals:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0262-vanuatu-pop.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0262-vanuatu-pop.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;And here’s the same method tweaked for the actual 2020 census total and with a rough adjustment made to fertility rates based on what was observed at the 2020 census:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0262-vanuatu-revised.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0262-vanuatu-revised.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Of course, this method delivers a full set of projections by age and sex, and we could construct life tables or any indicators we want from it. Here are population pyramids comparing the published UN projections for 2050 with my revised set. Not visually stunning in its comparison, but enough to prove that it’s possible:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0262-vanuatu-pyramid.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0262-vanuatu-pyramid.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;h2 id=&quot;reproducing-un-projections&quot;&gt;Reproducing UN projections&lt;/h2&gt;

&lt;p&gt;So here’s how I went about that.&lt;/p&gt;

&lt;p&gt;First, downloading all the data. The UN recommend bulk downloads of their CSV files. For my purposes I first need the fertility, mortality and population by sex and one year age groups. Of the population original data I am only going to use the 2020 year, and then project it forward myself based on fertility, mortality and migration; but I want the full set for comparison purposes. For migration, I couldn’t see in my hasty look at the UN site a dataset of migration projections by age and sex, so I just use the much simpler net migration rate (per thousand people) per year in the projection period. Here’s code to download all this UN data:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;patchwork&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dir.create&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;showWarnings&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#------------------download and import data for all countries from existing projections----------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;list.files&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;files&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;WPP2022_Fertility_by_Age1.zip&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;WPP2022_DeathsBySingleAgeSex_Medium_1950-2021.zip&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;WPP2022_DeathsBySingleAgeSex_Medium_2022-2100.zip&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;WPP2022_Population1JanuaryBySingleAgeSex_Medium_1950-2021.zip&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;WPP2022_Population1JanuaryBySingleAgeSex_Medium_2022-2100.zip&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;WPP2022_Demographic_Indicators_Medium.zip&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# let downloads take up to 10 minutes rather than 1 minute max, as files are&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# large (largest is fertility for single age, 78MB)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;options&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;timeout&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;600&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;file.exists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/WPP2022_Demographic_Indicators_Medium.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# download the zip files:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;files&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;download.file&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://population.un.org/wpp/Download/Files/1_Indicators%20(Standard)/CSV_FILES/{files[i]}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;destfile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/{files[i]}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mode&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;wb&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# unzip them&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;files&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unzip&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/{files[i]}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;exdir&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fert_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;    &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/WPP2022_Fertility_by_Age1.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_past&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/WPP2022_DeathsBySingleAgeSex_Medium_1950-2021.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_future&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/WPP2022_DeathsBySingleAgeSex_Medium_2022-2100.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_past&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;    &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/WPP2022_Population1JanuaryBySingleAgeSex_Medium_1950-2021.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_future&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/WPP2022_Population1JanuaryBySingleAgeSex_Medium_2022-2100.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/WPP2022_Demographic_Indicators_Medium.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_past&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_future&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_past&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_future&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# clean up&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_past&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_future&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_past&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_future&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Next, I wrote my own cohort component population projection function. I wanted to do this from scratch rather than using an existing demography package to make sure I understood what was happening (I’m not a demographer) and could make tweaks as necessary to match the UN approach. I used &lt;a href=&quot;https://farid-flici.github.io/tuto.html&quot;&gt;this tutorial&lt;/a&gt; by Farid Flici as my starting point; abstracted his code into a function for easy use with multiple countries, and added a net migration component.&lt;/p&gt;

&lt;p&gt;Because migrants’ ages tend to be dissimilar from the country they are migrating too - they are more likely to be in the prime of their working / family life I believe - I needed a way to set the ages of migrants. In the function below I defaulted to a normal distribution of ages, mean 28 and standard deviation 11, which worked well to get similar results to the UN in a few countries I tried. This was the hardest and most discretionary part of the exercise.&lt;/p&gt;

&lt;p&gt;My observation is that demographers seem to think in terms of matrices of numbers rather than database-oriented tidy data, and I have kept that matrix approach in this function.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#----------------------projections for one country-------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# adapting the implementation of the method for cohort component projection at &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# https://farid-flici.github.io/tuto.html&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param start_pop_m is a vector of population of males at single year age&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   periods beginning of year starting age 0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param start_pop_f as start_pop_m but for females&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param start_year first year ie the year for which the actual population&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   number refers to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param end_year end year for the population projection&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param fertility a matrix with rows for female agegroups 15 to 49 and a&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   column for each year, values are births per woman (not per thousand women)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   that year and age. Must have 35 rows and number of columns equal to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   end_year minus start_year plus one.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param mort_m a matrix with rows for age 0 to at least 50 and columns for&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   each year, for males. Must have number of columns equal to end_year minus&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   start_year plus one.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param mort_f as mort_m but for females. Must have same number of rows and&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   columns as mort_m.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param sex_ratio_birth number of boys born for every girl born, vector with&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   length of years to be projected.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param net_migration vector of proportions of the population that migrate to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   the country, net of those who leave. Defaults to a vector of zeroes.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param female_share_migration vector of proportions of net migrants that are&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   female. Defaults to a vector of 0.5s.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param migration_age_weights vector of relative distribution of the age of&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   net migrants. Must be 121 values representing relative proportion of net&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   migrants aged zero to 120. Defaults to a normal distribution of mean 28 and&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   standard deviation 11. Note that this distribution currently must be the&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   same through the projection period (unlike eg fertility and mortality,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   which have individual values for each age group and sex for each year).&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   Only the total net migration proportion can change over time, not its age&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   make-up.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @author Peter Ellis, expanding on a less-featured example by Farid Flici&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   https://farid-flici.github.io/tuto.html&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_proj&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fertility&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex_ratio_birth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.045&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;net_migration&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;female_share_migration&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migration_age_weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;28&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sd&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;11&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fertility&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;35&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migration_age_weights&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fertility&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;50&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;50&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;net_migration&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex_ratio_birth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;female_share_migration&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;min&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;female_share_migration&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;female_share_migration&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# fill in with high probability of death for ages beyond where we have mortality data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extra_deaths_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extra_deaths_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extra_deaths_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extra_deaths_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Create matrices for male and female population for the full projections&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# each row of the matrix is an age group, each column is a year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deaths&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rownames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rownames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;                &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# ages&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colnames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colnames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# first year gets populated with the actual population numbers that we have:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# subsequent years get modified by people aging one year, deaths, births, and net migration&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Age one and above:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deaths&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# migration. &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_migrants&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;net_migration&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migf&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_migrants&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;female_share_migration&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migration_age_weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migration_age_weights&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_migrants&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;female_share_migration&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migration_age_weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migration_age_weights&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migf&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Age zero ie births. Fertility rate by the number of women in the middle of the previous year.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Note that PopF rows 16:50 equates to women age 15:49&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reproducing_women&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;16&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;50&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;16&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;50&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop_boys&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex_ratio_birth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex_ratio_birth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;          
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop_girls&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop_boys&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reproducing_women&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%*%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fertility&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop_boys&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reproducing_women&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%*%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fertility&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop_girls&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Return a list of the two matrices&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deaths&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deaths&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Next, I wrote a function to extract the necessary fertility, mortality, migration rates and 2020 starting population from the UN data, feed it into my &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;pop_proj()&lt;/code&gt; function and return the result. Unlike &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;pop_proj()&lt;/code&gt;, which is to some degree fully portable, this function is very much specific to this particular project and is really just a convenience function for grabbing the data and turning it into the right units and shapes (vectors and matrices) needed for &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;pop_proj()&lt;/code&gt;.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;repeat_un_proj&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2020&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Country not found&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_fert&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fert_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Age-Specific Fertility rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ASFR&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# turn into proportions, not rates per 1000:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ASFR&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ASFR&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pivot_wider&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id_cols&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;names_from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values_from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ASFR&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# should be 35 rows ie fertilities for ages 15 to 49&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_fert&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;35&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# should be 81 columns, 1 column for each year from 2020 to 2100&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_fert&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Mortality is in numbers not a ratio so we need to join to the population data to turn it into a ratio&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopMale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopFemale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Time&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;AgeGrp&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# next step important because we will be sorting by AgeGrp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;100+&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;suppressWarnings&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
    
    
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# sometimes more deaths than people (eg 1 death, 0 people) so cap the death ratio at 1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathMale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pmin&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathMale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopMale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathMale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pivot_wider&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id_cols&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;names_from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values_from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathMale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathFemale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pmin&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathFemale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopFemale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathFemale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pivot_wider&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id_cols&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;names_from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values_from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathFemale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# check the years are correct, didn't get mangled or reordered&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;all&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colnames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;all&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colnames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;min&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# next step important because we will be sorting by AgeGrp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;100+&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;suppressWarnings&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# convert to units, not thousands of people:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopMale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopFemale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# reality check&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Population in millions; should be about 0.3 if the_country is Vanuatu, about 1400 if India:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1e6&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# net migration&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_cnmr&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;CNMR&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# sex ratio at birth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_srb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;SRB&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_proj&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_proj&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;min&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fertility&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_fert&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;net_migration&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_cnmr&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex_ratio_birth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_srb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_proj&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_pop_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_pop_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_fert&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_fert&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_mort_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_mort_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_cnmr&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_cnmr&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_srb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_srb&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Note that this function returns, in addition to the results of the population projection, the various inputs in their correct units and shape. This will be useful later when we want to modify some of those inputs.&lt;/p&gt;

&lt;p&gt;Now that we’ve got these functions, using them to do projections from 2020 and compare those projections to the published numbers is pretty straight forward. Here’s the code to do that for Vanuatu, which produces the first chart at the top of this blog post:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Vanuatu&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_proj&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;repeat_un_proj&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# total population&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2020&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`UN original`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;TPopulation1Jan&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Reproduction`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;apply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;apply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# First year should be an exact match:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_data&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`UN original`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_data&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Reproduction&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Attempt to re-create the UN population projections from population in 2020, fertility and mortality rates&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Population&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0262-vanuatu-pop.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0262-vanuatu-pop.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;As we can see the results are pretty much identical. Let’s look at my projected births and deaths based on fertility and mortality rates, and compare them to the published projected numbers&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# births&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2020&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`UN original`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Births&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Reproduction&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Attempt to re-create the UN population projections from population in 2020, fertility and mortality rates&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Births (thousands)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# deaths&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2020&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`UN original`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Deaths&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Reproduction&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deaths&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Attempt to re-create the UN population projections from population in 2020, fertility and mortality rates&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Deaths (thousands)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0262-vanuatu-births.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0262-vanuatu-births.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;
&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0262-vanuatu-deaths.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0262-vanuatu-deaths.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;I’m pretty happy with those. There’s definitely some discrepancies and a lag in the births which I suspect come down to how one treats populations of mothers - numbers on 1 January v 1 July, that sort of thing. But in the scheme of things these are very small.&lt;/p&gt;

&lt;p&gt;Now, Vanuatu is relatively easy because the UN assumed net zero migration in the projection period. We can see this by comparing the net migration rates in their Indicators dataset for a few countries, with this code:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;plot_mig&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;CNMR&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_vline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xintercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2022&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lty&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_hline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yintercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lty&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Net migration rate per thousand people&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Source: UN World Population Prospects 2022&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot_mig&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Vanuatu&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot_mig&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Fiji&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot_mig&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Australia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot_mig&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;India&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot_layout&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0262-migration.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0262-migration.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;For countries that have good data on it, migration is a big deal in the projections; but forecasting is hard, particularly of the future.&lt;/p&gt;

&lt;p&gt;My first few goes at reproducing the projections for Australia and India tended to be badly out because I had added in net migration evenly across the whole age distribution. My eventual solution, making net migration bell curved with an average age of 28, is a bit of a hack with the parameters chosen to make Australia’s projections come out right. Definitely a better method would be to have actual age-specific net migration forecasts. Whether such things are possible will very much depend on the country; it’s probably possible for Australia, but not for most of the countries I work with.&lt;/p&gt;

&lt;p&gt;Here’s the final result comparing UN projections with mine for a few interesting countries:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0262-Australia-pop.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0262-Australia-pop.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;
&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0262-China-pop.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0262-China-pop.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;
&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0262-India-pop.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0262-India-pop.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;h2 id=&quot;adjusting-the-starting-point&quot;&gt;Adjusting the starting point&lt;/h2&gt;

&lt;p&gt;Now, the whole point of this exercise was to see if we can plausibly adjust the starting point - say the population totals in 2020, or the forecast fertility rates - and say we are building on the UN’s projections to get our own.  Here’s my rough demo of how we might do that, again using the case of Vanuatu. Vanuatu’s 2020 census wasn’t available at the time of the UN’s 2022 population projections, so the actual population and fertility numbers for 2020 differ somewhat (of course) from what was projected.&lt;/p&gt;

&lt;p&gt;For the below, I am relying on &lt;a href=&quot;https://sdd.spc.int/digital_library/vanuatu-2020-national-population-and-housing-census-analytical-report-volume-2&quot;&gt;the analytical report of the Vanuatu census&lt;/a&gt;. On a quick glance I didn’t see a table of the actual total population by single year age and sex, so I just adjusted the UN’s projected totals by a factor to make them add up to the correct 2020 total males and females. Of course if we were doing this for real we’d get the real numbers straight from the census; I actually can do this easily enough at work but obviously for this blog wanted to use only easily accessible public data.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#-----------------changing one or two factors while keeping the rest the same--------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Say we had the 2020 Vanuatu census so we knew the real population then, and wanted&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# to redo the population projections with everything staying as they were before&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# file:///C:/Users/Peter/Downloads/Vanuatu_2020_Census_Analytical_report_Vol_2.pdf&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revision_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Vanuatu&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;repeat_un_proj&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revision_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# rough adjustment of starting population in 2020 to make totals match the Census totals.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# In principle could of course use the actual numbers by single age category.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_pop_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_pop_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;151597&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_pop_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_pop_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;148422&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# calculate the crude birth rate in the current projection&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_cbr_2020&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# make a rough adjustment of future fertility rates assuming they are &quot;out&quot; by the&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# same proportion that crude birth rate was out in 2020&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;adj_ratio&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;28.2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_cbr_2020&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_fert&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_fert&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;adj_ratio&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Refit the projection with the above rough adjustments:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_proj&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_proj&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2020&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fertility&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_fert&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex_ratio_birth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_srb&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;net_migration&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_cnmr&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;projected_pop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;apply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;apply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#-----------------Compare total population----------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revision_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2020&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`2022 UN projections`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;TPopulation1Jan&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Revised with 2020 census`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;projected_pop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p5&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2050&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revision_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Attempt to re-create the UN population projections from population in 2020, fertility and mortality rates&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Population&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#--------------Compare population pyramids-------------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;31&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;31&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;31&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;31&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Male&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Female&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Male&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Female&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;each&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Revised projections with 2020 census, made in 2024&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;UN projections, made in 2022&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;each&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;242&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;agef&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_reorder&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_rev&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Female&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;agef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;brown&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Male&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;orange&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_vline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xintercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Number of people&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Comparison of UN original and revised population projections&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Population age distribution in 2050&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-4000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-2000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labels&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;4,000&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Male&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Female&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;4,000&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;panel.grid&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;element_blank&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_discrete&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;20&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;And that’s what gets us these results:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0262-vanuatu-revised.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0262-vanuatu-revised.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0262-vanuatu-pyramid.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0262-vanuatu-pyramid.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;I’m hoping this might be actually useful for pragmatic updates of the UN population projections when more current data is available, without having to revise everything from scratch.&lt;/p&gt;

&lt;p&gt;OK, that’s all for today.&lt;/p&gt;
</description>
				<pubDate>Sun, 05 May 2024 00:00:00 +0000</pubDate>
				<link>https://freerangestats.info/blog/2024/05/05/pop-projections</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/05/05/pop-projections</guid>
			</item>
		
			<item>
				<title>The 'V20' group of vulnerable countries and the MVI by @ellis2013nz</title>
					<description>&lt;p&gt;Just a brief blog post that is basically an addendum to &lt;a href=&quot;/blog/2023/09/30/mvi&quot;&gt;the one I wrote a few weeks ago&lt;/a&gt; on the United Nations Multidimensional Vulnerability Index. I found myself wondering about the &lt;a href=&quot;https://www.v-20.org/about&quot;&gt;Vulnerable Twenty (V20) Group of Ministers of Finance of the Climate Vulnerable Forum&lt;/a&gt;, which is now (despite the ‘20’ in its name) a collection of 68 economies.&lt;/p&gt;

&lt;p&gt;How vulnerable do these 68 economies look on the proposed MVI? I imagine they and others will be interested. After all, while ‘vulnerable to climate change’ (for the V20) is not the same as ‘vulnerable to anything’ (for the MVI), they are both using the same word. Given how prominently and frequently ‘climate change’ and ‘vulnerability’ are connected in this context, I suspect some stakeholders would be expecting the two approaches to have a similar set of vulnerable countries&lt;/p&gt;

&lt;p&gt;It turns out that the V20 economies are (according to the MVI) only slightly more ‘vulnerable’ than the average of the 142 countries that have an MVI score. On the other hand, they are materially poorer.&lt;/p&gt;

&lt;p&gt;Consider this table of median values:&lt;/p&gt;

&lt;table class=&quot;table&quot; style=&quot;margin-left: auto; margin-right: auto;&quot;&gt;
 &lt;thead&gt;
  &lt;tr&gt;
   &lt;th style=&quot;text-align:left;&quot;&gt;    &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; Structural vulnerability &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; Lack of structural resilience &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; Multidimensional vulnerability &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; GDP per capita, PPP &lt;/th&gt;
  &lt;/tr&gt;
 &lt;/thead&gt;
&lt;tbody&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:left;&quot;&gt; Member of V20 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 49.2 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 57.1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 54.4 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; $5,500 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:left;&quot;&gt; Not a member of V20 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 48.4 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 55.0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 52.4 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; $13,700 &lt;/td&gt;
  &lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;

&lt;p&gt;In fact, many countries in the V20 have below-average MVI scores; and there are also countries that aren’t in the V20 but have notably above-average MVI scores. Consider this chart:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0257-discrepancies.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0257-discrepancies.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;The blue labelled countries are non-members of the V20 with an MVI of above 65. The red labelled countries are members of the V20 with an MVI below 50 (by construction, 50 is the average MVI score, so below 50 means below average). The GDP per capita, in purchasing power parity (PPP) terms in 2017 international dollars, is just for contextual information; it isn’t used for determining which countries are labelled or not.&lt;/p&gt;

&lt;p&gt;So clearly the countries that have self-identified as ‘vulnerable’ to climate change are not the same as the set you would choose based on the new Multidimensional Vulnerability Index with its broader concept of vulnerability. I’m not going to say which is right and which wrong, or whether both are right in their own way; just pointing out the fact.&lt;/p&gt;

&lt;p&gt;That’s really all for today.&lt;/p&gt;

&lt;p&gt;Here’s the R code that does this analysis. You can’t just copy it and run it in a fresh R session however, contrary to my usual practice; you need to have run the code for the last post’s analysis first, or be running this in a clone of my whole &lt;a href=&quot;https://github.com/ellisp/blog-source&quot;&gt;blog source repository&lt;/a&gt;.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# This is a comparison of membership of the V20 with the MVI and GDP per capita&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#------------Data prep-----------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;WDI&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;kableExtra&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clipr&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# run the script from the previous blog, to load in the MVI data in particular,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# and to load in all the standard R packages (tidyverse in particular)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;source&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;0255-mvi-pacific.R&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# get the world Banks' GDP data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# 	&quot;GDP per capita, PPP (constant 2017 international $)&quot;:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_ppp_pc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;WDI&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;NY.GDP.PCAP.PP.KD&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;drop_na&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;iso3c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# just get the latest year for each country:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;year&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;iso3c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;year&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_pc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;NY.GDP.PCAP.PP.KD&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# List of members of the V20 taken from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# https://www.v-20.org/v20-ministerial-dialogue-xi-communique#_ftnref5&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;v20&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Afghanistan&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Bangladesh&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Barbados&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Benin&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Bhutan&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Burkina Faso&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Cambodia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Chad&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Colombia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Comoros&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Costa Rica&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Côte D'Ivoire&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Democratic Republic of the Congo&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Dominica&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Dominican Republic&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Eswatini&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Ethiopia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Fiji&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Gambia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Ghana&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Grenada&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Guatemala&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Guinea&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Guyana&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Haiti&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Honduras&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Jordan&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Kenya&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Kiribati&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Kyrgyzstan&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Lebanon&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Liberia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Madagascar&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Malawi&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Maldives&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Marshall Islands&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Mongolia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Morocco&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Mozambique&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Namibia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Nepal&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Nicaragua&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Niger&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Pakistan&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Palau&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Palestine**&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Papua New Guinea&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Paraguay&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Philippines&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Rwanda&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Saint Lucia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Samoa&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Senegal&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Sierra Leone&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;South Sudan&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Sri Lanka&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Sudan&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;United Republic of Tanzania&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Timor-Leste&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Togo&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Tonga&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Trinidad and Tobago&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Tunisia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Tuvalu&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Uganda&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Vanuatu&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Viet Nam&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Yemen&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# A few hand edits were made so country names match those in the MVI (eg&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# D'Ivoire rather than d'Ivoire, etc)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# any non-matches? Yes, Palestine:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;v20&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;v20&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mvi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Palestine not in the mvi.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Mark which countries are in the V20, and join to the GDP data:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mvi2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mvi&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;v20&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Member of V20&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Not a member of V20&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_ppp_pc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;ISO&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;iso3c&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# note there are seven countries with no GDP PPP per capita values:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mvi2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;is.na&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_pc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#===============comparisons============&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#---------Table------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Average values by v20 membership or not:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mvi2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Structural vulnerability`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;median&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;svi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Lack of structural resilience`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;median&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lsri&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Multidimensional vulnerability`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;median&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`MVI - Score`&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`GDP per capita, PPP`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dollar&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;median&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_pc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;na.rm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;accuracy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rename&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`  `&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;kable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;digits&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;format.args&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;big.mark&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;,&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;align&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;l&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;r&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;r&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;r&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;r&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;kable_styling&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;write_clip&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# V20 members are very slightly more vulnerable, and notably poorer,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# than non-members&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#-------------Plot-------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pal&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Member of V20&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;darkred&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Not a member of V20&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;darkblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mvi2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_pc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`MVI - Score`&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_text_repel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mvi2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`MVI - Score`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;50&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Member of V20&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                                  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`MVI - Score`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;60&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Member of V20&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;label&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;800&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;37&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fontface&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;italic&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label.size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey90&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hjust&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Member of V20 but lower than average vulnerability according to MVI&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;label&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;800&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;73&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fontface&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;italic&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label.size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey90&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hjust&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Not a member of V20 but high vulnerability according to MVI&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_log10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dollar&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_colour_manual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;GDP per capita, PPP, 2017 prices&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Multidimensional Vulnerability Index&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Membership of the 'V20' group of vulnerable coutries&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Compared to scores on the UN's Multidimensional Vulnerability Index (MVI)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Source: UN https://www.un.org/ohrlls/mvi (MVI), World Bank World Development Indicators (GDP), analysis by freerangestats.info
Palestine is a member of V20 but does not have a MVI. Seven countries have an MVI but are missing GDP data.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

</description>
				<pubDate>Tue, 17 Oct 2023 00:00:00 +0000</pubDate>
				<link>https://freerangestats.info/blog/2023/10/17/mvi-and-v20</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2023/10/17/mvi-and-v20</guid>
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				<title>The UN's proposed Multidimensional Vulnerability Index by @ellis2013nz</title>
					<description>&lt;p&gt;The United Nations President of the General Assembly’s &lt;a href=&quot;https://www.un.org/ohrlls/mvi&quot;&gt;High-Level Panel&lt;/a&gt; on the development of a Multidimensional Vulnerability Index have (September 2023) released an &lt;a href=&quot;https://www.un.org/ohrlls/sites/www.un.org.ohrlls/files/final_mvi_report.pdf&quot;&gt;Advanced Unedited Version of their Final Report&lt;/a&gt;, and along with it the scores on that index for 142 countries. This is relevant for my work; it’s pretty complex; and I wanted to understand it fully so I downloaded it and drew some charts.&lt;/p&gt;

&lt;p&gt;A good measure of “vulnerability” has been sought for at least 30 years as a supplement to measures of economic prosperity, with a number of parties (particularly small island developing states) advocating for its inclusion in eligibility decisions for financial and other forms of assistance. Vulnerability has formally been a factor in determining Least Developed Country status for some time now. &lt;a href=&quot;https://www.un.org/ohrlls/mvi/history-of-mvi&quot;&gt;This High Level Panel was commissioned&lt;/a&gt; to resolve the contradictions in the various measures out there and address the “lack of international consensus on how to characterize and, by extension, mitigate their (small states’) vulnerabilities”.&lt;/p&gt;

&lt;p&gt;There is (I think) an awkward contradiction between acknowledging the multidimensional nature of vulnerability and the desire for a single index that can be used for financing decisions. The High Level Panel deal with this by creating an elegant hierarchy where increasingly abstract indexes are created out of other indexes, which themselves are made out of actual indicators that have been scaled from 0 to 100, with 100 meaning the most vulnerable (or least resilient). All the details are in the report linked to above.&lt;/p&gt;

&lt;p&gt;Numbers are combined to form the index in the next level up of the hierarchy by taking the square root of the mean of the squares of the original numbers, which has the effect of pushing the average a bit towards the higher numbers in the inputs; this is acknowledgement of the common judgement that you don’t need to be vulnerable against all the inputs to be vulnerable overall. Consider that the arithmetic mean of (2, 6) is 4; the geometric mean is 2.8 (pushed downwards); but the square root of the mean of the squares is 4.5 (pushed upwards). If that “6” meant you were vulnerable to something and the “2” you weren’t to something else, using 4.5 as your combined measure is giving a bit more importance to the variable that you are vulnerable to.&lt;/p&gt;

&lt;h2 id=&quot;visualising-the-results&quot;&gt;Visualising the results&lt;/h2&gt;

&lt;p&gt;First lets look at some results for the Pacific, the area I’m working on in my day job. Here are the values on the final, single Multidimensional Vulnerability Index (MVI) for the 12 Pacific Island countries in the dataset, compared to other countries in grey:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0255-overall-index.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-overall-index.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Unlike most of my posts, I haven’t interspersed my text and output with code creating it, because I want the results to be easy to focus on without distraction in talking about how the visualisations are done. But that code is still &lt;a href=&quot;https://github.com/ellisp/blog-source/blob/master/_working/0255-mvi-pacific.R&quot;&gt;available in one spot here&lt;/a&gt;.&lt;/p&gt;

&lt;p&gt;A few surprises in that first plot, for example that Vanuatu (often used as a case study of a vulnerable country, that has had a bad run recently with cyclones) is only very slightly above world median vulnerability.&lt;/p&gt;

&lt;p&gt;Next level down from the main index we have two important component indexes - “Structural Vulnerability” and “Lack of Structural Resilience”. The final index for each country is their average of scores on these two, so let’s look again at the Pacific countries on the two indexes at once, again with the rest of the world’s countries shown in grey:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0255-two-indexes.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-two-indexes.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Hmm ok, that’s interesting; for example we can see that Tuvalu gets it’s relatively high overall score (that we saw in the first plot where it was ranked second most vulnerable of Pacific countries) almost entirely from “lack of structural resilience” and in contrast is actually slightly below the median score for “Structural vulnerability”.&lt;/p&gt;

&lt;p&gt;The next two charts let us see what the concepts are behind these two indexes.&lt;/p&gt;

&lt;p&gt;First, here’s “structural vulnerability”. There are two levels of concepts I’ve merged into one chart here; the first three (economic, environmental and social vulnerability) are themselves made up of the more detailed remaining nine. We’re still dealing with abstracts, not down to any actual data here. But at least we can see what it is that conceptually is leading to high or low scores. In the case of Nauru for example we see a particularly high score for “exposure to entrance of international forced displacement of people”. Hmm, there’s some complicated history behind that one.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0255-sv-concepts.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-sv-concepts.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;And here’s the concepts making up “lack of structural resilience”:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0255-lsr-concepts.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-lsr-concepts.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;We can see that these “resilience” concepts are about the ability of a society to respond to shocks they will get as a result of vulnerability; for example the lack economies of scale, adequate water supply, or a resilient agricultural system.&lt;/p&gt;

&lt;p&gt;Finally, one more level down, here are the actual measureable indicators that are turned into those concepts, and ultimately the index itself:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0255-sv-variables.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-sv-variables.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0255-lsr-variables.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-lsr-variables.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;h2 id=&quot;visualising-the-hierarchy-of-concepts&quot;&gt;Visualising the hierarchy of concepts&lt;/h2&gt;

&lt;p&gt;But which variables feed in to which concepts, and how to grasp this all at once? To help me understand how the variables get aggregated into concepts, then into higher level concepts, then into two indexes and finally one index, I constructed a network graph to illustrate the hierarchy:&lt;/p&gt;

&lt;p&gt;&lt;a href=&quot;https:/freerangestats.info/img/0255-construction.png&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-construction.png&quot; width=&quot;100%&quot; /&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;This visualisation works for me - at least on a reasonable sized screen. But this, and all of the following network charts, are a bit fiddly to read in-line in the blog so I’d recommend opening them in another tab (with right mouse button) or clicking on them to navigate straight there (and then you can use the back key to come back here to the main text).&lt;/p&gt;

&lt;p&gt;This layout suggests a nice option - we can colour-code the nodes (representing variables, concepts or index scores) according to an actual country’s scores, and see what’s pushing the country’s vulnerability scores up and down and how they add up to the overall score. Here’s such visualisations for each of our 12 Pacific countries, grouped by subregion:&lt;/p&gt;

&lt;h3&gt;Melanesia&lt;/h3&gt;
&lt;h4&gt;Papua New Guinea&lt;/h4&gt;
&lt;p&gt;&lt;a href=&quot;https:/freerangestats.info/img/0255-mvi-Papua New Guinea.png&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-mvi-Papua New Guinea.png&quot; width=&quot;100%&quot; /&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Most vulnerable scores (bright yellow) for PNG coming from gender equity, and lack of access to sanitation. Non-vulnerable scores coming from no problem with drylands, coastal zones, and access to freshwater.&lt;/p&gt;

&lt;h4&gt;Fiji&lt;/h4&gt;
&lt;p&gt;&lt;a href=&quot;https:/freerangestats.info/img/0255-mvi-Fiji.png&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-mvi-Fiji.png&quot; width=&quot;100%&quot; /&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Fiji scores higher for vulnerability on exposure to natural hazards, rainfall shocks, low connectivity and low integration wutih international markets.&lt;/p&gt;

&lt;h4&gt;Vanuatu&lt;/h4&gt;
&lt;p&gt;&lt;a href=&quot;https:/freerangestats.info/img/0255-mvi-Vanuatu.png&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-mvi-Vanuatu.png&quot; width=&quot;100%&quot; /&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Vanuatu also scores high on vulnerability to natural hazards, low connectivity, and gender equity. But it is scored with a fairly high level of enviornmental resilience, and not too socially vulnerable.&lt;/p&gt;

&lt;p&gt;… and so on. I won’t write commentary on each individual country, but for completion they are all here:&lt;/p&gt;

&lt;h4&gt;Solomon Islands&lt;/h4&gt;
&lt;p&gt;&lt;a href=&quot;https:/freerangestats.info/img/0255-mvi-Solomon Islands.png&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-mvi-Solomon Islands.png&quot; width=&quot;100%&quot; /&gt;&lt;/a&gt;&lt;/p&gt;

&lt;h3&gt;Micronesia&lt;/h3&gt;
&lt;h4&gt;Federated States of Micronesia&lt;/h4&gt;
&lt;p&gt;&lt;a href=&quot;https:/freerangestats.info/img/0255-mvi-Micronesia (Federated States of).png&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-mvi-Micronesia (Federated States of).png&quot; width=&quot;100%&quot; /&gt;&lt;/a&gt;&lt;/p&gt;

&lt;h4&gt;Marshall Islands&lt;/h4&gt;
&lt;p&gt;&lt;a href=&quot;https:/freerangestats.info/img/0255-mvi-Marshall Islands.png&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-mvi-Marshall Islands.png&quot; width=&quot;100%&quot; /&gt;&lt;/a&gt;&lt;/p&gt;

&lt;h4&gt;Kiribati&lt;/h4&gt;
&lt;p&gt;&lt;a href=&quot;https:/freerangestats.info/img/0255-mvi-Kiribati.png&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-mvi-Kiribati.png&quot; width=&quot;100%&quot; /&gt;&lt;/a&gt;&lt;/p&gt;

&lt;h4&gt;Tuvalu&lt;/h4&gt;
&lt;p&gt;&lt;a href=&quot;https:/freerangestats.info/img/0255-mvi-Tuvalu.png&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-mvi-Tuvalu.png&quot; width=&quot;100%&quot; /&gt;&lt;/a&gt;&lt;/p&gt;

&lt;h4&gt;Nauru&lt;/h4&gt;
&lt;p&gt;&lt;a href=&quot;https:/freerangestats.info/img/0255-mvi-Nauru.png&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-mvi-Nauru.png&quot; width=&quot;100%&quot; /&gt;&lt;/a&gt;&lt;/p&gt;

&lt;h4&gt;Palau&lt;/h4&gt;
&lt;p&gt;&lt;a href=&quot;https:/freerangestats.info/img/0255-mvi-Palau.png&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-mvi-Palau.png&quot; width=&quot;100%&quot; /&gt;&lt;/a&gt;&lt;/p&gt;

&lt;h3&gt;Polynesia&lt;/h3&gt;
&lt;h4&gt;Samoa&lt;/h4&gt;
&lt;p&gt;&lt;a href=&quot;https:/freerangestats.info/img/0255-mvi-Samoa.png&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-mvi-Samoa.png&quot; width=&quot;100%&quot; /&gt;&lt;/a&gt;&lt;/p&gt;

&lt;h4&gt;Tonga&lt;/h4&gt;
&lt;p&gt;&lt;a href=&quot;https:/freerangestats.info/img/0255-mvi-Tonga.png&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0255-mvi-Tonga.png&quot; width=&quot;100%&quot; /&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;OK, that was interesting. I certainly feel I understand both a) how the index is constructed and b) the ratings it is giving to Pacific Island countries.&lt;/p&gt;
</description>
				<pubDate>Sat, 30 Sep 2023 00:00:00 +0000</pubDate>
				<link>https://freerangestats.info/blog/2023/09/30/mvi</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2023/09/30/mvi</guid>
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				<title>Finding a circle in a chart by @ellis2013nz</title>
					<description>&lt;p&gt;So &lt;a href=&quot;https://twitter.com/iwontoffendyou/status/1704935240907518367&quot;&gt;this tweet&lt;/a&gt; came across my feed. To save you going there it is about a selection exercise for a job for (I think) an IT start up, described proudly by its author as “insanely hard”; and the job is to find the radius of the brown circle in this diagram:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0254-circles.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0254-circles.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;At first it just annoyed me, and my reaction was a) I’d never want to work for this guy and b) this is a really dumb way to select your team, unless you’re recruiting for the maths department at a university. I strongly believe that recruitment exercises should be:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;as similar as possible to the real work required&lt;/li&gt;
  &lt;li&gt;easy enough that any reasonable candidate can do it, but with enough complexity that some of them will do it really well&lt;/li&gt;
  &lt;li&gt;reasonable to do in the time available&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;…and I don’t think this does any of those things.&lt;/p&gt;

&lt;p&gt;But then I got sufficiently interested to think about how I’d do the answer. I’m not good enough to work it out with pen and paper in a reasonable time frame. There is of course an optimal professional way to go about it, which is to throw all the relevant equations including that of the brown circle into a machine that can solve them for you, as posted by one user on Twitter &lt;a href=&quot;(https://www.wolframalpha.com/input?i=%28x-a%29%5E2%2B%28y-r%29%5E2%3Dr%5E2%2C+y%3Dx%5E2%2C+%28x-a%29%2F%28x%29%3D-2%28y-r%29%2F1%2C+a%5E2%2Br%5E2%3D%281-r%29%5E2%2C+a%3E0
)&quot;&gt;here with WolframAlpha working it out for you&lt;/a&gt;. The answer is r = 0.213842. (Am I dreaming or did WolframAlpha used to be called Mathematica? Or is that a different thing? Ah ok &lt;a href=&quot;https://en.wikipedia.org/wiki/WolframAlpha#:~:text=r%C9%99m%2D%2F%20WUULf%2Dr%C9%99m,answers%20from%20externally%20sourced%20data.&amp;amp;text=WolframAlpha%20was%20released%20on%20May,Mathematica%2C%20a%20technical%20computing%20platform.&quot;&gt;Wikipedia shows me&lt;/a&gt; that the answer is ‘sort of’ - WolframAlpha is based on the earlier product, Mathematica.)&lt;/p&gt;

&lt;p&gt;My own intuition on how to find the answer - because I’m not fluent in WolframAlpha, which I now see is by far the best way in absolute terms - was to find the centre point of the circle first. Basically use computing power to search for a point that has its shortest point equidistant from the three lines shown. This also made me think that the work isn’t &lt;em&gt;that&lt;/em&gt; different from what I can imagine being required in jobs I can think of in the real world (less likely to be mathematical, but I can imagine looking for the largest circle that fits between some boundaries), so perhaps I was a bit harsh in dismissing the exercise entirely.&lt;/p&gt;

&lt;p&gt;Here’s how I do this brute force method in R:&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Define some values of the red line (y = sqrt(x)) and purple line (unit circle)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# for 1000 values of x:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length.out&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;line1_y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;line2_y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# As above but finer grid, for 5000 values of x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d1_fine&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length.out&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;line1_y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;line2_y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Cut down versions of that for just the visually plausible part on the left of the chart&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2_fine&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d1_fine&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Function to search a rectangle (defined by first four parameters), calculate the&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# distance of each point in the rectangle from 3 lines (vertical at 0, y = sqrt(x),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# and unit circle) and return the 100 points with the highest value of the&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# lowest of the differences from the three lines&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;get_dc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x_from&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x_to&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y_from&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y_to&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;possibilities&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;expand_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x_from&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x_to&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length.out&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y_from&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y_to&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length.out&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Distances from the points on the y = sqrt(x) line&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distances1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;expand_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;possibilities&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance_1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;((&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;line1_y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance_1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance_1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Distances from the points in the unit circle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distances2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;expand_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;possibilities&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance_2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;((&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;line2_y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance_2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance_2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distances1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distances2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;can_x&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;can_y&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# add the distance to the vertical black line&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance_3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# which is the smallest of the three distances:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;smallest&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pmin&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance_1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance_2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance_3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;smallest&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# coarse search:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dc1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;get_dc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dc1&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That gets me these results. The first two columns are the candidate values for the centre of the brown circle. The next three columns are the shortest distance from that candidate center to a set of points on the lines for the red (y=sqrt(x)), purple (unit circle), and vertical black lines. The final column is the smallest of those three values - hence the smallest circle that could be drawn from that candidate point without crossing one of the three lines. The output is ordered so the candidate points with the highest value of &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;smallest&lt;/code&gt; come first:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;# A tibble: 100 × 6
   can_x can_y distance_1 distance_2 distance_3 smallest
   &amp;lt;dbl&amp;gt; &amp;lt;dbl&amp;gt;      &amp;lt;dbl&amp;gt;      &amp;lt;dbl&amp;gt;      &amp;lt;dbl&amp;gt;    &amp;lt;dbl&amp;gt;
 1 0.212 0.758      0.216      0.213      0.212    0.212
 2 0.217 0.758      0.213      0.212      0.217    0.212
 3 0.212 0.753      0.212      0.218      0.212    0.212
 4 0.222 0.758      0.209      0.211      0.222    0.209
 5 0.217 0.753      0.209      0.217      0.217    0.209
 6 0.212 0.763      0.220      0.208      0.212    0.208
 7 0.212 0.747      0.208      0.223      0.212    0.208
 8 0.207 0.742      0.207      0.229      0.207    0.207
 9 0.207 0.747      0.211      0.224      0.207    0.207
10 0.207 0.753      0.215      0.220      0.207    0.207
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;I’d deliberately written the work above into a function so I could easily iterate from this rough effort. Taking a much finer grid using the range of 100 candidate values of my first effort, and comparing to a more detailed set of points on the red and blue lines gets me a more precise estimate of the best circle centre, and ultimately the number we were asked to find, the radius of the cirle at that point that touches the three lines.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# more precise search:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dc2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;get_dc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;min&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dc1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dc1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;min&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dc1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dc1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2_fine&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dc2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;smallest&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Which gets me this value:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;[1] 0.2138219
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;This is the same as the answer from WolframAlpha to 4 decimal places. I will note that this took me about 40 lines of code whereas the WolframAlpha solution is done in 1 (or 4, depending how you lay it out). Which shows the value of using the right tool for the job… definitely WolframAlpha when you can express those boundary lines with mathematical formula. My method has advantage that it could work with arbitrary lines (e.g. country borders or something?), not just those expressed by an equation, which is actually a pretty big advantage in what seem to me anything like a realistic application.&lt;/p&gt;

&lt;p&gt;Now, to draw the plot. This is simple, the only trick is to note the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;parse = TRUE&lt;/code&gt; argument to &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;geom_text&lt;/code&gt; when annoting with equations, and a handy function (thanks Trevor!) for drawing a circle on a ggplot2 plot:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#----------------plot----------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# See Trevor's answer at&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# https://stackoverflow.com/questions/6862742/draw-a-circle-with-ggplot2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# for this convenient function for drawing circles in a ggplot2 chart:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gg_circle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;color&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;cos&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nb&quot;&gt;pi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length.out&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sin&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nb&quot;&gt;pi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length.out&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sin&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;nb&quot;&gt;pi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length.out&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;ribbon&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymin&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymin&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymax&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymax&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;color&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;color&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lw&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# linewidth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;line1_y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;linewidth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lw&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;line2_y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;purple&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;linewidth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lw&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_vline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xintercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;linewidth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lw&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gg_circle&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dc2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dc2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dc2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;can_x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;color&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;orange&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;orange&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coord_equal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Find the radius of the brown circle&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;text&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.95&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;y==sqrt(x)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;parse&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;text&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.70&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;x^2 + y^2 == 1&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;parse&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;purple&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That’s all for today. Take care out there.&lt;/p&gt;
</description>
				<pubDate>Sat, 23 Sep 2023 00:00:00 +0000</pubDate>
				<link>https://freerangestats.info/blog/2023/09/23/circle</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2023/09/23/circle</guid>
			</item>
		
			<item>
				<title>Model life tables by @ellis2013nz</title>
					<description>&lt;p&gt;
I am working to improve my knowledge of demography. This is something I've only had a relatively superficial engagement with but it's an important part of the responsibilities of the team where I work.
&lt;/p&gt;

&lt;p&gt;
A key tool in demography is a &quot;life table&quot;, which is basically a table where rows are ages or age groups and columns are different calculations relating to the chance of the cohort in that age group dying in a given year. The cumulative probability of survival lets you directly calculate life expectancy from any given age.
&lt;/p&gt;

&lt;p&gt;
In a country with a very effective statistical system, these life tables can be estimated pretty directly. You take the number of people dying at each age from the death registrations, and the denominator of people per age cohort from the latest population projection. 
&lt;/p&gt;

&lt;p&gt;
But if your death registry data is incomplete or your population projection is unreliable, it is a much tougher job. And death registration is incomplete in many countries; enough so that indicator 17.19.2 against the Sustainable Development Goal &quot;Partnerships for the Goals&quot; includes a target for 80% of deaths to be registered; a target that many countries are going to struggle to meet by 2030. 
&lt;/p&gt;

&lt;p&gt;
To meet this use case (very common in developing countries), the United Nations provides a set of &lt;a href = 'https://www.un.org/development/desa/pd/data/model-life-tables'&gt;model life tables&lt;/a&gt;:
&lt;/p&gt;

&lt;blockquote&gt;
&quot;The United Nations and the demographic research community at large commonly use two sets of standard model life table families to derive a variety of mortality indicators and underlying mortality patterns for estimation and projection (Coale-Demeny, 1966 and 1989; United Nations, 1982). These two sets of model life tables, designed primarily for use in developing countries or for estimating historic populations, are limited to mortality patterns for a life span from age 20 to 75. A revised set of model life tables, extending the initials sets from life expectancy at birth (e(0)) from age 75.0 up to 92.5, uses both a limited life table as an asymptotic pattern and the classic Lee-Carter approach to derive intermediate age patterns (Buettner, 2002).&quot;
&lt;/blockquote&gt;

&lt;p&gt;
The basic procedure to use these, as I understand it, is that you: 
&lt;ul&gt;
&lt;li&gt;get one or more statistics that you *can* measure - like infant or under five mortality (which can be estimated with care from survey or census data) and probability of surviving to age 60 if you are 15&lt;/li&gt; 
&lt;li&gt;use your judgement and demographer community wisdom to decide the &quot;family&quot; of life tables that is most appropriate for your country, and &lt;/li&gt;
&lt;li&gt;then from that family you pick the particular model life table that most closely matches the statistic or statistics that you do have. &lt;/li&gt;
&lt;/ul&gt;
&lt;/p&gt;

&lt;p&gt;Alternatively it is possible to model the curves (of death rate ~ age) directly with a logistic function of some sort - using the curves of the model tables as a starting point and modifying the parameters according to the statistics available. That takes me beyond today's scope I think.&lt;/p&gt;

&lt;p&gt;In essence, either way, you are relying on a &quot;typical&quot; shape for the mortality at the ages that you can't measure mortality directly.&lt;/p&gt;

&lt;p&gt;These are the families available, grouped under the two types of &quot;Coale-Demeny&quot; or &quot;United Nations&quot;:&lt;/p&gt;

&lt;pre&gt;
  type              family             n
  &lt;chr&gt;             &lt;chr&gt;          &lt;int&gt;
1 CD East           East           21222
2 CD North          North          21222
3 CD South          South          21222
4 CD West           West           21222
5 UN Chilean        Chilean        21222
6 UN Far_East_Asian Far_East_Asian 21222
7 UN General        General        21222
8 UN Latin          Latin          21222
9 UN South_Asian    South_Asian    21222
  &lt;/pre&gt;

&lt;p&gt;Those values of 'n' are the number of rows of data associated with each family. Each family has 81 complete life tables for each of male and female - a life table for each value of &quot;life expectancy at birth&quot; from 20 to 100. The life tables contain mortalities for 131 ages, from 0 to 130. And 81 * 131 * 2 = 21222:&lt;/p&gt;

&lt;pre&gt;
&gt; range(mlt_raw$age)
[1]   0 130
&gt; range(mlt_raw$e0)
[1]  20 100
&gt; 81*131*2
[1] 21222
&lt;/pre&gt;

&lt;p&gt;So this lets us do some interesting comparisons. For example, you can look at the relationship between infant mortality and life expectancy for each family of the model life tables:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data='https:/freerangestats.info/img/0252-facets.svg' width='100%'&gt;&lt;img src='https:/freerangestats.info/img/0252-facets.png' width='100%'&gt;&lt;/object&gt;

&lt;p&gt;Here's the code for everything so far - downloading the model life tables from the UN, reading them into R, counting the families and drawing that plot of life expectancies:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#-------------functionality and import data----------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;readxl&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;janitor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggrepel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;download.file&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://www.un.org/en/development/desa/population/publications/pdf/mortality/EMLT/MLT_UN2011_130_1y_complete.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;destfile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;model-life-tables.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mode&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;wb&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mlt_raw&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_excel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;model-life-tables.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sheet&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Sheet1&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clean_names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#---------------exploration-----------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mlt_raw&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;count&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Colors for genders. Not sure I'm happy with these, but I want something that's not blue and pink and&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# gives roughly same perceptual prominence to both colours.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pal&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;brown&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;darkblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Female&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Male&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# life expectancy plot:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mlt_raw&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# filter to age == 0 so we can see life expectancy at birth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mx1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mx1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;e0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_colour_manual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;mx1 for age zero i.e. raw infant mortality&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;life expectancy at birth&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Alternatively we can look at a more summarised version of the data by comparing a couple of particular demographic statistics, at a given life expectancy, for the different model life tables. Here's my attempt at repeating (but adding in a sex dimension) Figure 1 from &lt;a href = 'http://demographicestimation.iussp.org/content/introduction-model-life-tables'&gt;this UNFPA instructional site&lt;/a&gt;.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data='https:/freerangestats.info/img/0252-scatter.svg' width='100%'&gt;&lt;img src='https:/freerangestats.info/img/0252-scatter.png' width='100%'&gt;&lt;/object&gt;

&lt;p&gt;I like this plot. It lets you see at a glance how the different families of model life table vary in at least one or two important ways. For example, we can see that the &quot;South&quot; (Coale-Demeny) and &quot;South Asian&quot; (UN) tables have relatively high child mortality (and then of course compensating low adult mortality) for a given life expectancy.&lt;/p&gt;

&lt;p&gt;That was done with this code. Note my struggles in the comments with exactly what is meant age 5, age 60 etc. and hence which column to use from the life table; significant expertise with life tables involves understanding the exact ways adjustments are made for things like the difference between age x and the average age when people are age x; the fact that young people die earlier in their first year rather than later; and so on:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# recreate Figure 1 from &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# http://demographicestimation.iussp.org/content/introduction-model-life-tables&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Note that age means &quot;age from x to x + n&quot; where n is the age interval ie. 1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# so age==4 means 4 to 5&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# lx1 is the number of people at the beginning of that period. It starts at&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# 100000 for age 0. lx1_2 is sort of adjusted for people perhaps in the middle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# of hte period, or some adjustment for different birthdays?&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# so taking this together, lx1 when age == 5 will be the number of people alive&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# at the beginning of age 5 to age 6&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;u5m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mlt_raw&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;e0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;60&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type_mlt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;under_five&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lx1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type_mlt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;under_five&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;am&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mlt_raw&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;e0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;60&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type_mlt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;adult&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lx1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;60&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lx1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;u5m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;am&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;family&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;sex&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;type_mlt&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;under_five&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;adult&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type_mlt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_text_repel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2.7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Under five mortality&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Adult mortality&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Model life table type&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Comparison of different families of Model Life Table&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Life expectancy = 60&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;OK so that's a nice representation of the overall life expectancy, but what about the gritty detail of the mortality rates at each individual age? One way to look at this is via an animation:&lt;/p&gt;

&lt;img src='https:/freerangestats.info/img/0252-model-life-tables.gif' width = '100%'&gt;

&lt;p&gt;I quite like this for giving you an overview of the mortality rates of the different families and different life expectancies, but it's not great for comparing say two different families with the same life expectancy. &lt;/p&gt;

&lt;p&gt;The animation was produced with this code:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#------------animation---------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;types&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mlt_raw&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;e0s&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mlt_raw&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;e0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dir.create&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;tmp_mlt&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;types&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;e&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;e0s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mlt_raw&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;e0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;e&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mx1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_colour_manual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Age&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Death rate&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Model life table type = {the_type}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Life expectancy = {e}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme_minimal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;base_family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Calibri&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;png&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;tmp_mlt/model-{the_type}_le-{e}.png&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;width&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1300&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;height&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;800&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;300&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;cairo&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dev.off&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Convert all the single frames into a GIF.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Requires ImageMagick to be installed. Can uncomment and run it here or do &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# it directly in a system / shell window&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;projdir&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;setwd&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;tmp_mlt&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;system&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;'magick -loop 0 -delay 30 *.png &quot;model-life-tables.gif&quot;'&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;setwd&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;projdir&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;To better make the comparisons that I felt the animation wasn't good at (particularly family to family of model life table), I made a shiny app. See below, or:
&lt;ul&gt;
  &lt;li&gt;&lt;a href = &quot;https://ellisp.shinyapps.io/model-life-tables/&quot;&gt;Full screen version&lt;/a&gt;&lt;/li&gt;
  &lt;li&gt;&lt;a href = &quot;https://github.com/ellisp/blog-source/tree/master/_working/0252-model-life-tables&quot;&gt;Source code&lt;/a&gt;&lt;/li&gt;
&lt;/ul&gt;


&lt;/p&gt;
&lt;!-- wrapper style for resizing iframes from http://www.collaboration133.com/how-to-scale-iframe-content-in-ie-chrome-firefox-and-safari/2717/ --&gt;
&lt;style&gt;
  #scaled-frame { width: 850px; height: 530px; border: 0px; }
  #scaled-frame {
  zoom: 0.75;
  -moz-transform: scale(0.75);
  -moz-transform-origin: 0 0;
  -o-transform: scale(0.75);
  -o-transform-origin: 0 0;
  -webkit-transform: scale(0.75);
  -webkit-transform-origin: 0 0;
  overflow: hidden;
  }

  @media screen and (-webkit-min-device-pixel-ratio:0) {
  #scaled-frame  { zoom: 1;  }
  }
&lt;/style&gt;
&lt;div style=&quot;height: 404px&quot;&gt;
  &lt;iframe id=&quot;scaled-frame&quot; src=&quot;https://ellisp.shinyapps.io/model-life-tables/&quot; style = &quot;overflow-y: hidden;&quot;&gt;&lt;/iframe&gt;
&lt;/div&gt;

&lt;P&gt;If I had a bit more oomph I would add some tooltips and stuff, but I think I've done enough to feel I'm getting the hang of this thing.&lt;/P&gt;

&lt;p&gt;As mentioned earlier, this isn't an area of deep expertise for me. I'm quite likely to have got some details of the terminology wrong, for example, so use the above with caution, and please add comments for anything you spot that I can correct.&lt;/P&gt;
</description>
				<pubDate>Sun, 06 Aug 2023 00:00:00 +0000</pubDate>
				<link>https://freerangestats.info/blog/2023/08/06/model-life-tables</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2023/08/06/model-life-tables</guid>
			</item>
		
			<item>
				<title>Log transforms, geometric means and estimating population totals by @ellis2013nz</title>
					<description>&lt;h2 id=&quot;motivation&quot;&gt;Motivation&lt;/h2&gt;

&lt;p&gt;I usually talk about statistical inference as having one of three purposes:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;estimating some finite and defined population parameter, like the mean income of mechanical apprentices in 1890s London or the number of 5-9 year olds living in Melbourne. Typically the results could in principle be measured directly by a census and the statistical modelling and inference comes in as a way to deal with sampling, measurement, and other sources of error.&lt;/li&gt;
  &lt;li&gt;predicting or forecasting some value. This might be a straight out forecast (“80% chance of between 3 and 6mm of rain tomorrow”) or some kind of hypothetical scenario modelling (“if you do A, you will probably see X happen; but if you do B you are more likely to get Y”)&lt;/li&gt;
  &lt;li&gt;generating insight into some underlying physical or social process (“women are paid less than men because of systemic sexism, not their ability to do the job”)&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Of course a bit of probing shows that the categories aren’t as clear cut as I’m making out. For example, estimating a population parameter for an official statistic will often be more effective if we have a good model of the underlying processes; and we would want to test that model by making predictions and forecasts if we can. One epistemological school of thought insists that all three methods are actually variants on predictive inference.&lt;/p&gt;

&lt;p&gt;But I find it useful schema to think through why I am ever doing some particular bit of inference.&lt;/p&gt;

&lt;p&gt;Today’s blog is about a situation where it particularly helps to think through &lt;em&gt;why&lt;/em&gt; we’re doing our inference. Because we’re trying to estimate a population total we need to make some different modelling choices from what we’d do if we were after understanding and describing an underlying process.&lt;/p&gt;

&lt;p&gt;Specifically, it’s also how we need to not just reflexively do a log transform on the data even when it clearly helps the assumptions justifying ordinary least squares estimation look better. If we’re just after an unbiased population estimate, it might be simpler and less risky to use the original scale of the data.&lt;/p&gt;

&lt;h2 id=&quot;estimating-average-diamond-price&quot;&gt;Estimating average diamond price&lt;/h2&gt;

&lt;h3 id=&quot;exploratory-data-analysis&quot;&gt;Exploratory data-analysis&lt;/h3&gt;

&lt;p&gt;Imagine we own (or know someone who owns…) the 53,940 diamonds described in the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;diamonds&lt;/code&gt; dataset that comes built into the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;ggplot2&lt;/code&gt; R package (or has it been moved now? either way, everyone has it I’m sure). Unfortunately our copy of &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;ggplot2&lt;/code&gt; is corrupted so we don’t have the price column in the data. Pricing diamonds is time-consuming and expensive so we can only do it for a sample of 1,000 of the diamonds. Our task - a classic piece of statistical inference - is to estimate the total value on the market of our diamonds collection from that sample - or equivalently, the mean price which we can multiply by 53,940 to get the total.&lt;/p&gt;

&lt;p&gt;We take our sample of 1,000 diamonds and plot the statistical density of price. This gives us this right-skewed distribution, similar to what we’d see with many economic and financial variables.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0204-distribution.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0204-distribution.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;The right skew is an important part of what follows because it is both part of the reason for doing a log transform in a common model, and the reason why it is problematic for our main inference purpose.&lt;/p&gt;

&lt;p&gt;That plot was produced with this code&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;patchwork&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Source: example analysis of the diamonds dataset in the ggplot2 R package&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;321&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;samp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diamonds&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;samp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;price&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_density&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_rug&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Distribution of diamond price is skewed rightwards&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;(Like many economic and financial variables)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Price of an individual diamond&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dollar&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;We’ve got the information on weight (in carats) and colour on our whole population of diamonds; it’s only price that’s missing. It makes a lot of sense to use this to create a model of price using our sample diamonds and apply that model to the full dataset to get a set of predicted prices.&lt;/p&gt;

&lt;p&gt;Following the methods taught in any statistics course introducing regression modelling, we start with a scatter plot of weight on the horizontal axis and price on the vertical axis.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0204-untransformed.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0204-untransformed.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;I’ve actually drawn a likely model in place here. Anyone who paid attention in that statistics course will recognise that the data set suffers from heteroskedasticity. As noted in the plot’s subtitle this breaches the Gauss-Markov assumptions, one of which is that variance is constant regardless of the expected value of the response variable. In this case - as is the case for many economic variables - the variance increases as the expected price increases, causing a familiar characteristic fanning out of the data. This means we can’t rely on Gauss-Markov to show that ordinary least squares (OLS) is the best linear unbiased estimator. In fact, we can be confident that OLS &lt;em&gt;isn’t&lt;/em&gt;, and that even if we kept the same structure of our model we would be better off with some kind of weighted least squares estimator.&lt;/p&gt;

&lt;p&gt;What you do here will depend on exactly what kind of statistics course taught you regression. Some people would reach for the weighted-least squares estimator. Others like me will instinctively say “there’s no particular reason for the relationship between weight and price to be linear in the original scale. Perhaps the marginal return for an extra carat of weight changes as we go along. If we take logarithms of both weight and price we might get something that is at least as good a model, and is likely to deal with the heteroscedasticity as a fortunate side effect”.&lt;/p&gt;

&lt;p&gt;If you were trained in this school of thought you are likely to replot the data with transformed axes, and fit a different model that is linear in the logs. Illustrated by this:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0204-transformed.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0204-transformed.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Boom, the heteroscedasticity is gone; and not only that the lines for each colour are nearly parallel. For various reasons, some philosophical and some more aesthetic, I am actually pretty sure this “linear in the logs” model is a better model for the purpose of understanding the relationship between price and weight.&lt;/p&gt;

&lt;p&gt;Those charts were produced with this chunk of code.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;samp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;price&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;carat&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;color&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_smooth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lm&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dollar_format&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;accuracy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;A linear model of the untransformed price has heteroskedasticity challenges&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Model of form price ~ carat * cut.\nResiduals are right-skewed and variance increases as expected price increases, 
breaking the Gauss-Markov assumptions.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Weight of the diamond (carats)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Price of an individual diamond&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;samp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;price&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;carat&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;color&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_log10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_log10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dollar_format&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;accuracy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_smooth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lm&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;A linear model after log transforms is better from a model-building perspective&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Model of form log(price) ~ log(carat) + cut.\nStandard assumptions justifying ordinary least squares make more sense after transformation.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Weight of the diamond (carats)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Price of an individual diamond&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;h3 id=&quot;fitting-a-model-and-choosing-transformation-based-on-diagnostics&quot;&gt;Fitting a model and choosing transformation based on diagnostics&lt;/h3&gt;

&lt;p&gt;The conviction that the log transform is a good one is reinforced when we look at the diagnostic plots of the two models (or rather, of two similar models - in the charts below I have simplified the models to remove an interaction between colour and weight).  Here are the four most common diagnostic plots based on the residuals of the untransformed linear model:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0204-diagnostics-untransformed.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0204-diagnostics-untransformed.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;The code for fitting the model and drawing these diagnostics is going to come a bit later&lt;/p&gt;

&lt;p&gt;If you were trained the way I was, you see plenty of red flags here suggesting the linear model isn’t a great fit, the residuals don’t have the homoskedasticity needed to guarantee ordinary least squares being the best estimator, and the data isn’t normal which will cause a problem for conventional statistical tests:&lt;/p&gt;
&lt;ul&gt;
  &lt;li&gt;the ‘residuals versus fitted’ plot in the top left shows clear structure in the residuals, not the white noise we hope for. They curve up at the left (so some non-linearity going on), plus they fan out to the right (heteroskedasticity)&lt;/li&gt;
  &lt;li&gt;the absolute scale of the residuals definitely increases as the expeted value increases (scale-location plot on bottom left), a definite indicator of heteroskedasticity&lt;/li&gt;
  &lt;li&gt;the normal quantile-quantile plot strongly indicates the residuals aren’t normal, but has fat tails (e.g. when they theoretically would be about 3 on the standardised scale, they are about 5 - much higher)&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Quick digression - in case you want a visual key of how things look in a quantile-quantile plot, here’s one I knocked up earlier and might as well have here as anywhere. The thing we are looking for in a Q-Q plot is that if the observed data is similar to the reference distribution (in this case a normal distribution), we’d see a basically straight line.&lt;/p&gt;
&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0204-qqnorm-demo.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0204-qqnorm-demo.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Produced with this quick excursion into base graphics:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;300&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;exp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;exp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;par&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mfrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bty&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;l&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Roboto&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;qqnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;main&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Normal&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;qqnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;main&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Right-skewed&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;qqnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;main&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Left-skewed&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lapply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;density&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;main&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)})&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;OK, back to our diamonds; so we have fat-tailed, heteroskedastic residuals, a right-skewed original variable, and a bit of a curve in the data. We reach for our log-transform and refit the model and get this &lt;em&gt;much&lt;/em&gt; better set of diagnostic plots:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0204-diagnostics-transformed.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0204-diagnostics-transformed.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;The issues of concern are fixed:&lt;/p&gt;
&lt;ul&gt;
  &lt;li&gt;the curved shape and the fanning in the residuals-v-fitted plot has gone and we’re left with something looking much more like white noise&lt;/li&gt;
  &lt;li&gt;the scale-location plot looks like solid homoskedasticity&lt;/li&gt;
  &lt;li&gt;The normal quantile-quantile plot looks a lot more “normal” (ie straight line) and apart from a few outliers the values of the standardised residuals are what you’d expect them to be for a normal distribution&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Conventionally, this looks way better. And if we wanted a model to understand the relationship of diamonds’ price to their weight and colour, I would say this was a better model.&lt;/p&gt;

&lt;h3 id=&quot;using-a-model-to-predict-total-price&quot;&gt;Using a model to predict total price&lt;/h3&gt;

&lt;p&gt;But understanding that isn’t the primary purpose of the exercise. We want the total expected price of the actual collection of 53,490 diamonds based on the model we’ve fit to 1,000 of them. So I’m going to compare three ways of doing this&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;‘Model with no transformations’ - this is the first model described above, the one with diagnostic plots suggesting heteroskedasticity&lt;/li&gt;
  &lt;li&gt;‘Model with log transformation’ - this is the second model described above, where I have taken the logarithm of price and of weight in carats before using ordinary least squares again for parameter estimates&lt;/li&gt;
  &lt;li&gt;‘Simple average’ - I just take the average price of my sample of 1,000 and extrapolate that to the full population.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;The ‘simple average’ approach doesn’t use the information we have on the full population’s weight and colour at all, so we expect it to be a lot less efficient, but it should still be unbiased. Formally it’s equivalent to the ‘Model with no transformations’ but with the regression coefficients other than the intercept all constrained to be zero.&lt;/p&gt;

&lt;p&gt;For each of the three approaches I take the model that has been fit on my sample and apply it to the full population of 53,490 to estimate each diamond’s price, and add them up to get the average price I can expect for &lt;em&gt;all&lt;/em&gt; my diamonds (remembering that’s the point of the exercise). Then, I’m looking at the answers, and comparing that result to the true average price. Then I do this again with a different sample of 1,000 diamonds; and plot the overall results:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0204-model-comparison-density.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0204-model-comparison-density.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Well, I sort of gave this away at the beginning; but the model with the &lt;em&gt;untransformed&lt;/em&gt; price variable in the sample performs much better. Basically because it’s unbiased. All that heteroskedasticity and non-normality and a bit of curve doesn’t matter; if fixing them means you bias the total result substantially downwards.&lt;/p&gt;

&lt;p&gt;Note that taking the simple average of the sample is of course also unbiased like the untransformed model, but as expected is inefficient and has a much wider sampling error than either of the models which make better use of the auxiliary information available.&lt;/p&gt;

&lt;p&gt;Here’s a scatter plot of those results. Each black point is one of the estimates from one of these estimation methods (multiple methods fit to a given sample of 1,000 diamonds) and the red point is the true value.&lt;/p&gt;
&lt;ul&gt;
  &lt;li&gt;The chart on the left compares the simple average (of an untransformed price variable) to the model also with the untransformed price variable. Both methods on average get the right result, but the ‘simple average’ method has lots of points far too low or far too high (which is why the aspect ratio of that plot is the way it is)&lt;/li&gt;
  &lt;li&gt;The chart on the right compares the two models, with untransformed and transformed price variables. We see that given the same sample of 1,000 diamonds, one method is much better than the other.&lt;/li&gt;
&lt;/ul&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0204-model-comparison-scatter.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0204-model-comparison-scatter.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Of course as soon as you think about what’s happening here it’s obvious. With the log-transformed price variable, when I predict new values for the out-of-sample diamonds on the log scale, those are unbiased estimates &lt;em&gt;of the logarithm of the unobserved diamonds’ prices&lt;/em&gt;. When I say e to the power of those estimates, I get biased estimates of the price on their original scale. It’s actually exactly as though I took the geometric mean of the sample - which in a right-skewed variable will be on average lower than the arithmetic mean - and tried to scale it up to the populatio level just by multiplying it by N/n.&lt;/p&gt;

&lt;p&gt;Of course you could implement some fancy method to adjust for that bias, but I would say why bother? Look at the nice properties of the predictions from the model with no transformations. It turns out that a bit of curvature and non-normal non-heteroskedastic residuals may not matter as much as one might think - if your model is for a very particular, constrained purpose.&lt;/p&gt;

&lt;p&gt;So it seems obvious - but twice now, ten years apart, I’ve found myself nearly making this mistake in important circumstances. I had a sample, and needed to estimate a population total for which I had auxiliary information. I fit a model, which was obviously greatly improved (in terms of removing heteroskedasticity and non-normality of residuals) by taking log transforms. And I reflexively did this, and used the model to predict population totals, before (luckily) something at the back of my mind warned me of pretty much the content of this blog, before I went and made a fool of myself (or just as likely the mistake wouldn’t have been found - peer review is never as good as it should be within government and consulting, which were the two environments here - so we would have just blithely under estimated the variable of interest and never known). And this happened exactly the same way, twice (one example was international visitor spend in New Zealand, the other was revenue of space companies in Australia).&lt;/p&gt;

&lt;p&gt;So we always found the mistake and did it right and the last time was a few years ago. But I’m writing this in the hope I don’t head down the wrong path again in another 10 years or so.&lt;/p&gt;

&lt;p&gt;Here’s the code that did all that sampling, model fitting and population estimation. I did this with 1,000 samples of 1,000 diamonds each. The sets of diagnostic plots shown earlier are from just the last version of &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;mod1&lt;/code&gt; and &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;mod2&lt;/code&gt; fit to the last of those 1,000 samples (which is why I’ve delayed the code for them down to here).&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reps&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;log_trans&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reps&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;untransformed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reps&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;naive&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reps&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reps&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;samp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diamonds&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;log&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;price&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;log&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;carat&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;color&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;samp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;predict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;newdata&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diamonds&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;price&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;carat&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;color&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;samp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;predict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;newdata&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diamonds&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;samp&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;price&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;exp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Plot the standard diagnostics of the two models for the last example&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p7&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;par&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mfrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bty&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;l&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# conventionally good&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p8&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;par&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mfrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bty&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;l&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# unsatisfactory - residuals with curvature, non-normal, and increase as values increase&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diamonds&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;price&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;naive&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;untransformed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coord_equal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;point&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;label&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;True value&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hjust&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Based on simple average from sample&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Based on untransformed model&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Simple average leads to higher variance than model-based estimate&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Comparison of different estimation methods from a simple random sample&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p5&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;log_trans&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;untransformed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coord_equal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;point&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;text&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;True value&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hjust&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Based on log-log model&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Based on untransformed model&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Log-log model leads to underestimates&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p6&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&amp;gt;%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&amp;gt;%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;log_trans&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Model with log transformation&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;naive&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Simple average of sample&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;untransformed&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Model with no transformation&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&amp;gt;%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_density&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_vline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xintercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;text&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.0005&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;True value&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hjust&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;vjust&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_color_brewer&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;palette&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Set2&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_fill_brewer&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;palette&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Set2&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Comparison of different estimation methods from a simple random sample&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Estimated value&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;log-log model is biased low; simple average has higher variance; model with no transformation is best.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Now, if was doing this for real, I probably wouldn’t stop here. Because I’m a bit worried about the heteroskedasticity, even though I don’t want the complications that come from taking a log transformation of price to make it go away. Ordinary least squares isn’t the best estimator when the residuals have variance that varies with the expected value of price (in this case, as price goes up the variance of the residuals go up). I’d probably want to down-weight the observations with higher expected values. An elegant way to do this is to move from ordinary least squares to a generalized linear model with an identity link function, normal family response, but a variance function that increases as the mean does (rather than staying constant). But I actually don’t think it would make that much difference, and becomes a different point for this blog anyway.&lt;/p&gt;

&lt;p&gt;And I have other stuff to do so OK, that’s all for now.&lt;/p&gt;

</description>
				<pubDate>Sun, 30 Jul 2023 00:00:00 +0000</pubDate>
				<link>https://freerangestats.info/blog/2023/07/30/log-transforms</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2023/07/30/log-transforms</guid>
			</item>
		
			<item>
				<title>Weighted versus unweighted percentiles by @ellis2013nz</title>
					<description>&lt;p&gt;This interesting paper came out recently: &lt;a href=&quot;https://www.demographic-research.org/volumes/vol48/26/&quot;&gt;A test of the predictive validity of relative versus absolute income for self-reported health and well-being in the United States&lt;/a&gt;, by David Brady, Michaela Curran and Richard Carpiano. It uses a large sample of longitudinal data to exploit between-individual and within-individual (across time) variation in absolute and relative income to explore which one is more associated with well-being. Seems that in the USA, relative income is more important.&lt;/p&gt;

&lt;p&gt;“Relative income” here is defined as your position (percentile rank) in the national distribution of incomes in a particular year. The average relative income should be 50 by definition. I noticed that the mean percentile reported in table A.1 was 57, not 50, which led to this &lt;a href=&quot;https://twitter.com/ellis2013nz/status/1671996132757757953&quot;&gt;slightly confusing (for me, anyway) exchange on Twitter&lt;/a&gt;. It sparked my interest in a broader question, not really related to the paper itself - what is the relationship between weighted and unweighted percentile rank when using survey data?&lt;/p&gt;

&lt;h2 id=&quot;weighted-percentile-rank&quot;&gt;Weighted percentile rank&lt;/h2&gt;

&lt;p&gt;I was surprised to find (well actually, I had discovered this before, so really should say to be “reminded”) that there doesn’t seem to be a weighted version of dplyr’s &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;percent_rank()&lt;/code&gt; function in dplyr or other R packages. There are several packages with functions that will estimate quantiles from weighted data, but if you want to then turn the original vector of continuous variables into percentiles you need to take those quantiles and &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;cut&lt;/code&gt; the original variable using them as breaks.&lt;/p&gt;

&lt;p&gt;I started by making my own convenience function to do this. To calculate the breaks, I am using the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;wtd.quantile&lt;/code&gt; function from Mark Hancock’s &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;reldist&lt;/code&gt; package, which was notably faster than its competition (more on this later).&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DescTools&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;microbenchmark&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;modi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reldist&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sampling&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;kableExtra&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' Weighted percent rank&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param x a vector&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param weights weights eg survey or frequency weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param q quantiles to compute as the basis for cutting x, &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#'   passed through to reldist::wtd.quantile &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @details&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' This is a bit of a hack, a homemande version of dplyr::percent_rank&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' I haven't thought through things like NAs, ties, etc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @importFrom reldist wtd.quantile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @export&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wt_percent_rank&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;q&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# fastest method to calculate breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reldist&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wtd.quantile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weight&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;q&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;q&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# problem sometimes with ties in the breaks. This forces a way through:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;Inf&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# cut the data where the quantile breaks are:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cut&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labels&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# labels = FALSE for speed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# convert this into a number from 0 to 1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# for unweighted data, wt_percent_rank should give identical&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# results to percent_rank&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;homemade&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wt_percent_rank&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dplyr&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent_rank&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;difference&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;homemade&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dplyr&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;abs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;difference&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That little test of whether I get the same result as &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;dplyr::percent_rank()&lt;/code&gt; with unweighted data turns out ok:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;  homemade dplyr difference
      &amp;lt;dbl&amp;gt; &amp;lt;dbl&amp;gt;      &amp;lt;dbl&amp;gt;
 1    0.189 0.189          0
 2    0.557 0.557          0
 3    0.869 0.869          0
 4    0.944 0.944          0
 5    0.704 0.704          0
 6    0.448 0.448          0
 7    0.411 0.411          0
 8    0.121 0.121          0
 9    0.584 0.584          0
10    0.307 0.307          0
# ℹ 990 more rows
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;h2 id=&quot;simulating-a-population-and-drawing-a-sample&quot;&gt;Simulating a population and drawing a sample&lt;/h2&gt;

&lt;p&gt;So to actually explore the difference between weighted and unweighted percentiles, I wanted to simulate a complex survey with unequal probabilities of observations being in the sample, and a skewed variable (income a good example) that was also related, directly or indirectly, to the probability of an observation being in the sample. I did this by creating a educatin variable with 7 different levels, and said&lt;/p&gt;
&lt;ul&gt;
  &lt;li&gt;the higher the education the higher the mean income&lt;/li&gt;
  &lt;li&gt;the higher the education, the lower the chance of being sampled (or, which is effectively the same for my purpose, the lower the chance of completing a response to the survey)&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Once I have drawn the sample I am going to use standard post-stratification methods to calculate weights that mean the points in the sample between them represent the original population fully.&lt;/p&gt;

&lt;p&gt;Whether or not this sampling or non-response effect is realistic doesn’t matter much. Obviously in reality things will be much more complicated. I just wanted something where survey weights were going to vary in a known way (that could be inferred from the data) that correlates with the variable of interest.&lt;/p&gt;

&lt;p&gt;Actual income is simulated as coming from a log-normal distribution.&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#----------------define population-----------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1e4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1e7&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;population&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;educ&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;replace&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;income&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;exp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;9&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;educ&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob_sample&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;educ&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# marginal totals we will use later for weighting&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;marginal_total&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;population&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;count&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;educ&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Freq&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# check the income is indeed higher for people in higher education groups:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;population&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;educ&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;income&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;This is what the mean income looks like for my different education levels:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;   educ `mean(income)`
  &amp;lt;int&amp;gt;          &amp;lt;dbl&amp;gt;
1     1         16307.
2     2         19982.
3     3         24369.
4     4         29761.
5     5         36302.
6     6         44355.
7     7         54212.
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;
&lt;p&gt;OK, now I need to draw a sample. I originally did this with &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;dplyr::slice_sample()&lt;/code&gt; but found it slow. The &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;strata()&lt;/code&gt; function in Yves Tillé and Alina Matei’s &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;sampling&lt;/code&gt; package is about twice as fast (benchmarking later in this post) so I used that instead. The code below draws that sample, does the post-stratification weighting (by simply joining the sample to the marginal totals data frame calculated earlier and creating weights forced to add up to those marginal totals) and then calculates the percentile ranks (both weighted and unweighted).&lt;/p&gt;

&lt;p&gt;BTW, the reason I was particularly attuned to speed in this overall exercise is that I had heard that Stata was very slow in calculating the weighted percentile ranks. But apart from drawing the sample, everything I was doing in R turned out to be very fast; even calculating percentile ranks on 200,000 observations grouped by year (so similar to the actual data in the original paper) only takes a few seconds.&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#---------------draw a sample------------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Sampling, with unequal probabilities&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# the slow part here is actually the sampling with unequal probabilities. Not&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# sure why so slow. See benchmarking later for why we use this strata()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# function.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample_rows&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sampling&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;strata&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;population&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pik&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;population&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob_sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;systematic&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;population&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample_rows&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ID_unit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# join with marginal totals and calculate weights necessary&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# so weights add up to the marginal totals&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;marginal_total&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;educ&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;educ&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wt&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Freq&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# calculate unweighted and weighted percentile ranks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;uw_perc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent_rank&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;income&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w_perc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wt_percent_rank&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;income&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;difference&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w_perc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;uw_perc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# check that the weights add up to the population number&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# visual check that the sum of weights in each category&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# of educ matches the population totals (they do):&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;count&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;educ&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wt&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Sum of weights&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;marginal_total&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;educ&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;h2 id=&quot;comparing-the-weighted-and-unweighted-percentiles&quot;&gt;Comparing the weighted and unweighted percentiles&lt;/h2&gt;

&lt;p&gt;OK, time to do the actual comparisons. Here’s a scatter plot of the weighted and unweighted percentiles. Note the very very high correlation - 0.999! - but also the visually clear slight curvature:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0250-scatter1.svg&quot; width=&quot;90%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0250-scatter1.png&quot; width=&quot;90%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;That curvature is more obvious if we plot the difference between the two estimates:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0250-scatter2.svg&quot; width=&quot;90%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0250-scatter2.png&quot; width=&quot;90%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Why do we get this pattern? It’s simple enough if you consider three people - at the bottom, the top and the middle of the sample’s income distribution. For the person at the bottom, whether we are weighting the observations or not, we know that this person has the lowest observed income. They’re in the bottom percentile either way, and the weights of all the people above them simply don’t matter. This applies in reverse for the person at the top. So the weighted and unweighted percentiles are identical in these extremes.&lt;/p&gt;

&lt;p&gt;For the person in the middle however it does matter. We look at the unweighted sample and say “right, you’re bang in the middle, you’re the 50th percentile”. But then we look at the weights and say “hold on, the half of the sample that has incomes below you has relatively low weights (because, in my simulation, people with lower education were more likely to be in the survey so get lower weights to compensate). So while it looks like you’ve got higher income than 50% of people, that 50% of the sample actually only represents 45% of the population.” So we get a material difference between weighted and unweighted percentile estimates in the middle, but not at the ends, of the distribution.&lt;/p&gt;

&lt;p&gt;There’s nothing inherent that says the weighted percentiles in the middle will always be lower than unweighted as in my simulation - that’s only the case because the lower income people on average had lower weights. With other sampling designs or non-response rates, this could be reversed. But however the weights work out, there will be more discrepancy between the weighted and unweighted percentiles in the middle of the distribution than the ends.&lt;/p&gt;

&lt;p&gt;Here’s the simple code for drawing those plots.&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#-----------visual comparisons-------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Comparison between weighted and unweighted percentiles 
                        in a simulated right-skewed variable from a complex 
                        survey.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;80&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# it *looks* like the unweighted and weighted percentils are indistinguishable:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;uw_perc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w_perc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_abline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slope&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;intercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_title&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_squish&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Correlation = 
                       {round(cor(sample$uw_perc, sample$w_perc), 3)}.
                       Blue line shows equality.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Unweighted percentile&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Weighted percentile&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Note  this line is slightly curved.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# at the higher incomes and lower incomes there is no difference -&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# we know you are the richest or the poorest. But for people in the&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# middle it makes a real difference:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;uw_perc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;difference&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.11&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Unweighted percentile&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Weighted minus\nunweighted percentile&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_title&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;A couple of other interesting observations. The unweighted mean of the unweighted percentile is 50; and so is the weighted mean of the weighted percentile. But the unweighted mean of the weighted percentile is low - about 47 - and the weighted mean of the unweighted percentile is high - about 53. This all makes sense based on the above reasoning.&lt;/p&gt;

&lt;p&gt;Here are some summary stats on the sample:&lt;/p&gt;

&lt;table class=&quot;table&quot; style=&quot;margin-left: auto; margin-right: auto;&quot;&gt;
 &lt;thead&gt;
  &lt;tr&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; educ &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; mean(wt) &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; mean(income) &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; mean(uw_perc) &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; mean(w_perc) &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; n() &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; sum(wt) &lt;/th&gt;
  &lt;/tr&gt;
 &lt;/thead&gt;
&lt;tbody&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 690.4403 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 16980.99 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 37.66090 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 34.70156 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 2067 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1427140 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 2 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 762.0085 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 21147.84 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 42.89254 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 39.78324 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1873 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1427242 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 3 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 842.4838 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 25146.92 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 48.18424 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 45.01756 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1697 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1429695 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 4 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 989.3329 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 31281.51 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 54.45006 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 51.25665 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1445 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1429586 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 5 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1175.7494 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 35873.40 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 58.81986 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 55.67827 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1217 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1430887 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 6 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1500.4732 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 42648.17 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 62.77626 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 59.67482 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 951 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1426950 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 7 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1904.6667 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 50135.01 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 66.77896 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 63.82556 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 750 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1428500 &lt;/td&gt;
  &lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;

&lt;p&gt;And the code to make those summary stats and compare the different means of the different percentiles.&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# check that the average weights are higher for the higher&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# education groups, as we wanted it to be&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;educ&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;income&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;uw_perc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w_perc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;kable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;kable_styling&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# unweighted mean of the unweighted percentile is 50:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;uw_perc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# but unweighted mean of the weighted percentile isn't!&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w_perc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Weighted mean of the weighted percentile is 50:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weighted.mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w_perc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# but weighted mean of the unweighted percentile isn't!&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weighted.mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;uw_perc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;h2 id=&quot;some-speed-benchmarking&quot;&gt;Some speed benchmarking&lt;/h2&gt;

&lt;p&gt;Finally, here are some speed tests that were behind some of the design choices above. As it turns out, none of it matters very much. As mentioned above, R can calculate the weighted percentiles for hundreds of thousands of observations grouped by year in only a couple of seconds, so I didn’t need to worry about that. But in case it matters, here’s my core conclusions from running the benchmarking below&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;reldist::wtd.quantile&lt;/code&gt; is about twice as fast as &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;Desctools::Quantile&lt;/code&gt; in calculating weighted quantiles and gives near-identical results.&lt;/li&gt;
  &lt;li&gt;setting &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;labels = FALSE&lt;/code&gt; in a call to &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;cut()&lt;/code&gt; makes the function about four times as fast&lt;/li&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;dplyr::slice_sample&lt;/code&gt; takes about the same time to do a sample with unequal probability of selection as a home made function using &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;base::sample&lt;/code&gt;, but &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;sampling:strata&lt;/code&gt; function is about twice as fast.&lt;/li&gt;
  &lt;li&gt;my final homemade function &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;wt_percent_rank&lt;/code&gt; for weighted percentile ranks is about four times slower than the unweighted &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;dplyr::percent_rank&lt;/code&gt;, but is still plenty fast enough (&amp;lt;5 seconds for 200,000 observations).&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Benchmarking code:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#===================speed tests==================&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Some tests done here behind some of the decisions of what is actually used in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# the script above.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#---------------calculating the breaks at various points in quantile---------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;321&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;exp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;exp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# percentiles to use in benchmarking qunatile functions&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# note modi::weighted.quantile(x, w = w, prob = 0.5) only works with a scalar value of prob&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;microbenchmark&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unweighted&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quantile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;probs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Desctools&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DescTools&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Quantile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;probs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reldist&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reldist&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wtd.quantile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;q&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weight&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# reldist is notably faster than DescTools in fact it is nearly as fast as&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# the unweighted base r calculation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# the results are a tiny bit different but usually not:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dt&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DescTools&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Quantile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;probs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rd&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reldist&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wtd.quantile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;q&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weight&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diff&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rd&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;abs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diff&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#-------------------actually assigning the percentiles to the original data-----------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wtd.quantile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;q&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weight&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;microbenchmark&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cut&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cut&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labels&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# much faster (about 4x) with labels = FALSE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#-------------------sampling with unequal probabilities---------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# sampling a bunch of rows at random with unequal probabilities using&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# slice_sample is surprisinglyu slow. Is there a better way?&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# try my home made sampler, and also one from the sampling package&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;eg_d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_sampler&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weight_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rows&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weight_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rows&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;microbenchmark&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unweighted&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice_sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;eg_d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`weighted dplyr`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;    &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice_sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;eg_d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weight_by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`weighted homemade`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_sampler&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;eg_d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weight_by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;eg_d&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`sampling package`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;eg_d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;strata&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;eg_d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pik&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;eg_d&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;systematic&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ID_unit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# times for weighted dplyr and home made funcionts are very similar &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# for this_n of 100, 1000, 10000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# the sampling::strata() function is about twice as fast&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# unweighted sampling is much much faster&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#---------comparing speed of unweighted and weighted percentile calculation------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;microbenchmark&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wt_percent_rank&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;probs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent_rank&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Note that the weighted version is 3-4 times slower even after&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# not using labels for cut and using the quickest quantile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# calculation we have.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#----------percentile rank for large sample-----------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# simulate similar size data to that which is in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# https://www.demographic-research.org/volumes/vol48/26/48-26.pdf&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;psid_eg&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;exp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;exp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;replace&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;system.time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;({&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;psid_eg&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;year&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wpr&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;wt_percent_rank&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;})&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# &amp;lt; 5 seconds&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That’s all, cheerio.&lt;/p&gt;

&lt;h3 id=&quot;versioning-of-this-blog-post&quot;&gt;Versioning of this blog post&lt;/h3&gt;

&lt;p&gt;An earlier version of this post had some speculation about the original referenced paper without me adequately checking what was actually happening. As it’s not relevant for my main point that paragraph has been removed.&lt;/p&gt;
</description>
				<pubDate>Sat, 24 Jun 2023 00:00:00 +0000</pubDate>
				<link>https://freerangestats.info/blog/2023/06/24/weighted-percentiles</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2023/06/24/weighted-percentiles</guid>
			</item>
		
			<item>
				<title>Simpler drawing of Pacific choropleth maps by @ellis2013nz</title>
					<description>&lt;h2 id=&quot;simple-use-of-the-draw_pac_map-function&quot;&gt;Simple use of the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;draw_pac_map()&lt;/code&gt; function&lt;/h2&gt;

&lt;p&gt;Last year &lt;a href=&quot;/blog/2022/10/13/pacific-map&quot;&gt;I blogged about making a choropleth map of the Pacific&lt;/a&gt;, dealing with a few technical details like getting the map centred on the Pacific, using exclusive economic zones (EEZs) to colour things in for good visibility, adding the international dateline, etc.&lt;/p&gt;

&lt;p&gt;Soon after that blog post I abstracted that work and added it to the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;frs&lt;/code&gt; R package, my miscellaneous collection of bits and pieces associated with this Free Range Statistics blog. I reasoned that as I will be in the Pacific for at least a few years I am likely to want to keep drawing these maps, and they might be useful at work too. You can see:&lt;/p&gt;
&lt;ul&gt;
  &lt;li&gt;&lt;a href=&quot;https://github.com/ellisp/frs-r-package/blob/master/pkg/R/draw_pac_map.R&quot;&gt;the code for the function itself&lt;/a&gt;; and&lt;/li&gt;
  &lt;li&gt;&lt;a href=&quot;https://github.com/ellisp/frs-r-package/blob/master/processing/prep-pac-map.R&quot;&gt;preparation of the various data objects called by the function&lt;/a&gt;.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;It’s easy to install with &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;remotes::install_github(&quot;ellisp/frs-r-package/pkg&quot;)&lt;/code&gt;.&lt;/p&gt;

&lt;p&gt;Here’s the minimal demo of the function. By default it will draw all the countries’ EEZs grey:&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;frs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# needs to be version 0.8.2 or higher&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;draw_pac_map&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country_label_size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0249-blank-map.svg&quot; width=&quot;90%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0249-blank-map.png&quot; width=&quot;90%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;For a more realistic use case, here’s a map showing the &lt;a href=&quot;https://hdr.undp.org/data-center/human-development-index#/indicies/HDI&quot;&gt;Human Development Index&lt;/a&gt; or HDI for 2021, for all Pacific countries that have one:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0249-hdi-map.svg&quot; width=&quot;90%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0249-blank-map.png&quot; width=&quot;90%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;You can see that the HDI, which is produced by the UNDP, only is calculated for countries that are full UN members; so members of the Realm of New Zealand (Niue, Tokelau and Cook Islands) or French overseas territories (New Caledonia, French Polynesia and Wallis and Futuna) for example aren’t included.&lt;/p&gt;

&lt;p&gt;On the other hand, Nauru &lt;em&gt;is&lt;/em&gt; a full member of the UN, but it is one of four such countries that lack an HDI score because it is missing data on one of the components - in Nauru’s case, mean years of education. The other countries missing an HDI score for similar reasons are North Korea, Monaco and Somalia.&lt;/p&gt;

&lt;p&gt;Here’s the code that downloads that HDI data and draws the map:&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# Human Development Index&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;readxl&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ISOcodes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;download.file&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://hdr.undp.org/sites/default/files/2021-22_HDR/HDR21-22_Statistical_Annex_HDI_Table.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;destfile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;hdi.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mode&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;wb&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hdi&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_excel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;hdi.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;skip&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sheet&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Table 1&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rename&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`VERY HIGH HUMAN DEVELOPMENT`&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hdi2021&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;...3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hdi2021&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hdi2021&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hdi2021&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;is.na&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hdi2021&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ISO_3166_1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;iso3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Alpha_3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Name&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# check to see if any failed to match names&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#View(filter(d, is.na(iso3)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;draw_pac_map&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill_df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hdi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;join_col&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;iso3&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill_col&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;hdi2021&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill_col_label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Human Development\nIndex 2021&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Calibri&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country_label_col&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_fill_viridis_c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;option&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;A&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;You can see that it’s non trivial to get a good combination of colours that works complete with the country labels. There are options for most of the things you’d want to change in the initial call to &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;draw_pac_map&lt;/code&gt; itself; and because it outputs a &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;ggplot2&lt;/code&gt; object you can modify it by adding scales, labels, themes etc the usual way (as I do in the above by adding &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;scale_fill_viridis_c()&lt;/code&gt; to set the scale of colours to use for the fill aesthetic).&lt;/p&gt;

&lt;p&gt;Mental note to self - it would be entirely possible to use the data on SDGs 3, 4.3, 4.4 and 8.5 we have in the Pacific Data Hub to create estimates of HDI for at least some of the territories that the UNDP doesn’t calculate it for. But that’s enough of a job to leave for another time.&lt;/p&gt;

&lt;h2 id=&quot;combining-with-pacific-data-hub-data-for-an-animation-of-sdgs&quot;&gt;Combining with Pacific Data Hub data for an animation of SDGs&lt;/h2&gt;

&lt;p&gt;OK, here’s a more complex use case. I wanted to show off the Pacific Data Hub at a presentation last year to attendees of the Pacific Community’s Committee of Representatives of Governments and Administrations - effectively the board of the Pacific Community or SPC, where I work. I decided it would be nice to do this while also showing that there is a lot of work to be done to improve official statistics production in the region. So I opted to use a sequence of maps, pulling data from the Pacific Data Hub for all the Sustainable Development Goal (SDG) indicators that the Pacific had agreed were priorities to measure. I limited it to those that made sense to represent on a one-dimensional choropleth map (e.g. where a single number for each country is a meaningful thing to compare).&lt;/p&gt;

&lt;p&gt;So the result was this GIF, which basically becomes eye candy in the background while I talked about the issues:&lt;/p&gt;

&lt;p&gt;&lt;img src=&quot;https:/freerangestats.info/img/0249-pac-maps-sdgs.gif&quot; width=&quot;100%&quot; /&gt;&lt;/p&gt;

&lt;p&gt;I’m going to present all the code that does this animation in one chunk rather than try to explain it line by line. Hopefully it makes sense as-is. The key, like any animation, is to save one frame at a time; then I use ImageMagick to knit them all up into an actual animation. You’ll see I opted to not clutter up this particular set of maps with country labels.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#==================animated map===================&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#-----------------------Get the SDG data and metadata---------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rsdmx&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;janitor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dataflow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;DF_SDG&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# metadata ie code lists. Warning this next function, in frs, is a bit of a hack&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# and is subject to change!&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d_code_list&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;get_pdh_codelists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dataflow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dataflow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;version&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;3.0&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d_raw&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;readSDMX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;providerId&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;PDH&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;resource&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;flowRef&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dataflow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as_tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# join the data to the code list so we have labels&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d_raw&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d_code_list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category_en&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Codelist for SDG indicators&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;INDICATOR&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;id&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category_en&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category_fr&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rename&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator_full&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clean_names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# for simplicity, limit ourselves just to indicators that are totals eg all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# age groups, not individual age groups, not disaagregated by sex, etc.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# this loses a lot of data of course but is ok because it is just for illustration&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# to coose the indicators most suitable for choropleth matps&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d_totals&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;_T&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;_T&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;urbanization&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;_T&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;income&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;_T&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;education&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;_T&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;occupation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;_T&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;composite_breakdown&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;_Z&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;disability&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;_T&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator_full&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geo_pict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator_summary&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d_totals&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator_full&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;countries&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geo_pict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# what indicators are we going to use, in a simple vector&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;inds&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator_summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator_full&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#--------------------draw the maps------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dir.create&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;pac-maps&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;showWarnings&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;inds&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_ind&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;inds&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d_totals&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator_full&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_ind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;units&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;count&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unit_measure&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sort&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unit_measure&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;units&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;warning&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;More than 1 unit measure, choosing the most common&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_data&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unit_measure&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;units&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unit_title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stringr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_to_title&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gsub&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;_&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot; &quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;units&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_replace&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Usd&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;USD&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_replace&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Ha &quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Hectares &quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_replace&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Ha$&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Hectares&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_replace&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Bool$&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Yes (1)/No (0)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_replace&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Km&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Kilometres&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;draw_pac_map&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_data&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Calibri&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;base_size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;14&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country_labels&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill_col&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;obs_value&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill_col_label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unit_title&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ylim&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-40&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;30&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_fill_viridis_c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_ind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;paste&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sort&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_data&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;collapse&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;, &quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;axis.text&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;element_blank&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;png&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;pac-maps/{i + 1000}-{indicator_summary[i, ]$indicator}.png&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1500&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;900&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;150&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;cairo-png&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dev.off&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Convert all the single frames into a GIF.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Requires ImageMagick to be installed. Can uncomment and run it here or do &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# it directly in a system / shell window&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# projdir &amp;lt;- setwd(&quot;pac-maps&quot;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# system('magick -loop 0 -delay 150 *.png &quot;pac-maps-sdgs.gif&quot;')&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# setwd(projdir)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That’s it! See you next time.&lt;/p&gt;
</description>
				<pubDate>Sat, 17 Jun 2023 00:00:00 +0000</pubDate>
				<link>https://freerangestats.info/blog/2023/06/17/pacific-map-in-package</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2023/06/17/pacific-map-in-package</guid>
			</item>
		
			<item>
				<title>Simulating confounders, colliders and mediators by @ellis2013nz</title>
					<description>&lt;h2 id=&quot;motivation-and-key-points&quot;&gt;Motivation and key points&lt;/h2&gt;

&lt;p&gt;So &lt;a href=&quot;https://twitter.com/cremieuxrecueil/status/1662882966857547777&quot;&gt;this tweet&lt;/a&gt; came across my feed. It refers to &lt;a href=&quot;https://journals.sagepub.com/doi/10.1177/25152459221095823&quot;&gt;this article, &lt;em&gt;Statistical Control Requires Causal Justification&lt;/em&gt;&lt;/a&gt; by Wysocki, Lawson and Rhemtulla, published in June last year (2022) in &lt;em&gt;Advances in Methods and Practices in Psychological Science&lt;/em&gt;.  I was struck by the clarity of Figure 4 from that paper. It is such a great tool for at once explaining the difference between a confounder, collider and mediator and showing how they have different impacts on a regression; and hence lead to different decisions about whether to include the confounder, collider or mediator variable in the model. What I’m showing here isn’t the original, but my own version of it, which I’ll eventually come to describing how I did it later in this post:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0247-results-with-dags.svg&quot; width=&quot;90%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0247-results-with-dags.png&quot; width=&quot;90%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Here is the original caption for Figure 4 from Wysocki et al:&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;“Partial and simple regression coefficients under three causal structures. In each graph, the x-axis depicts the population value of the direct effect that connects the control variable and the predictor (a), and the y-axis depicts the value of the regression coefficient of Y on X. The direct effect of X on Y and the value of the direct effect connecting Y and C are held constant across the results. Solid lines represent the partial (yellow line) and simple (green line) regression coefficients. The dashed line represents the total X → Y causal effect.”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;The idea is that in all three cases we are interested in the degree to which X causes Y. A third variable C is also present and is correlated with both X and Y. Should we ‘control’ for it, ie include it as another variable on the right hand side of a regression? The answer is, it depends on whether C is a confounder, collider or mediator - and there will be no way to tell from your data which it is. So some kind of reasoning based on prior knowledge and theorising is required, or making (and documenting) some explicit assumptions. Some sciencing is required before you do the actual model specification and fitting.&lt;/p&gt;

&lt;p&gt;The basic interpretation of these terms is as follows. For illustrative purposes I am going to use examples from a hypothetical situation where we want to understand the impact of education on income; that is, X is education and Y is income:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;C is a &lt;em&gt;confounder&lt;/em&gt; if it impacts on both X and Y. For example, gender is known to lead to different education outcomes, and it also leads to differences in income in various complicated ways. Another example might be innate ability, if such a thing exists and could be measured. You want to control for confounders, because if you don’t include them in the regression, some of the coefficient in front of X is going to be picking up causal impacts that are really down to the confounder. In the chart on the right of the figure we see that the green ‘simple coefficient’ - the coefficient we’d estimate for X in a simple regression of Y ~ X - gets the wrong result. The correct value of the impact of X on Y is 0.15, and to get this you need the yellow ‘partial coefficient’ ie the coefficient for X you get when you control for C, in a regression of Y ~ X + C. How far out the simple coefficient is depends on the size of a, the causal effect of C on X; if a is zero then both methods get you the same answer&lt;/li&gt;
  &lt;li&gt;C is a &lt;em&gt;collider&lt;/em&gt; is when both X and Y impact on C, but it itself does not impact on either of them. I find it harder to think of pure examples of these, but in my case I’m going to use hobbies as one. The hobbies you’ve got definitely depend on what you were exposed to in your education; and whether you can afford them financially and timewise will depend on your income (in complicated ways). True, hobbies will also potentially impact on your income but I’m hoping (for purposes of illustration) in ways that are less important than the Y -&amp;gt; C part of the diagram here. You &lt;em&gt;don’t&lt;/em&gt; want to control for colliders in a regression (see how the green ‘simple coefficient’ line correctly estimates the true causality of X -&amp;gt; Y, whereas the yellow ‘partial coefficient’ line doesn’t). The intuition for this is that because C doesn’t really impact on Y, when you include it in the regression some of the true impact of X on Y is falsely attributed to C just because of its correlation with both the important variables. (Side note - I’m not thrilled with the example of ‘hobbies’ here, but it serves the point. I also considered ‘suburb you live in’ and ‘sports team you support’, none of which are pefect either.)&lt;/li&gt;
  &lt;li&gt;C is a &lt;em&gt;mediator&lt;/em&gt; is when X causes C and C causes Y. A good example in our case could be occupation; your education has a big impact on your occupation, which in turn has a big impact on your income. You don’t want to control for a mediator if you are interested in the full effect of X on Y! Because a huge part of how X impacts on Y is precisely through the mediation of C, in our case choice of and access to occupation, given a certain level of education. If you ‘control’ for occupation you will be greatly underestimating the importance of education.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Incidentally this last point is behind one of the great perpetual culture war debates, whether to control for occupation and experience when estimating the gender pay gap. Much of the gender pay gap goes away when we do so, so does this mean there is no problem here? No, because occupation and experience are mediator variables which are precisely the means by which gender leads to a pay gap. By controlling for them you are missing out on the actual mechanism of precisely in what you are interested.&lt;/p&gt;

&lt;p&gt;However there is a time when you might be interested in controlling for occupation and experience in a regression of pay gap on gender. This is when you are seeking to measure the direct impact, and only the direct impact, of gender in pay decisions - most likely in the context of gender discrimination at the final stage of remuneration determination. In this case, occupation and experience are no longer mediators, they are if anything confounders, because X is no longer “gender” but “gender discrimination in the current workplace making a pay decision” (for which we use actual, observed gender as a proxy). Now C impacts on Y and might impact on X. So if you want to estimate that final “equal pay for equal work” step of the chain then yes it is legitimate to control for occupation and experience. Just don’t mix this up with the total impact of gender on the pay gap, which &lt;em&gt;is&lt;/em&gt; mediated through occupation and experience.&lt;/p&gt;

&lt;h2 id=&quot;simulating-data&quot;&gt;Simulating data&lt;/h2&gt;

&lt;p&gt;So I wanted to simulate this data fitting these three types of relationships, and re-create the original Figure 4 from Wysocki et al’s paper. I think this is pretty simple, so long as I am happy with getting the substance right and not matching exact details. First, I made three functions, one each for the situation where the nuisance variable is a confounder, collider or mediator.&lt;/p&gt;

&lt;p&gt;In the code below I call the nuisance variable (which is C in the original diagrams) z to avoid creating a conflict with the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;C()&lt;/code&gt; or &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;c()&lt;/code&gt; functions in R (this would be completely workable, but goes against highly ingrained habits on my part to not use frequently-used base R functions as names for other objects). Each of the three functions starts the data generation with the one variable that is exogenous to the system - isn’t causally impacted on by either of the other two. Then it uses that variable to generate the variable that is impacted on by the first variable, and then finally creates the final variable. Each variable has a standard (ie mean = 0, variance = 1) Gaussian randomness built in, and the relationship between variables has simple linear coefficients but no intercept terms. For example, in the confounder example, Y = b1.X + b2.C + noise.&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# attempt to recreate Figure 4 from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# https://journals.sagepub.com/doi/10.1177/25152459221095823&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggdag&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;patchwork&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#----------------simulating data-------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# In the below I call a variable `z` that in the diagrams is called `C`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' Simulate a 3 variable situation where a nuisance variable is a confounder&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param zx coefficient for impact of z on x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param xy coefficient for impact of x on y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param zy coefficient for impact of z on y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param n sample size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param seed random seed set for reproducibility&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @returns a tibble of three variables x, y and z. x causes y &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' and z is a confounder ie it impacts on both x and y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_confounder&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zx&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zx&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' Simulate a 3 variable situation where a nuisance variable is a collider&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param xz coefficient for impact of x on z&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param xy coefficient for impact of x on y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param yz coefficient for impact of y on z&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param n sample size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param seed random seed set for reproducibility&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @returns a tibble of three variables x, y and z. x causes y &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' and z is a collider ie it is impacted on by both x and y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_collider&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xz&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yz&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xz&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yz&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' Simulate a 3 variable situation where a nuisance variable is a mediator&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param xz coefficient for impact of x on z&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param xy coefficient for impact of x on y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param zy coefficient for impact of z on y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param n sample size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @param seed random seed set for reproducibility&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' @returns a tibble of three variables x, y and z. x causes y &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' and z is a mediator ie x impacts on z and z impacts on y, so some of the &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#' impact of x on y comes via z&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_mediator&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xz&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xz&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;The idea with these functions is that they work with multiple values of “a” (as per the original terminology of the diagrams). But for illustrative purposes we can use these functions to generate data where a=0.3 (a reasonably material relationship) and see how the correlations come out. This helps make it obvious that you can’t tell from the data alone (at least with this sort of snapshot, observational data) which way the causality is going:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; # Correlations of example different datasets:
&amp;gt; round(cor(sim_confounder(0.3, n = 10000)), 2)
     x    y    z
x 1.00 0.26 0.29
y 0.26 1.00 0.49
z 0.29 0.49 1.00
&amp;gt; round(cor(sim_collider(0.3, n = 10000)), 2)
     x    y    z
x 1.00 0.15 0.34
y 0.15 1.00 0.47
z 0.34 0.47 1.00
&amp;gt; round(cor(sim_mediator(0.3, n = 10000)), 2)
     x    y    z 
x 1.00 0.28 0.29
y 0.28 1.00 0.49
z 0.29 0.49 1.00
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Basically the correlations tell us nothing about the direction of causality.&lt;/p&gt;

&lt;p&gt;So my next step is to generate the data for many different values of a, for each of the three relationship types. There are plenty of ways to do this efficiently, here is mine. For those curious, &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;res&lt;/code&gt; standards for “results”, in my object names. Naming things is hard!&lt;/p&gt;

&lt;p&gt;At the end of the code below I have an object &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;res&lt;/code&gt; which has values of &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;a&lt;/code&gt; and estimates of the simple coefficient and partial coefficient in front of X (in the regression of Y on X) for that relationship type, as well as the true causal impact of X on Y.&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#-----------generate data and fit regressions, for various values of a&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_a&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;from&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-0.7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length.out&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res_conf&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lapply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_a&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_confounder&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bind_rows&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_a&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;each&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Confounder&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Simple coefficient`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))[[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;'x'&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Partial coefficient`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))[[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;'x'&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Causal effect`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res_coll&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lapply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_a&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_collider&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bind_rows&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_a&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;each&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Collider&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Simple coefficient`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))[[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;'x'&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Partial coefficient`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))[[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;'x'&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Causal effect`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res_medi&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lapply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_a&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_mediator&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bind_rows&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_a&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;each&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Mediator&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Simple coefficient`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))[[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;'x'&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Partial coefficient`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))[[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;'x'&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Causal effect`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Simple coefficient`&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bind_rows&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res_conf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res_coll&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res_medi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;h2 id=&quot;drawing-a-plot-of-regression-estimates&quot;&gt;Drawing a plot of regression estimates&lt;/h2&gt;

&lt;p&gt;It only remains to draw and polish the chart, which is basic ggplot2 stuff. Probably the only noteworthy trick in the below is a slight improvement on the original diagram where I move the Y axis and its labels to the right of the chart, to avoid clutter near where I am going to place the DAGs in the end figure.&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#------------------draw plot-------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_font&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Calibri&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tg&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;guide_legend&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;direction&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;vertical&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;keywidth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;cm&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_relevel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Causal effect&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;after&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;Inf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_relevel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Confounder&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_recode&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Causal effect (X causing Y)&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Causal effect&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;linetype&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;linewidth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-0.7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;right&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_linetype_manual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;guide&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tg&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_colour_manual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;yellow&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lightgreen&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;guide&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tg&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;linetype&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;expression&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Coefficient &quot;&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;beta&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Reproducing (inexactly) a figure by Wysocki, Lawson and Rhemtulla in \n'Statistical Control Requires Causal Justification'. https:/freerangestats.info&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme_light&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;base_family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_font&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot.caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;element_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey50&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;strip.text&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;element_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;face&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;bold&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;top&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;h2 id=&quot;drawing-directed-acyclic-graphs-dags&quot;&gt;Drawing Directed Acyclic Graphs (DAGs)&lt;/h2&gt;

&lt;p&gt;Next comes drawing the DAGs themselves. I’m not happy with how I’ve done this, but it works… The tricky part was getting labels in the edges connecting each node. This is possible (as indeed I do in the code below) but it involved using trial and error to work out which labels went with each connecting edge. I don’t like this at all! There must be a way to do this “properly” with mapping an aesthetic to some variable but I couldn’t work it out.&lt;/p&gt;

&lt;p&gt;DAGs as drawn with ggdag seem to have a random orientation, so setting the seed below each time just before drawing a DAG is critical for having them all in the same orientation, with X and Y both at the bottom of the image.&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#------------------draw DAGs---------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrow_col&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey70&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dagify&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;C&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;C&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&amp;gt;%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xend&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yend&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_dag_edges_fan&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;a&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;edge_colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrow_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label_colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_dag_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme_dag&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dagify&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;C&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&amp;gt;%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xend&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yend&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_dag_edges_fan&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;a&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;edge_colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrow_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label_colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_dag_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme_dag&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dagify&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;C&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;C&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&amp;gt;%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xend&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yend&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#geom_dag_point() +&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_dag_edges_fan&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;a&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;edge_colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrow_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label_colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_dag_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme_dag&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;So that code defines the three DAGs in the objects &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;d1&lt;/code&gt;, &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;d2&lt;/code&gt;, &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;d3&lt;/code&gt; but doesn’t draw them.&lt;/p&gt;

&lt;h2 id=&quot;pulling-the-combined-figure-together&quot;&gt;Pulling the combined figure together&lt;/h2&gt;

&lt;p&gt;The final step is to use the magic of &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;patchwork&lt;/code&gt; package by Thomas Pedersen to lay out the diagrams on the actual page. The design argument to &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;plot_layout()&lt;/code&gt; lets you take total control of how multiple &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;ggplot2&lt;/code&gt; objects (in our case, &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;p&lt;/code&gt;, &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;d1&lt;/code&gt;, &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;d2&lt;/code&gt; and &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;d3&lt;/code&gt;) are layed out in a figure, so I make use of that here.&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;design&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot_layout&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;design&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;design&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot_annotation&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Statistical control requires causal justification&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Only when the variable is a confounder is it correct to 'control' for it in a regression.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;text&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;element_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_font&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
   &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;h2 id=&quot;comparison&quot;&gt;Comparison&lt;/h2&gt;

&lt;p&gt;So as a reminder, here’s a repeat of the final image:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0247-results-with-dags.svg&quot; width=&quot;90%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0247-results-with-dags.png&quot; width=&quot;90%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;… and here’s the original version from Wysocki et al:&lt;/p&gt;

&lt;p&gt;&lt;img src=&quot;https:/freerangestats.info/img/0247-original-diagram-shrunk.png&quot; alt=&quot;Figure 4 from Wysocki et al&quot; /&gt;&lt;/p&gt;

&lt;p&gt;There’s an obvious difference! My “simple coefficient” for X in the presence of a confounder has a gentle S curve whereas their’s is a straight line; and their “partial coefficient” for X in the presence of a collider has a gentle S curve whereas mine is in a straight line. Why? I don’t know! Obviously it is to do with our different simulated data. Mine is generated as some simple linear relationships; their’s is described as:&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;“We used path tracing (i.e., Wright’s rules; Alwin &amp;amp; Hauser, 1975; see Appendix A for an example) to obtain a population correlation matrix for each causal structure and calculated regression coefficients from each population correlation matrix using the formula β=Σ−1xxΣxy, in which Σxx is the p×p
 correlation matrix of predictors and Σxy is a p×1 vector that contains correlations between each predictor and the outcome”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;That’s clearly a bit different from what I did. I’m not going to delve into exactly why that makes the results a bit different. The difference doesn’t change the substantive point about how you can go wrong with the wrong choice of partial or simple coefficient (corresponding to whether you “control” for the nuisance variable or not).&lt;/p&gt;

&lt;p&gt;That’s all folks. Remember not to include collider and mediator variables in your controls in a regression, at least when you are interested in causal explanations (predictions being something else again)!&lt;/p&gt;
</description>
				<pubDate>Sun, 04 Jun 2023 00:00:00 +0000</pubDate>
				<link>https://freerangestats.info/blog/2023/06/04/causality-sims</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2023/06/04/causality-sims</guid>
			</item>
		
			<item>
				<title>Covid-19 vaccination rates in the Pacific by @ellis2013nz</title>
					<description>&lt;p&gt;During the Covid-19 global health crisis, the organisation where I work - the Pacific Community, or SPC - compiled and published weekly updates on Covid-19 incidence, mortality and vaccination rates. This only recently stopped with the WHO determining in May that the global health emergency was ended, after more than three years.&lt;/p&gt;

&lt;p&gt;We published &lt;a href=&quot;https://php.spc.int/covid-19&quot;&gt;weekly narrative updates&lt;/a&gt; as well as a “data flow” on the Pacific Data Hub “dot Stat” implementation at &lt;a href=&quot;https://stats.pacificdata.org/&quot;&gt;https://stats.pacificdata.org/&lt;/a&gt;.&lt;/p&gt;

&lt;p&gt;For some reason I can’t now recall, a few months back (it’s just taken me that long to write it up) I was interested in the roll-out of vaccination rates. Here’s how they look over time:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0246-vaccination-time.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0246-vaccination-time.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Some countries had a slow start, but they mostly ended up with quite high rates of getting at least two Covid vaccination shots. The global proportion of people that are fully vaccinated &lt;a href=&quot;https://www.nytimes.com/interactive/2021/world/covid-vaccinations-tracker.html&quot;&gt;is 67%&lt;/a&gt;, according to New York Times with data from Our World in Data. Only two Pacific Island countries and territories seem to be lower than that.&lt;/p&gt;

&lt;p&gt;Here’s the code to download that from the Pacific Data Hub and draw the chart:&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#------------------data prep-------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rsdmx&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;janitor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ISOcodes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggrepel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;boot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Download the vaccination rates&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;vaccination&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;readSDMX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;providerId&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;PDH&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;resource&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;flowRef&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;DF_COVID_VACCINATION&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as_tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clean_names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# have a look at all the various indicators available. There are lots...&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;count&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;vaccination&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;count&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;vaccination&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unit_measure&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# we will choose just one:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;v2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;vaccination&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# rate of sectond booster administered&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;COVIDVACAD2RT&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.Date&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;covid_2shot_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#--------------------draw time series chart----------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Analysis by freerangestats.info; source: Pacific Data Hub, https://stats.pacificdata.org/&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;v2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;inner_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ISO_3166_1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;geo_pict&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Alpha_2&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_reorder&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;covid_2shot_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;covid_2shot_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Proportion of eligible population with 2 or more Covid vaccination shots&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Covid vaccination rates over time in the Pacific&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;axis.text.x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;element_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hjust&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;angle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;45&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;The striking thing here is that the slowest roll-out was in Papua New Guinea and Solomon Islands, the two poorest of the larger Pacific Island countries. This made me wonder, is there a general relationship between economic production (say GDP per capita) and vaccination rate in the Pacific? Are the poorer countries the least vaccinated?&lt;/p&gt;

&lt;p&gt;That led to this chart:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https:/freerangestats.info/img/0246-vaccination-v-gdp.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https:/freerangestats.info/img/0246-vaccination-v-gdp.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Papua New Guinea and Solomons do indeed stand out in the bottom left of the chart, but not all countries with low GDP per capita have low vaccination rates. Notably Kiribati, the poorest Pacific Island country and territory of them all (in GDP per capita terms - which might be a bit misleading as they don’t include some fishing license revenue, but it is certainly true that Kiribati is poor by Pacific standards) has a high vaccination rate.&lt;/p&gt;

&lt;p&gt;In fact, having done this, I think I would say that vaccination rates in the Pacific are low in countries that are poor AND have population dispersed not only over a wide area (few populations are as dispersed over as many square kilometres as Kiribati’s) but in very numerous inland locations. But I’m not sure that really gets us any further than just saying “Papua New Guinea and Solomon Islands have the lowest vaccination rates”.&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#==================comparison to GDP==================&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#-----------------Data prep---------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Download some summary information on each country and territory from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# the &quot;pocket summary&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pocket&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;readSDMX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;providerId&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;PDH&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;resource&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;flowRef&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;DF_POCKET&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as_tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clean_names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# check, what indicators do we have? (lots):&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pocket&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distinct&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# extract just the most recent observed vaccination rate for each country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# from our vaccination time series:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;latest_vaccination&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;v2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geo_pict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geo_pict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;covid_2shot_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# extract just the GDP per capita from the pocket summary:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pocket&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# GDP per capita, current prices, US dollars&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;GDPCPCUSD&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geo_pict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_pc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geo_pict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# extract the latest population:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pocket&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# GDP per capita, current prices, US dollars&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;MIDYEARPOPEST&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geo_pict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;population&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geo_pict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# combine the GDP, population, vaccination and country names:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;geo_pict&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;latest_vaccination&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;geo_pict&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ISO_3166_1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;geo_pict&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Alpha_2&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#------------------------------draw chart-----------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_pc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;covid_2shot_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# returns a misleading warning that can be ignored, see https://github.com/tidyverse/ggplot2/issues/5053:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_smooth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weight&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;population&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;glm&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;formula&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method.args&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;quasibinomial&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey70&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey85&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;population&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_text_repel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_size_area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_log10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dollar_format&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;accuracy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;limits&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;GDP per capita, US dollars (logarithmic scale)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Proportion of eligible population with 2 or more Covid vaccination shots&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Population:&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Covid vaccination and GDP per capita in the Pacific&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Grey line is from population-weighted logistic regression. Relationship is not statistically significant.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;How do I conclude that the relationship isn’t “statistically significant”? Well, in the first case, we can see that the shaded area for the curved line from the model that’s been fit in the plot could have a horizontal line that fits within the shaded area - imagine a horizontal line drawn at the 25% level. That model is a generalized linear model with a quasibinomial family response. The curve comes not from a smoothing term but just the logit link function that is default for a quasibinomial family and works with the response constrained to be within 0 and 1.&lt;/p&gt;

&lt;p&gt;I wanted to check so I also explicitly fit the model myself (rather than relying on ggplot doing it on the fly), which is done in the code below. The conventional t statistic for the coefficient in front of &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;log(GDP)&lt;/code&gt;  is 0.07, which is narrowly “not significant” and matches the visual. However the F statistic from an Analysis of Deviance table &lt;em&gt;is&lt;/em&gt; significant, 0.02. Is it ok to use an F statistic to test a generalized linear model with a quasibiomial family, with a response that is proportions of a population? To be honest I’m not sure. Most sources seem to say it is, but perhaps I am missing something about the context here.&lt;/p&gt;

&lt;p&gt;I’m using the wrong “population” figure for weights in this model - I’ve just used the whole population for each country and territory, rather than the population of people eligible for the vaccine for reporting purpose (which varies by country). But I am almost certain that’s not material, either for the size of dots on the chart or for the model. Anyway, I can’t be bothered to go back and fix that.&lt;/p&gt;

&lt;p&gt;I &lt;em&gt;was&lt;/em&gt; worried about the small sample size and possible influence of a few points. Knock out Kiribati and Pitcairn and perhaps the model would look significant. Knock out PNG and it certainly wouldn’t be. So I decided to use brute force i.e. the bootstrap to generate a confidence interval for the slope of &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;log(GDP)&lt;/code&gt;. This will give us a better sense of the robustness of evidence here. Some of the bootstrap examples will in fact knock out PNG, Kiribati and Pitcairn; or include them multiple times. Let’s see on average what that does to the estimates.&lt;/p&gt;

&lt;p&gt;So the confidence intervals for that slope, from all four methods of calculating bootstrap confidence intervals (“Normal”, “Basic”, “Percentile” and “BCa”) included zero. Case closed, we can’t say there’s statistically significant evidence of a relationship between GDP per capita and Covid vaccination rates, based just on these Pacific countries and territories. Something else more complicated might be happening instead. Or in other words, it might just be a PNG thing.&lt;/p&gt;

&lt;p&gt;All this modelling is based on a thought experiment that the 22 countries and territories in the data are a random sample from a hypothetical meta-population. That’s the only way “statistically significant” can mean anything in this sort of situation, when you have observations on the entire real population.&lt;/p&gt;

&lt;p&gt;Here’s the code for the modelling part:&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Post continues after R code&lt;/em&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#--------------modelling---------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;covid_2shot_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;log&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_pc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;quasibinomial&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;population&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# why is the t test not significant but F  test is? (and Chi square too)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;anova&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;test&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;F&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# well all of these are various approximations to things... and the sample size is small&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Definitely not significant by conventional tests if we drop Papua New Guinea&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;anova&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;covid_2shot_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;log&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_pc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Papua New Guinea&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;quasibinomial&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;population&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;test&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;F&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Let's use bootstrap to resolve this propblem&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mf&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;covid_2shot_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;log&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_pc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;quasibinomial&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;population&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;42&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;booted&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;boot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;R&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;999&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;boot.ci&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;booted&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That’s all for today. Take care out there.&lt;/p&gt;
</description>
				<pubDate>Sun, 28 May 2023 00:00:00 +0000</pubDate>
				<link>https://freerangestats.info/blog/2023/05/28/covid-vaccinations</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2023/05/28/covid-vaccinations</guid>
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