README.Rmd

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# `nonprobsvy`: an R package for modern statistical inference methods based on non-probability samples

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## Basic information

The goal of this package is to provide R users access to modern methods
for non-probability samples when auxiliary information from the
population or probability sample is available:

-   inverse probability weighting estimators with possible calibration
    constraints [@chen2020],
-   mass imputation estimators based in nearest neighbours [@yang2021],
    predictive mean matching and regression imputation [@kim2021],
-   doubly robust estimators with bias minimization [@chen2020,
    @yang2020].

The package allows for:

-   variable section in high-dimensional space using SCAD [@yang2020],
    Lasso and MCP penalty (via the `ncvreg`, `Rcpp`, `RcppArmadillo`
    packages),
-   estimation of variance using analytical and bootstrap approach (see
    @wu2023),
-   integration with the `survey` package when probability sample is
    available [@Lumley2004, @Lumley2023],
-   different links for selection (`logit`, `probit` and `cloglog`) and
    outcome (`gaussian`, `binomial` and `poisson`) variables.

Details on use of the package be found:

-   on the draft (and not proofread) version the book [Modern inference
    methods for non-probability samples with
    R](https://ncn-foreigners.github.io/nonprobsvy-book/),
-   example codes that reproduce papers are available at github in the
    repository [software
    tutorials](https://github.com/ncn-foreigners/software-tutorials).

## Installation

You can install the recent version of `nonprobsvy` package from main branch
[Github](https://github.com/ncn-foreigners/nonprobsvy) with:

```{r, eval=FALSE}
remotes::install_github("ncn-foreigners/nonprobsvy")
```

or install the stable version from [CRAN](https://CRAN.R-project.org/package=nonprobsvy)

```{r, eval=FALSE}
install.packages("nonprobsvy")
```


or development version from the `dev` branch

```{r, eval=FALSE}
remotes::install_github("ncn-foreigners/nonprobsvy@dev")
```

## Basic idea

Consider the following setting where two samples are available:
non-probability (denoted as $S_A$ ) and probability (denoted as $S_B$)
where set of auxiliary variables (denoted as $\boldsymbol{X}$) is
available for both sources while $Y$ and $\boldsymbol{d}$ (or
$\boldsymbol{w}$) is present only in probability sample.

| Sample                  |           | Auxiliary variables $\boldsymbol{X}$ | Target variable $Y$ | Design ($\boldsymbol{d}$) or calibrated ($\boldsymbol{w}$) weights |
|---------------|--------------:|:-------------:|:-------------:|:-------------:|
| $S_A$ (non-probability) |         1 |             $\checkmark$             |    $\checkmark$     |                                 ?                                  |
|                         |       ... |             $\checkmark$             |    $\checkmark$     |                                 ?                                  |
|                         |     $n_A$ |             $\checkmark$             |    $\checkmark$     |                                 ?                                  |
| $S_B$ (probability)     |   $n_A+1$ |             $\checkmark$             |          ?          |                            $\checkmark$                            |
|                         |       ... |             $\checkmark$             |          ?          |                            $\checkmark$                            |
|                         | $n_A+n_B$ |             $\checkmark$             |          ?          |                            $\checkmark$                            |


## Basic functionalities

Suppose $Y$ is the target variable, $\boldsymbol{X}$ is a matrix of
auxiliary variables, $R$ is the inclusion indicator. Then, if we are
interested in estimating the mean $\bar{\tau}_Y$ or the sum $\tau_Y$ of
the of the target variable given the observed data set
$(y_k, \boldsymbol{x}_k, R_k)$, we can approach this problem with the
possible scenarios:

-   unit-level data is available for the non-probability sample $S_{A}$,
    i.e. $(y_{k}, \boldsymbol{x}_{k})$ is available for all units
    $k \in S_{A}$, and population-level data is available for
    $\boldsymbol{x}_{1}, ..., \boldsymbol{x}_{p}$, denoted as
    $\tau_{x_{1}}, \tau_{x_{2}}, ..., \tau_{x_{p}}$ and population size
    $N$ is known. We can also consider situations where population data
    are estimated (e.g. on the basis of a survey to which we do not have
    access),
-   unit-level data is available for the non-probability sample $S_A$
    and the probability sample $S_B$, i.e.
    $(y_k, \boldsymbol{x}_k, R_k)$ is determined by the data. is
    determined by the data: $R_k=1$ if $k \in S_A$ otherwise $R_k=0$,
    $y_k$ is observed only for sample $S_A$ and $\boldsymbol{x}_k$ is
    observed in both in both $S_A$ and $S_B$,

### When unit-level data is available for non-probability survey only

<table class='table'>
<tr> <th>Estimator</th> <th>Example code</th> <tr>
<tr>
<td>
Mass imputation based on regression imputation
</td>
<td>
```{r, eval = FALSE}
nonprob(
  outcome = y ~ x1 + x2 + ... + xk,
  data = nonprob,
  pop_totals = c(`(Intercept)`= N,
                 x1 = tau_x1,
                 x2 = tau_x2,
                 ...,
                 xk = tau_xk),
  method_outcome = "glm",
  family_outcome = "gaussian"
)
```
</td>
<tr>
<tr>
<td>
Inverse probability weighting
</td>
<td>
```{r, eval = FALSE}
nonprob(
  selection =  ~ x1 + x2 + ... + xk, 
  target = ~ y, 
  data = nonprob, 
  pop_totals = c(`(Intercept)` = N, 
                 x1 = tau_x1, 
                 x2 = tau_x2, 
                 ..., 
                 xk = tau_xk), 
  method_selection = "logit"
)
```
</td>
<tr>
<tr>
<td>
Inverse probability weighting with calibration constraint
</td>
<td>
```{r, eval = FALSE}
nonprob(
  selection =  ~ x1 + x2 + ... + xk, 
  target = ~ y, 
  data = nonprob, 
  pop_totals = c(`(Intercept)`= N, 
                 x1 = tau_x1, 
                 x2 = tau_x2, 
                 ..., 
                 xk = tau_xk), 
  method_selection = "logit", 
  control_selection = controlSel(est_method_sel = "gee", h = 1)
)
```
</td>
<tr>
<tr>
<td>
Doubly robust estimator
</td>
<td>
```{r, eval = FALSE}
nonprob(
  selection = ~ x1 + x2 + ... + xk, 
  outcome = y ~ x1 + x2 + …, + xk, 
  pop_totals = c(`(Intercept)` = N, 
                 x1 = tau_x1, 
                 x2 = tau_x2, 
                 ..., 
                 xk = tau_xk), 
  svydesign = prob, 
  method_outcome = "glm", 
  family_outcome = "gaussian"
)
```
</td>
<tr>
</table>                                                                                                                                                                                                                                                                              

### When unit-level data are available for both surveys

<table class='table'>
<tr> <th>Estimator</th> <th>Example code</th> <tr>
<tr>
<td>
Mass imputation based on regression imputation
</td>
<td>
```{r, eval = FALSE}
nonprob(
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "glm", 
  family_outcome = "gaussian"
)
```
</td>
<tr>
<tr>
<td>
Mass imputation based on nearest neighbour imputation
</td>
<td>
```{r, eval = FALSE}
nonprob(
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "nn", 
  family_outcome = "gaussian", 
  control_outcome = controlOutcome(k = 2)
)
```
</td>
<tr>
<tr>
<td>
Mass imputation based on predictive mean matching
</td>
<td>
```{r, eval = FALSE}
nonprob(
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "pmm", 
  family_outcome = "gaussian"
)
```
</td>
<tr>
<tr>
<td>
Mass imputation based on regression imputation with variable selection (LASSO)
</td>
<td>
```{r, eval = FALSE}
nonprob(
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "pmm", 
  family_outcome = "gaussian", 
  control_outcome = controlOut(penalty = "lasso"), 
  control_inference = controlInf(vars_selection = TRUE)
)
```
</td>
<tr>
<tr>
<td>
Inverse probability weighting
</td>
<td>
```{r, eval = FALSE}
nonprob(
  selection =  ~ x1 + x2 + ... + xk, 
  target = ~ y, 
  data = nonprob, 
  svydesign = prob, 
  method_selection = "logit"
)
```
</td>
<tr>
<tr>
<td>
Inverse probability weighting with calibration constraint
</td>
<td>
```{r, eval = FALSE}
nonprob(
  selection =  ~ x1 + x2 + ... + xk, 
  target = ~ y, 
  data = nonprob, 
  svydesign = prob, 
  method_selection = "logit", 
  control_selection = controlSel(est_method_sel = "gee", h = 1)
)
```
</td>
<tr>
<tr>
<td>
Inverse probability weighting with calibration constraint with variable selection (SCAD)
</td>
<td>
```{r, eval = FALSE}
nonprob(
  selection =  ~ x1 + x2 + ... + xk, 
  target = ~ y, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "pmm", 
  family_outcome = "gaussian", 
  control_inference = controlInf(vars_selection = TRUE)
)
```
</td>
<tr>
<tr>
<td>
Doubly robust estimator
</td>
<td>
```{r, eval = FALSE}
nonprob(
  selection = ~ x1 + x2 + ... + xk, 
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "glm", 
  family_outcome = "gaussian"
)
```
</td>
<tr>
<tr>
<td>
Doubly robust estimator with variable selection (SCAD) and bias minimization
</td>
<td>
```{r, eval = FALSE}
nonprob(
  selection = ~ x1 + x2 + ... + xk, 
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob,
  method_outcome = "glm", 
  family_outcome = "gaussian", 
  control_inference = controlInf(
    vars_selection = TRUE, 
    bias_correction = TRUE
  )
)
```
</td>
<tr>
</table>                                                                                      

## Examples

Simulate example data from the following paper: Kim, Jae Kwang, and
Zhonglei Wang. "Sampling techniques for big data analysis."
International Statistical Review 87 (2019): S177-S191 [section 5.2]

```{r example, message=FALSE}
library(survey)
library(nonprobsvy)

set.seed(1234567890)
N <- 1e6 ## 1000000
n <- 1000
x1 <- rnorm(n = N, mean = 1, sd = 1)
x2 <- rexp(n = N, rate = 1)
epsilon <- rnorm(n = N) # rnorm(N)
y1 <- 1 + x1 + x2 + epsilon
y2 <- 0.5*(x1 - 0.5)^2 + x2 + epsilon
p1 <- exp(x2)/(1+exp(x2))
p2 <- exp(-0.5+0.5*(x2-2)^2)/(1+exp(-0.5+0.5*(x2-2)^2))
flag_bd1 <- rbinom(n = N, size = 1, prob = p1)
flag_srs <- as.numeric(1:N %in% sample(1:N, size = n))
base_w_srs <- N/n
population <- data.frame(x1,x2,y1,y2,p1,p2,base_w_srs, flag_bd1, flag_srs)
base_w_bd <- N/sum(population$flag_bd1)
```

Declare `svydesign` object with `survey` package

```{r}
sample_prob <- svydesign(ids= ~1, weights = ~ base_w_srs, 
                         data = subset(population, flag_srs == 1))
```

Estimate population mean of `y1` based on doubly robust estimator using
IPW with calibration constraints.

```{r}
result_dr <- nonprob(
  selection = ~ x2,
  outcome = y1 ~ x1 + x2,
  data = subset(population, flag_bd1 == 1),
  svydesign = sample_prob
)
```

Results

```{r}
summary(result_dr)
```

Mass imputation estimator

```{r}
result_mi <- nonprob(
  outcome = y1 ~ x1 + x2,
  data = subset(population, flag_bd1 == 1),
  svydesign = sample_prob
)
```

Results

```{r}
summary(result_mi)
```

Inverse probability weighting estimator

```{r}
result_ipw <- nonprob(
  selection = ~ x2,
  target = ~y1,
  data = subset(population, flag_bd1 == 1),
  svydesign = sample_prob)
```

Results

```{r}
summary(result_ipw)
```

## Funding

Work on this package is supported by the National Science Centre, OPUS
22 grant no. 2020/39/B/HS4/00941.

## References (selected)