models_pdf.Rmd

---
title: "models_pdf.rmd"
author: "Billiau/Eng/Li"
date: "12/6/2019"
output: pdf_document
---

## Set up
```{r setup}
source('packages.R')
source('styleguide.R')
source('helpers.R')
source('cleaner.R')

# Read in Clean DF 
df.clean <- add_time("complete_data_clean.csv")
df.clean <- add_coach_change(df.clean)
df.tourney <- add_time("tourney_data_clean.csv")
df.tourney <- add_coach_change(df.tourney)

# Check dimensions - len(unique schools) * len(unique years) must equal # of rows
dim_checker(df.clean)
dim_checker(df.tourney)
```

## Models 

### Model 1: pool all teams together, OLS model for 3PAr change over time

```{r cars}
### Model 1: pool all teams together, OLS model for 3PAr change over time
lm1 <- lm(X3PAr ~ time, df.tourney)
summary(lm1)

# Model 1: Control for team no pooling, OLS model for 3PAr change over time
lm1a <- lm(X3PAr ~ time, df.tourney)
summary(lm1a)

p <- ggplot(df.tourney, aes(x = time + 2003, y = X3PAr)) + 
  geom_point() +
  stat_smooth(method = "lm", col = '#EE3838', se = F) +   
  labs(title="3PAr Over Time - Mod 1") +
  xlab("Year") +
  ylab("3PAr") +
  #ylim(c(0,0.6)) + 
  theme_hodp()
p
```

### Model 2 & 3
Model 2 - Mixed Model, fixed effect of time, random intercept stratified on School
Model 3 - Mixed Model, fixed effect of time, random slopes and intercept stratified on School
 
Includes model comparison 

```{r}
lmer2 <- lmer(X3PAr ~ time + (1 | School), data=df.tourney)
summary(lmer2)
head(coef(summary(lmer2)))
head(coef(lmer2)$School)

# split into categories -- > see if there are  
# binary indicator as for if they were a winning team or losing team for most of the seasons

#Fitting a random slopes, random intercepts model may fail to converge
lmer3 <- lmer(X3PAr ~ time + (1  + time|School), data=df.tourney) # fails to converge

### Nelder_Mead converges successfully!!! - but only for the df.tourney
lmer3d <- update(lmer3, control=lmerControl(optimizer="Nelder_Mead"))

# Summary
summary(lmer3d)

### COMPARE 
# Fixed coefs
coef(summary(lmer2))
coef(summary(lmer3d))

# Look at differences b/t individual schools coefs
head(coef(lmer2)$School)
head(coef(lmer3d)$School)

# Unsurprisingly, our random slopes and intercepts model is significantly better than 
# our simple random intercepts model. It may be even more overfit though. 
anova(lmer2,lmer3d)

### Let's do some plots
# Get coefficients 
year <- 2003:2017
intercept.mm <- summary(lmer3d)$coef[1,1]
slope.mm <- summary(lmer3d)$coef[2,1]
lmer3fn <- function(year) { 
  return(intercept.mm + (year - 2003) * slope.mm)
}

p <- ggplot(df.tourney, aes(x = time + 2003, y = X3PAr)) + 
  geom_point() +
  stat_smooth(method = "lm", col = '#EE3838', se = F) +  
  geom_segment(aes(x = 2003, y = lmer3fn(2003), xend = 2017, yend = lmer3fn(2017), colour = "yellow"), 
               data = df.tourney) + 
  labs(title="3PAr Over Time - Pooled") +
  xlab("Year") +
  ylab("3PAr") +
  #ylim(c(0,0.6)) + 
  theme_hodp()
p
# Very similar fixed effects vs. Completely pooled OLS
```

## SEGMENTED REGRESSION with OLS
Using the segmented package to explore segmenting the OLS model 

```{r}
# Using the segmented package
# have to provide estimates for breakpoints.
# apriori guess of 3 based on when the rule change was announced,   
seg4 <- segmented(lm1, 
                  seg.Z = ~ time, 
                  psi =c(3,10))
# display the summary
summary(seg4)

# get breakpoints
seg4$psi
# get the slopes
head(slope(seg4))

# get the fitted data
my.fitted <- fitted(seg4)
my.model <- data.frame(year = df.tourney$year, X3PAr = my.fitted)

# plot the fitted model
ggplot(my.model, aes(x = year, y = X3PAr)) + geom_line()

# Replot things
cols <- c("Simple OLS"='#EE3838',"Segmented OLS"='#78C4D4')
p <- ggplot(df.tourney, aes(x = time + 2003, y = X3PAr)) + 
  geom_point() +
  stat_smooth(method = "lm", aes(col = '#EE3838'), se = F,size=1) + 
  geom_line(data = my.model, aes(x = year, y = X3PAr, color = '#78C4D4'),
            linetype = "solid", size=1) + 
  scale_colour_identity(name="Model Type", 
                        breaks = c('#EE3838','#78C4D4'),
                        labels = c("Simple OLS", "Segmented OLS break between 9 and 10"),
                        guide = "legend")  +   
  labs(title="3PAr Over Time - Pooled") +
  xlab("Year") +
  ylab("3PAr") +
  theme_hodp()
p 


# Method Test if breakpoints are significant
davies.test(lm1, ~time)

seg6 <- segmented(lm1, 
                  seg.Z = ~ time, 
                  psi = list(time = c(9.3)))
# Check for existence of one breakpoint using the pscore.test command

davies.test(seg6, ~time)

seg7 <- segmented(lm1, 
                  seg.Z = ~ time, 
                  psi = list(time = c(3.1, 9.3)))
summary(seg7)
davies.test(seg7, ~time)




# 2 Breakpoints
# between the 2006-2007 and 2007-08 seasons - Rule change was announced in May 2007 
# https://www.espn.com/mens-college-basketball/news/story?id=2859065

# Then another, more significant breakpoint between 2012-2013 and 2013-2014 seasons 
# Curry sets record for NBA 3's in 2012-13
# get breakpoints
seg7$psi
# get the slopes
slope(seg7)

# get the fitted data
seg7.fitted <- fitted(seg7)
seg7.fitted.df <- data.frame(year = df.tourney$year, X3PAr = seg7.fitted)

# plot the fitted model
ggplot(seg7.fitted.df, aes(x = year, y = X3PAr)) + geom_line()

p <- ggplot(df.tourney, aes(x = time + 2003, y = X3PAr)) + 
  geom_point() +
  stat_smooth(method = "lm", aes(col = '#EE3838'), se = F,size=1) + 
  geom_line(data = seg7.fitted.df, aes(x = year, y = X3PAr, color = '#78C4D4'),
            linetype = "solid", size=1) + 
  scale_colour_identity(name="Model Type", 
                        breaks = c('#EE3838','#78C4D4'),
                        labels = c("Simple OLS", "Segmented OLS: 2 breaks"),
                        guide = "legend")  +   
  labs(title="3PAr Over Time - Pooled") +
  xlab("Year") +
  ylab("3PAr") +
  theme_hodp()
p 

```

### Mixed model - segmented, unsuccessful aside exploring teams
Unsuccessful Aside playing around with same coach

```{r}
# Mixed Model Segmented - coaching change
lmer8 <- lmer(X3PAr ~ time*same.coach + (1  + time|School) , data=df.tourney) # fails to converge
# Use all fit to find a model that converges
# Source: https://joshua-nugent.github.io/allFit/
ncores <- detectCores()
diff_optims <- allFit(lmer8, maxfun = 1e6, parallel = 'multicore', ncpus = ncores)
is.OK <- sapply(diff_optims, is, "merMod")
diff_optims.OK <- diff_optims[is.OK]
lapply(diff_optims.OK,function(x) x@optinfo$conv$lme4$messages)

# Nelder_Mead for convergence
lmer8a <- update(lmer8, control=lmerControl(optimizer="Nelder_Mead"))

# Summary
summary(lmer8a)

### COMPARE 
# Fixed coefs
coef(summary(lmer8a))

# Look at differences b/w individual schools coefs
head(coef(lmer8a)$School)

### Let's do some plots
# Get coefficients 
year <- 2003:2017
intercept.false <- summary(lmer8a)$coef[1,1]
slope.false <- summary(lmer8a)$coef[2,1]
intercept.true <- summary(lmer8a)$coef[3,1]
slope.true <- summary(lmer8a)$coef[4,1]
lmer8fn <- function(year, true_flag) { 
  return(intercept.false + (year - 2003) * slope.false + 
           intercept.true*true_flag + slope.true * (year - 2003) * true_flag)
}

p <- ggplot(df.tourney, aes(x = time + 2003, y = X3PAr)) + 
  geom_point() +
  geom_segment(aes(x = 2003, y = lmer8fn(2003,0), xend = 2017, yend = lmer8fn(2017,0), 
                   colour = '#EE3838'), 
               data = df.tourney) + 
  geom_segment(aes(x = 2003, y = lmer8fn(2003,1), xend = 2017, yend = lmer8fn(2017,1), 
                   colour = '#78C4D4'), 
               data = df.tourney) +
  scale_colour_identity(name="Model Type", 
                        breaks = c('#EE3838','#78C4D4'),
                        labels = c("Coaching Change", "Same Coach"),
                        guide = "legend") +
  labs(title="3PAr Over Time - Pooled") +
  xlab("Year") +
  ylab("3PAr") +
  theme_hodp()
p
```


## Model 9 - Final mode
Mixed model by era 

```{r}
df.tourney$era <- as.factor((df.tourney$year > 2006) + (df.tourney$year > 2012))

# Mixed Model Segmented - coaching change
lmer9 <- lmer(X3PAr ~ time*era + (1  + time|School), data=df.tourney) # fails to converge
# Use all fit to find a model that converges
# Source: https://joshua-nugent.github.io/allFit/
ncores <- detectCores()
diff_optims <- allFit(lmer9, maxfun = 1e6, parallel = 'multicore', ncpus = ncores)
is.OK <- sapply(diff_optims, is, "merMod")
diff_optims.OK <- diff_optims[is.OK]
lapply(diff_optims.OK,function(x) x@optinfo$conv$lme4$messages)

### T tests to determine whether or not slopes are significantly different
# https://influentialpoints.com/Training/simple_linear_regression-principles-properties-assumptions.html

# Nelder_Mead for convergence
lmer9a <- update(lmer9, control=lmerControl(optimizer="Nelder_Mead"))
summary(lmer9a)

# draws the mean number of threes point attempts per year across years

### Compare

### COMPARE 
# Fixed coefs
coef(summary(lmer9a))

# Look at differences b/w individual schools coefs
head(coef(lmer9a)$School)

### Let's do some plots
# Get coefficients 
year <- 2003:2017
intercept.0 <- summary(lmer9a)$coef[1,1]
slope.0 <- summary(lmer9a)$coef[2,1]
intercept.1 <- summary(lmer9a)$coef[3,1]
intercept.2 <- summary(lmer9a)$coef[4,1]
slope.1 <- summary(lmer9a)$coef[5,1]
slope.2 <- summary(lmer9a)$coef[6,1]
lmer9fn <- function(year, era1, era2) { 
  return(intercept.0 + (year - 2003) * slope.0 + 
           intercept.1*era1 + slope.1 * (year - 2003) * era1 + 
           intercept.2*era2 + slope.2 * (year - 2003) * era2)
}

p <- ggplot(df.tourney, aes(x = time + 2003, y = X3PAr)) + 
  geom_point() +
  geom_segment(aes(x = 2003, y = lmer9fn(2003,0,0), xend = 2006, yend = lmer9fn(2006,0,0), 
                   colour = '#EE3838'), 
               data = df.tourney) + 
  geom_segment(aes(x = 2007, y = lmer9fn(2007,1,0), xend = 2012, yend = lmer9fn(2012,1,0), 
                   colour = '#78C4D4'), 
               data = df.tourney) +
  geom_segment(aes(x = 2013, y = lmer9fn(2013,0,1), xend = 2018, yend = lmer9fn(2018,0,1), 
                   colour = '#4B5973'), 
               data = df.tourney) +
  geom_vline(xintercept = 2012.5) +
  geom_vline(xintercept = 2006.5) +
  scale_colour_identity(name="Model Type:", 
                        breaks = c('#EE3838','#78C4D4','#4B5973'),
                        labels = c("2003-2006", "2007-2012", "2013-2018"),
                        guide = "legend") +
  annotate(geom="label", x = 2012.5, y = 0, label = "NBA Revolution", fill ="#F2F2F2", color = "black") +
  annotate(geom="label", x = 2006.5, y = 0, label = "NCAA Rule Change", fill ="#F2F2F2", color = "black") +
  labs(title="3PAr Over Time - Segmented Mixed Model") +
  xlab("Year") +
  ylab("3PAr") +
  theme_hodp()
p

# lmer9a vs. lmer3d anova test
anova(lmer3d, lmer9a)
```

## Contrast t-tests 


```{r}
# Test if slopes are significantly different
# Create a vector of coefficients to test if difference between slope for 
# era0*time and era1*time is 0 or not
coefs <- summary(lmer9a)$coef[,1]

# Construct our vector of differences C
C = c(0,0,0,0,-1,0)
contrast_test_lmer9a(C,coefs)
# slopes are different

# Test if difference between slope for era0*time and era2*time is 0 or not
# Construct our vector of differences C
C = c(0,0,0,0,0,-1)
contrast_test_lmer9a(C,coefs)
# slopes are different

# Test if difference between slope for era0*time and era2*time is 0 or not
# Construct our vector of differences C
C = c(0,0,0,0,0,-1)
contrast_test_lmer9a(C,coefs)
# slopes are different

# Test if difference between slope for era1*time and era2*time is 0 or not
# Construct our vector of differences C
C = c(0,0,0,0,1,-1)
contrast_test_lmer9a(C,coefs)
# slopes are different

# Pairwise tests were performed. The problem of multiple comparisons is ignored here 
# (a) because our pvalues are <<0.001 and (b) because we are only doing 3 tests to
# compare all of the slopes.



### AIC TABLE ### 
# Make AIC Table
simple_OLS_AIC = AIC(lm1)
lmer2_AIC = AIC(lmer2)
lmer3d_AIC = AIC(lmer3d)
lin_seg_AIC = AIC(seg7)
mm_final_AIC = AIC(lmer9a)

AICs = c(simple_OLS_AIC,lmer2_AIC, lmer3d_AIC, lin_seg_AIC,mm_final_AIC)
titles = c("Simple OLS",
           "Simple Random Intercepts", 
           "Simple Random Slopes and Intercepts", 
           "Segmented OLS", 
           "Segmented Random Slopes and Intercepts")
data.frame(titles, AICs)

########## EOF ########## 
```