assignment5.Rmd

---
title: "Homework 5"
author: "Allan Kimaina"
header-includes:
- \usepackage{pdflscape}
- \newcommand{\blandscape}{\begin{landscape}}
- \newcommand{\elandscape}{\end{landscape}}
output:
  pdf_document: default
  html_document: default
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = F)

# loadl lm  package
library(plyr)
library(dplyr)
library("tibble")
library(car)
library(sjPlot)
library(sjmisc)
library(sjlabelled)
library(ggpubr)
library(ggpmisc)
library(gridExtra)
library(stargazer)
library(e1071)
library(jtools)
library(effects)
library(multcompView)
library(ggplot2)
library(ggrepel)
library(MASS)
library(broom)
library(ggcorrplot)
library(leaps)
library(relaimpo)
library(olsrr)

# aestetics
library(xtable)# latex table
library(texreg) # for latex table

# load GLM packages
library(ROCR)
library(arm)
library(foreign)
library(nnet)
library(VGAM)
library(ordinal)
library(ModelGood)
library(InformationValue)
library(rms)
library(AER) # for overdispersion
library(metRology)
library(hett)
library(lme4)




# load required data
riskyBehaviours.df=read.csv("data/risky_behaviors.csv")
# Data wrangling
riskyBehaviours.df$couples <- factor(riskyBehaviours.df$couples)
riskyBehaviours.df$women_alone <- factor(riskyBehaviours.df$women_alone)

# https://stackoverflow.com/questions/14510277/scale-back-linear-regression-coefficients-in-r-from-scaled-and-centered-data
# scaling
riskyBehaviours.df$cBupacts <- (riskyBehaviours.df$bupacts - mean(riskyBehaviours.df$bupacts)) / (2 * sd(riskyBehaviours.df$bupacts))

#hist(riskyBehaviours.df$fupacts)

```

\onecolumn
# Chapter 7 Question 2:

Continuous probability simulation: the logarithms of weights (in pounds) of men in the United States are approximately normally distributed with mean 5.13 and standard deviation 0.17; women with mean 4.96 and standard deviation 0.20. Suppose 10 adults selected at random step on an elevator with a capacity of 1750 pounds.

``What is the probability that the elevator cable breaks?``

```{r results='asis', echo=FALSE, warning=FALSE}
set.seed(111)
men_weight <- 5.13
men_sd <- .17
prop_female <- .52
women_weight <- 4.96
women_sd <- .20
n <- 10
num_sims <- 1000
capacity <- 1750
elevatorCableBreaks <- rep(NA, num_sims)
for (j in 1:num_sims) {
  weights <- rep(NA, n)
  is_female <- rbinom(n=n, size=1, prob=prop_female)
  for (i in 1:n) {
    log_weight <- ifelse(is_female[i]==1,
                         rnorm(n=1, mean=women_weight, sd=women_sd),
                         rnorm(n=1, mean=men_weight, sd=men_sd)
                         )
    weights[i]<- exp(log_weight)
  }
  elevatorCableBreaks[j] <- ifelse(capacity<sum(weights), 1, 0)
}
sum(elevatorCableBreaks) / num_sims


```

 We assumed the proportion of women to men is .52 (provided in the chapter but not in this question)
\onecolumn

# Chapter 7 Question 8:
Inference for the ratio of parameters: a (hypothetical) study compares the costs and effectiveness of two different medical treatments. 

 * In the first part of the study, the difference in costs between treatments A and B is estimated at $600 per patient, with a standard error of $400, based on a regression with 50 degrees of freedom.
 
 *  In the second part of the study, the difference in effectiveness is estimated at 3.0 (on some relevant measure), with a standard error of 1.0, based on a regression with 100 degrees of freedom. 
 
 *   For simplicity, assume that the data from the two parts of the study were collected independently.

Inference is desired for the incremental cost-effectiveness ratio: the difference between the average costs of the two treatments, divided by the difference between their average effectiveness. (This problem is discussed further by heitjan, Moskowitz, and Whang, 1999.)

## Part A:
`` Create 1000 simulation draws of the cost difference and the effectiveness difference, and make a scatterplot of these draws. ``

```{r results='asis', echo=FALSE, warning=FALSE}
set.seed(111)
num_sims <- 1000

# initialize with NA
cost_diff = rep(NA, num_sims)
effect_diff = rep(NA, num_sims)

n_cost = 52
n_effect = 102

for (i in 1:num_sims) {
  # cost difference
  treatment = rbinom(n = n_cost, size = 1, prob = 0.5)
  X = cbind(1, treatment)
  sigma = sqrt(400 ^ 2 / solve((t(X) %*% X))[2, 2])
  error = rnorm(n_cost, mean = 0, sd = sigma)
  cost = 100000 + 600 * treatment + error
  Cost_model = lm(cost ~ treatment)
  cost_diff[i] = coef(Cost_model)[2]
  
  # effect difference
  treatment = rbinom(n = n_effect, size = 1, prob = 0.5)
  X = cbind(1, treatment)
  sigma = sqrt(1 / solve((t(X) %*% X))[2, 2])
  error = rnorm(n_effect, mean = 0, sd = sigma)
  effect = 100000 + 3 * treatment + error
  effect_model = lm(effect ~ treatment)
  effect_diff[i] = coef(effect_model)[2]
  
}




costEffect.df <- data.frame(cost=cost_diff, effect=effect_diff)

ggplot(data=costEffect.df, aes(x=cost, y=effect)) + 
      geom_point() +
      labs(x="Cost Difference", y="Effectiveness Difference") +
      theme_classic()+ 
      guides(fill=FALSE)



```


## Part B: 
``Use simulation to come up with an estimate, 50% interval, and 95% interval for the incremental cost-effectiveness ratio. ``

```{r results='asis', echo=FALSE, warning=FALSE}


costEffect.df$ratio <- costEffect.df$cost/costEffect.df$effect

hist(costEffect.df$ratio, 
     col="blue",
     main = "Cost Effectiveness", xlim = c(-2000,3000), xlab="ratio",
     breaks=300)


```

```{r results='asis', echo=FALSE, warning=FALSE}
quantile(costEffect.df$ratio, c(.25, .75))
quantile(costEffect.df$ratio, c(.025, .975))

#stargazer(summary(costEffect.df))


```

## Part C:
``Repeat this problem, changing the standard error on the difference in effectiveness to 2.0.``

```{r results='asis', echo=FALSE, warning=FALSE}
num_sims <- 1000
effect_diff = rep(NA, num_sims)
n_cost = 52
n_effect = 102

for (i in 1:num_sims) {
  # effect difference
  treatment = rbinom(n = n_effect, size = 1, prob = 0.5)
  X = cbind(1, treatment)
  sigma = sqrt(1 / solve((t(X) %*% X))[2, 2])
  error = rnorm(n_effect, mean = 0, sd = sigma)
  effect = 100000 + 2 * treatment + error
  effect_model = lm(effect ~ treatment)
  effect_diff[i] = coef(effect_model)[2]
  
}


costEffect.df$ratio <- costEffect.df$cost/costEffect.df$effect
hist(costEffect.df$ratio, 
     col="blue",
     main = "Cost Effectiveness", xlim = c(-2000,3000), xlab="ratio",
     breaks=300)

```



```{r results='asis', echo=FALSE, warning=FALSE}
quantile(costEffect.df$ratio, c(.25, .75))
quantile(costEffect.df$ratio, c(.025, .975))

```


\onecolumn

# Chapter 8 Question 1:
Fitting the wrong model: suppose you have 100 data points that arose from the following model: y =3+0 .1x1+0.5x2+error, with errors having a t distribution with mean 0, scale 5, and 4 degrees of freedom.We shall explore the implications of fitting a standard linear regression to these data.


## Part A:
``Simulate data from this model. For simplicity, suppose the values of x1 are simply the integers from 1 to 100, and that the values of x2 are random and equally likely to be 0 or 1.3 Fit a linear regression (with normal errors) to these data and see if the 68% confidence intervals for the regression coefficients (for each, the estimates ±1 standard error) cover the true values. ``

```{r results='asis', echo=FALSE, warning=FALSE}
set.seed(111)
x1 <- 1:100
x2 <- rbinom(100, 1, 0.5)
error <- rnorm(100, 0, 1)

y = 3 + .1*x1 + .5*x2 + error

model_8.1.a <- lm(y ~ x1 + x2)
texreg(list(model_8.1.a), 
       custom.model.names = c("Model 8A"),
       single.row=TRUE,  float.pos = "h")


```

```{r results='asis', echo=FALSE, warning=FALSE}
coverage_test = c(3, 0.1, 0.5)
regression_coef = as.data.frame(summary(model_8.1.a)$coefficients)
int_coverage = cbind(regression_coef$Estimate-regression_coef$`Std. Error`,
                   regression_coef$Estimate+regression_coef$`Std. Error`)

int_coverage_test = cbind(coverage_test>=int_coverage[,1]) & (coverage_test<=int_coverage[,2])
rownames(int_coverage) <- c("Intercept","X1","X2")
rownames(int_coverage_test) <- c("Intercept","X1","X2")
test_matrix <- merge(int_coverage, int_coverage_test, by = "row.names", all = TRUE)
colnames(test_matrix) <- c("Coef","Lower","Upper","Coverage")
xtable(test_matrix, comment=FALSE)
```
After generating 68% confidence intervals for the regression coefficient, all the point estimates except Intercept were not contained in the 68% confidence intervals. However the CI band was too wide making high uncertainty in the point estimate. Repeating this simulation without setting seed would definitely result in difference coverage conclusion due to this wide CI

## Part B:
`` Put the above step in a loop and repeat 1000 times. Calculate the confidence coverage for the 68% intervals for each of the three coefficients in the model. ``

```{r results='asis', echo=FALSE, warning=FALSE}
set.seed(111)
coefs <- array(NA, c(3, 1000))
se <- array(NA, c(3, 1000))

for (i in 1:ncol(coefs)) {
  x1 <- 1:100
  x2 <- rbinom(100, 1, 0.5)
  error <-rnorm(100, 0, 1)
              
  y = 3 + 0.1*x1 + 0.5*x2 + error
  
  lm.model <- summary(lm(y ~ x1 + x2))
  coefs[1,i] <- tidy(lm.model)[1,2]
  coefs[2,i] <- tidy(lm.model)[2,2]
  coefs[3,i] <- tidy(lm.model)[3,2]
  
  se[1,i] <- tidy(lm.model)[1,3]
  se[2,i] <- tidy(lm.model)[2,3]
  se[3,i] <- tidy(lm.model)[3,3]
}

mean_coef <- rowMeans(coefs)
mean_se <- rowMeans(se)

int_coverage<- cbind(mean_coef + (-1 * mean_se), 
                     mean_coef + (1 * mean_se))

int_coverage_test = cbind(coverage_test>=int_coverage[,1]) & (coverage_test<=int_coverage[,2])
rownames(int_coverage) <- c("Intercept","X1","X2")
rownames(int_coverage_test) <- c("Intercept","X1","X2")
test_matrix <- merge(int_coverage, int_coverage_test, by = "row.names", all = TRUE)
colnames(test_matrix) <- c("Coef","Lower","Upper","Coverage")
xtable(test_matrix, comment=FALSE)

```

After simulating the previous step 1000 times, we calculated the mean of all the point estimates and standard error. All the 3 estimates were contained in the 68% confidence intervals.

## Part C:
``Repeat this simulation, but instead fit the model using t errors (see Exercise 6.6).``

```{r results='asis', echo=FALSE, warning=FALSE}
set.seed(111)
coefs <- array(NA, c(3, 1000))
se <- array(NA, c(3, 1000))

for (i in 1:ncol(coefs)) {
  x1 <- 1:100
  x2 <- rbinom(100, 1, 0.5)
  error <- rt.scaled(100, df = 4, mean = 0, sd = 5)
  y = 3 + 0.1*x1 + 0.5*x2 + error
  
  lm.model <-  summary(tlm(y ~ x1 + x2))
 
  coefs[1,i] <- lm.model$loc.summary$coefficients[1,1]
  coefs[2,i] <- lm.model$loc.summary$coefficients[2,1]
  coefs[3,i] <- lm.model$loc.summary$coefficients[3,1]
  
  se[1,i] <- lm.model$loc.summary$coefficients[1,2]
  se[2,i] <- lm.model$loc.summary$coefficients[2,2]
  se[3,i] <- lm.model$loc.summary$coefficients[3,2]
}

mean_coef <- rowMeans(coefs)
mean_se <- rowMeans(se)

int_coverage<- cbind(mean_coef + (-1 * mean_se), 
                     mean_coef + (1 * mean_se))

int_coverage_test = cbind(coverage_test>=int_coverage[,1]) & (coverage_test<=int_coverage[,2])
rownames(int_coverage) <- c("Intercept","X1","X2")
rownames(int_coverage_test) <- c("Intercept","X1","X2")
test_matrix <- merge(int_coverage, int_coverage_test, by = "row.names", all = TRUE)
colnames(test_matrix) <- c("Coef","Lower","Upper","Coverage")
xtable(test_matrix, comment=FALSE)

```
After  the previous step 1000 times using t errors, we calculated the mean of all the point estimates and standard error. All the 3 estimates were contained in the 68% confidence intervals.
However the standard error were inflated making the CI band  too wide compared to the previous model
\onecolumn

# Chapter 8 Question 4:
Model checking for count data: the folder risky.behavior contains data from a study of behavior of couples at risk for HIV; see Exercise 6.1.

## Part A:
``Fit a Poisson regression model predicting number of unprotected sex acts from baseline hIV status. Perform predictive simulation to generate 1000 datasets and record both the percent of observations that are equal to 0 and the percent that are greater than 10 (the third quartile in the observed data) for each. Compare these values to the observed value in the original data. ``

### Fit a Poisson regression model
```{r results='asis', echo=FALSE, warning=FALSE}
riskyBehaviours.glm1 <- glm(fupacts ~ bs_hiv, family=poisson, data=riskyBehaviours.df)
texreg(list(riskyBehaviours.glm1), 
       custom.model.names = c("Model 1: Poisson"),
       single.row=TRUE,  float.pos = "h")

```
This model does not fit well, the rule of thumb dictates that the ratio of residual deviance to degree of freedom should be close to 1, however we get a value which is way far from 1.

### 1000 Simulations
```{r results='asis', echo=FALSE, warning=FALSE}
# data wrangling and cleaning
riskyBehaviours.df$bs_hiv_bin <- ifelse(riskyBehaviours.df$bs_hiv=="negative",0,1)
X = cbind(1,as.numeric(riskyBehaviours.df$bs_hiv_bin))

# simulate 1000
n.sims <- 1000
riskyBehaviours.sims1 <- arm::sim(riskyBehaviours.glm1, n.sims)
n<- length(riskyBehaviours.df$fupacts)
y.rep <- array(NA, c(n.sims,n))
beta <- coef(riskyBehaviours.sims1)

# do 1000 simulations
for(i in 1:n.sims){
  y.hat <- exp(X%*%beta[i,])
  y.rep[i,]<-rpois(n,y.hat)
}

# test statistics
test.rep <- rep(NA, n.sims)
test.rep.gt10 <- rep(NA, n.sims)
for (i in 1:n.sims){
  test.rep[i]<- mean(y.rep[i,]==0)
  test.rep.gt10[i]<-mean(y.rep[i,]>10)
}

#summary(rowSums(y.hat == 0)/ncol(y.hat))
#summary(rowSums(y.hat > 10)/ncol(y.hat))

real.gt.0= mean(riskyBehaviours.df$fupacts == 0)
real.gt.10= mean(riskyBehaviours.df$fupacts > 10)

# summary(test.rep)
# summary(test.rep.gt10)
par(mfrow = c(1, 2)) 
hist(test.rep, main="# Plot A ( 1K Simulations)", xlab="proportion of sex acts = 0",  col="blue")
hist(test.rep.gt10, main="# Plot B (1K Simulations)", xlab="proportion of of sex acts > 10",  col="blue")

```

### Plot A:  Frequency of number of unprotected sex acts at followup = 0

```
##     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 0.000e+00 0.000e+00 0.000e+00 1.843e-05 0.000e+00 2.304e-03 
```

The actual dataset had 29% of the observation having  0 number of unprotected sex acts at followup. However, after performing predictive simulation to generate 1000 datasets, we found out that the percentage of hypothesized observations with  number of unprotected sex acts at followup equal to 0 ranged from 0% to 0.2% with most of replication at 0% - all of which were much much lower than the observed test statistic of 29%.

### Plot B:  Frequency of  number of unprotected sex acts at followup > 10

```
##  Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## 0.7995  0.8364  0.8479  0.8485  0.8594  0.9101 
```
 
The actual dataset had 36% of the observations having number of unprotected sex acts at followup that is greater than 10. However, after performing predictive simulation to generate 1000 datasets, we found out that the percentage of hypothesized observations with  number of unprotected sex acts at followup greater than 10 ranged from 84% `(1stIQR)` to 86% `(3rdIQR)` with an average of 85% - all of which were much much higher than the observed test statistic of 36%. 
 
## Part B:
``Repeat (a) using an overdispersed Poisson regression model.``

### Fit an overdispersed Poisson regression model
```{r results='asis', echo=FALSE, warning=FALSE}
riskyBehaviours.glm2 <- glm(fupacts ~ bs_hiv, family=quasipoisson, data=riskyBehaviours.df)
texreg(list(riskyBehaviours.glm2), 
       custom.model.names = c("Model 2: overdispersed Poisson"),
       single.row=TRUE,  float.pos = "h")



```
Much better model but it does not fit as well, the ratio of residual deviance to degree of freedom is still far from 1.

### 1000 Simulations
```{r results='asis', echo=FALSE, warning=FALSE}

# simulate 1000
n.sims <- 1000
riskyBehaviours.sims2 <- arm::sim(riskyBehaviours.glm2, n.sims)
n<- length(riskyBehaviours.df$fupacts)
y.rep <- array(NA, c(n.sims,n))
beta <- coef(riskyBehaviours.sims2)

# do 1000 simulations
overdisp <- summary(riskyBehaviours.glm2)$dispersion
for(i in 1:n.sims){
  y.hat <- exp(X%*%beta[i,])
  a <- y.hat/(overdisp-1) # dispersion param
  y.rep[i,]<-rnegbin(n,y.hat, a)
}

# test statistics
test.rep <- rep(NA, n.sims)
test.rep.gt10 <- rep(NA, n.sims)
for (i in 1:n.sims){
  test.rep[i]<- mean(y.rep[i,]==0)
  test.rep.gt10[i]<-mean(y.rep[i,]>10)
}

#summary(rowSums(y.hat == 0)/ncol(y.hat))
#summary(rowSums(y.hat > 10)/ncol(y.hat))

real.gt.0= mean(riskyBehaviours.df$fupacts == 0)
real.gt.10= mean(riskyBehaviours.df$fupacts > 10)

# summary(test.rep)
# summary(test.rep.gt10)
par(mfrow = c(1, 2)) 
hist(test.rep, main="# Plot C ( 1K Simulations)", xlab="proportion of sex acts = 0",  col="blue")
hist(test.rep.gt10, main="# Plot D (1K Simulations)", xlab="proportion of of sex acts > 10",  col="blue")

```

### Plot C: Frequency of number of unprotected sex acts at followup = 0

```
##  Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## 0.1290  0.2166  0.2373  0.2388  0.2604  0.3364 
```

After performing predictive simulation based on overdispersed model, we found out that the percentage of hypothesized observations with  number of unprotected sex acts at followup equal to 0 varied from 21% to 26%, with most of replication averaging at 24% - all of which were reasonably much closer to the observed test statistic of 29% compared to the previous model. In summary, this tells us that this model reasonably fits this aspect of data  well, however other aspects of this data might not be well fit by this model.

### Plot D: Frequency of  number of unprotected sex acts at followup > 10

```
##  Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## 0.2834  0.3664  0.3871  0.3881  0.4124  0.4862  
```
 
The actual dataset had 36% of the observations having number of unprotected sex acts at followup that is greater than 10. After the predictive simulation, we found out that the percentage of hypothesized observations with  number of unprotected sex acts at followup greater than 10 varied from 37% `(1stIQR)` to 41% `(3rdIQR)` with an average of 39% - all of which were reasonably much closer to the observed test statistic of 36% than the previous model. This tells us that this model reasonably fits this aspect of data  well, however other aspects of this data might not be well fit by this model.


## Part C:
``Repeat (b), also including ethnicity and baseline number of unprotected sex acts as input variables.``

### Fit an overdispersed Poisson regression model
```{r results='asis', echo=FALSE, warning=FALSE}
riskyBehaviours.glm3 <- glm(fupacts ~ bs_hiv + bupacts, family=quasipoisson, data=riskyBehaviours.df)
texreg(list(riskyBehaviours.glm3), 
       custom.model.names = c("Model 3: overdispersed Poisson"),
       single.row=TRUE,  float.pos = "h")



```
Much  better model than the previous model but it does not meet the standard, the ratio of residual deviance to degree of freedom is 24 which is still far from 1.

### 1000 Simulations
```{r results='asis', echo=FALSE, warning=FALSE}

# simulate 1000
n.sims <- 1000
 riskyBehaviours.sims3<- arm::sim(riskyBehaviours.glm3, n.sims)
n<- length(riskyBehaviours.df$fupacts)
y.rep <- array(NA, c(n.sims,n))
beta <- coef(riskyBehaviours.sims3)
X = cbind(1,as.numeric(riskyBehaviours.df$bs_hiv_bin),riskyBehaviours.df$bupacts)

# do 1000 simulations
overdisp <- summary(riskyBehaviours.glm3)$dispersion
for(i in 1:n.sims){
  y.hat <- exp(X%*%beta[i,])
  a <- y.hat/(overdisp-1) # dispersion param
  y.rep[i,]<-rnegbin(n,y.hat, a)
}

# test statistics
test.rep <- rep(NA, n.sims)
test.rep.gt10 <- rep(NA, n.sims)
for (i in 1:n.sims){
  test.rep[i]<- mean(y.rep[i,]==0)
  test.rep.gt10[i]<-mean(y.rep[i,]>10)
}

#summary(rowSums(y.hat == 0)/ncol(y.hat))
#summary(rowSums(y.hat > 10)/ncol(y.hat))

real.gt.0= mean(riskyBehaviours.df$fupacts == 0)
real.gt.10= mean(riskyBehaviours.df$fupacts > 10)

# summary(test.rep)
# summary(test.rep.gt10)
par(mfrow = c(1, 2)) 
hist(test.rep, main="# Plot E ( 1K Simulations)", xlab="proportion of sex acts = 0",  col="blue")
hist(test.rep.gt10, main="# Plot F (1K Simulations)", xlab="proportion of of sex acts > 10",  col="blue")

```

### Plot E: Frequency of number of unprotected sex acts at followup = 0

```
##    Min.  1st Qu. Median  Mean    3rd Qu. Max 
##  0.1198  0.1843  0.2028  0.2041  0.2235  0.2857 
```

After performing predictive simulation based on overdispersed model with added predictor (bupacts), we found out that the percentage of hypothesized observations with  number of unprotected sex acts at followup equal to 0 varied from 18% to 22%, with most of replication averaging at 20% - most of which were closer to the observed test statistic of 29%

### Plot F: Frequency of  number of unprotected sex acts at followup > 10

```
##   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## 0.3111  0.3802  0.4032  0.4033  0.4263  0.4908  
```
 
The actual dataset had 36% of the observations having number of unprotected sex acts at followup that is greater than 10. After the predictive simulation, we found out that the percentage of hypothesized observations with  number of unprotected sex acts at followup greater than 10 varied from 38% `(1stIQR)` to 42% `(3rdIQR)` with an average of 40% - all of which were closer to the observed test statistic of 36%.


In summary this model really does well in compromising and balancing between the 2 aspect of data we have accessed. The upshot  is that this model fit the data much much better compared to the previous 2 model


\onecolumn

# Chapter 11 Question 4:
The folder cd4 has CD4 percentages for a set of young children with HIV who were measured several times over a period of two years. The dataset also includes the ages of the children at each measurement.

### Time Function
```{r results='asis', echo=FALSE, warning=FALSE, }
# load required data
cd4.df=read.csv("data/cd4.csv")
cd4.df$newpid= as.character(cd4.df$newpid)
cd4.df$treatmnt= as.factor(cd4.df$treatmnt)

# transfrom data
time<-cd4.df$visage-cd4.df$baseage
cd4.df<-cbind(cd4.df,time)

# define ggplot alpha level
cd4.df = ddply(cd4.df, .(newpid), function(x){
  x$alpha = ifelse(runif(n = 1) > 0.9, 1, 0.1)
  x$is_test = factor(ifelse(x$newpid<5, 0, 1))
  x
})

# define timne function
time_function_plot <- ggplot(cd4.df, aes(x = time, y = cd4pct) )+
                    geom_point()+
                    labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+
                    aes(alpha=alpha, color=factor(newpid)) +
                    theme_classic()+
                    theme(legend.position="none") 
time_function_plot
```


We define our time function as visit_age - base_Age. This ensured that all children visit started from day 0 and had a serial/sequential progression of events as shown in the graph above. Some data points (children) have been grey scaled for interchangeability.

It is also interesting to note that most visit occurred quarterly


## Part A: 
``Graph the outcome (the CD4 percentage, on the square root scale) for each child as a function of time.``



### Overall Plot - all 254 children
```{r results='asis', echo=FALSE, warning=FALSE}


cd4_c11_q4.plot<- ggplot(cd4.df, aes(x = time, y = cd4pct)) +
    guides(colour=FALSE) +
    labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+
    theme_classic()+ guides(fill=FALSE)+
    theme(legend.position="none") +
    scale_y_sqrt()
    #geom_smooth(se=FALSE, colour="black", method = "lm", size=2)
    #facet_wrap(~person_id)

 cd4_c11_q4.plot +
       geom_line()+
   geom_point()+
 aes(color  = factor(newpid)) 


```

The plot above is difficult synthesize and inter prate, let's try plotting 5% of the children.
The below shows 5% of the children (colored) and the remaining 90% of children (grey scale).

### Plot for 5% of all children
```{r results='asis', echo=FALSE, warning=FALSE}


cd4_c11_q4a.plot.2<-  cd4_c11_q4.plot+
      geom_line(aes(alpha=alpha, color=factor(newpid)) )+
   geom_point(aes(alpha=alpha, color=factor(newpid)) )+
  guides(alpha=FALSE)  
  #geom_smooth(se=T, colour="black", method = "lm", size=1, linetype=6)

subset <- c(3,4,6,7,10,11,12,15,17,19,62,63)
cd4_c11_q4a.plot.3<- ggplot(cd4.df[cd4.df$newpid %in% subset,],
                            aes(x = time, y = cd4pct)) +
    guides(colour=FALSE) +
    labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+
    theme_bw()+
    theme(legend.position="none") +
    geom_line()+
    geom_point()+
    aes(color  = factor(newpid)) +
  # geom_smooth(se=T, colour="red", method = "lm",  linetype=6)+
    facet_wrap(~factor(newpid)) 

 #grid.arrange(cd4_c11_q4a.plot.2, cd4_c11_q4a.plot.3, nrow = 2.)   
cd4_c11_q4a.plot.2
cd4_c11_q4a.plot.3
```





## Part B:
``Each child’s data has a time course that can be summarized by a linear fit. Estimate these lines and plot them for all the children.``


```{r results='asis', echo=FALSE, warning=FALSE}

cd4_c11_q4b.plot.3<- ggplot(cd4.df, aes(x = time, y = cd4pct, color=factor(newpid)) )+
                    #  geom_point()+
                    geom_smooth(se=F,method = "lm", linetype=1) +
                    labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+
                    theme_classic()+
                    theme(legend.position="none") 

cd4_c11_q4b.plot.3
```

From the above plot, it is difficult to decipher the magnitude and direction of each regression line. The next page provides a plot for only a subset of these children which is more interpretable.

```{r results='asis', echo=FALSE, warning=FALSE, out.width="93%"}

cd4_c11_q4b.plot.4 <- ggplot(cd4.df)+
                    #geom_point(
                    #  aes(x = time, y = cd4pct, alpha=alpha, color=factor(newpid) ) )+
                    geom_line(stat = "smooth", 
                              method = lm, 
                              aes(x = time, y = cd4pct, alpha=alpha, color=factor(newpid))
                              ) +
                    labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+
                     coord_cartesian(ylim = c(0,60), xlim = c(0, 2))+
                    theme_classic()+
                    theme(legend.position="none") 

cd4_c11_q4b.plot.4
```


```{r results='asis', echo=FALSE, warning=FALSE, out.width="93%"}

# selective subsetting
subset <- c(3,4,6,7,10,11,12,15,17,19,62,63)
cd4_c11_q4b.plot.2<- ggplot(cd4.df[cd4.df$newpid %in% subset,],
                            aes(x = time, y = cd4pct)) +
    guides(colour=FALSE) +
    labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+
    theme_bw()+
    theme(legend.position="none") +
    geom_line()+
    geom_point()+
    aes(color  = factor(newpid)) +
  # geom_smooth(se=T, colour="red", method = "lm",  linetype=6)+
    facet_wrap(~factor(newpid)) +
    geom_smooth(se=F, colour="black", method = "lm",  linetype=6)
cd4_c11_q4b.plot.2
```

For some children, cd4 percent increases over time. However, for most children, cd4 percent decreases with time  indicating that time is negatively associated with  the outcome.

This is rather odd! we expect that most of the children to have increasing cd4 percent over time. Lets take a look at the general  trend of all children in general (Group level regression) as shown below.

```{r results='asis', echo=FALSE, warning=FALSE}
cd4_c11_q4b.plot<- ggplot(cd4.df, aes(x = time, y = cd4pct)) +
 geom_line() +
 # geom_point()+
    guides(colour=FALSE) +
    labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+
    theme_classic()+ guides(fill=FALSE)+
    theme(legend.position="none") +
    aes(alpha=alpha, color=factor(newpid)) +
    geom_smooth(se=T, colour="black", method = "lm", linetype=6)

cd4_c11_q4b.plot
```

The dotted line is the population level regression and apparently it has a negative slope. It's likely that  some treatment/ARVs works better than others or maybe other child level factors might be confounded with the design variables. This is beyond the scope of this analysis, therefore we shall not make any adjustment or include other predictors.


## Part C:
``Set up a model for the children’s slopes and intercepts as a function of the treatment and age at baseline. Estimate this model using the two-step procedure–first estimate the intercept and slope separately for each child, then fit the between-child models using the point estimates from the first step.``

```{r results='asis', echo=FALSE, warning=FALSE}
# step 1
df_1 = cd4.df %>% na.omit()  %>% 
  group_by(newpid) %>%
  dplyr::select(newpid,cd4pct,time) %>% 
  mutate(intercept = coef(lm(cd4pct~time))[1]) %>% 
  mutate(slope = coef(lm(cd4pct~time))[2]) %>% 
  dplyr::select(newpid, intercept,slope)

# step 2
df_2 = inner_join(cd4.df,df_1,by="newpid")

model.intercept = lm(intercept~baseage+treatmnt, data =df_2 )
model.slope = lm(slope~baseage+treatmnt, data =df_2 )

texreg(list(model.intercept,model.slope), 
       custom.model.names = c("Intercept Model", "Slope Model"),
       single.row=TRUE,  float.pos = "h")

```

\onecolumn

# Chapter 12 Question 2: 
Continuing with the analysis of the CD4 data from Exercise 11.4:

## Part A:
``Write a model predicting CD4 percentage as a function of time with varying intercepts across children. Fit using lmer() and interpret the coefficient for time.``

```{r results='asis', echo=FALSE, warning=FALSE}
model_chap12_q2a = lmer(cd4pct~1+time+(1|newpid),data=cd4.df,REML=F) # varying intercepts

texreg(list(model_chap12_q2a), 
       custom.model.names = c("Model 12.2 A"),
       single.row=TRUE,  float.pos = "h")
#summary(model_chap12_q2a)
#coef(model_chap12_q2a)

```

Given that this model is a random intercept model, it only accounts for "idiosyncratic" variation that is due to individual level differences in baseline cd4 percent. It also assumes that time effects on cd4 percent is fixed meaning that every child cd4 percent varies systematically with time. 

* **Time** -  Given that time is a fixed effect on every child, each year on the study is associated with a decreased in cd4 percentage by 3 % on average. In this model, time was statistically significant and had a substantial effect on cd4 percentage.

## Part B:
``Extend the model in (a) to include child-level predictors (that is, group-level predictors) for treatment and age at baseline. Fit using lmer() and interpret the coefficients on time, treatment, and age at baseline.``



```{r results='asis', echo=FALSE, warning=FALSE}
#model_chap12_q2b=lmer(cd4pct~1+time+(1+treatmnt+baseage|newpid),data=cd4.df,REML=F) 
model_chap12_q2b=lmer(cd4pct~1+time+treatmnt+baseage+(1|newpid),data=cd4.df,REML=F) 



texreg(list(model_chap12_q2b), 
       custom.model.names = c("Model 12.2 B"),
       single.row=TRUE,  float.pos = "h")
#summary(model_chap12_q2b)
#coef(model_chap12_q2b)
#toLatex(mtable(model_chap12_q2b))
```

In general, adding child-level predictors (i.e, group-level predictors) definitely improves the model. In fact it increased the precision of our estimates. Note that the variance of our random effect estimate has reduced from 128 to 123 meaning some of the variations has been explained by addition of fixed effect predictors like base age and treatment. Conversely, residual has not reduced by much  possibly because we haven't added another random effect predictor that is due to child-specific idiosyncrasies.

### Interpretation

* **Treatment** - Children who were on second treatment had 1.2125% higher cd4 percent on average  than children on first treatment. This predictor had substantial effect on cd4 percent however it was insignificant.

*  **Baseline Age** - 1 year difference in baseline age is associate with changing cd4 percentage by 0.9485 % on average. Meaning children who were enrolled into the study early had higher cd4 percentage. This predictor was highly significant.

* **Time** -  Each year on treatment was associated with a decreased in cd4 percentage by 2.9615 % on average. This predictor was highly significant.

## Part C:
``Investigate the change in partial pooling from (a) to (b) both graphically and numerically.``




### Graphically

```{r results='asis', echo=FALSE, warning=FALSE}

# complete pooling
complete_pooling.fit<-lm(cd4pct~time, data=cd4.df)

df_complete_pooling <- data_frame(
  model = "Complete pooling",
  newpid = unique(cd4.df$newpid),
  intercept = coef(complete_pooling.fit)[1], 
  slope_time = coef(complete_pooling.fit)[2])



# model for question 2 A
df = coef(model_chap12_q2a)[["newpid"]]
df$intercept = df$'(Intercept)'
df_chap12_q2a <- df %>% 
   dplyr::select(intercept, slope_time = time)  %>% 
  as_tibble() %>% 
   rownames_to_column("newpid") %>% 
  add_column(model = "Model 12.2 A")


# model for question 2 B
df = coef(model_chap12_q2b)[["newpid"]]
df$intercept = df$'(Intercept)'
df_chap12_q2b <- df %>% 
   dplyr::select(intercept, 
         slope_time = time,
         treatmnt_slope= treatmnt2, baseage_slope= baseage )  %>% 
  as_tibble() %>% 
   rownames_to_column("newpid") %>% 
  add_column(model = "Model 12.2 B")


# bind data
df_models <- bind_rows(df_chap12_q2a, df_chap12_q2b, df_complete_pooling) %>% 
  left_join(cd4.df, by = "newpid")

# create subset
subset <- c(67,72,77,78,79,11,12,19,27,23,62,63)

# construct plot
model_comparison <- ggplot(df_models[df_models$newpid %in% subset,]) + 
  aes(x = time, y = cd4pct) + 
  geom_abline(aes(intercept = intercept, slope = slope_time, color = model),
              size = .75) + 
   #theme_bw()+
    #geom_line( linetype=6)+
    geom_point()+
  facet_wrap("newpid") +
  labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+
  scale_x_continuous(breaks = 0:4 * 2) + 
  scale_color_brewer(palette = "Dark2") + 
  theme(legend.position = "top")

model_comparison

```


A few observation to make.

* From the plot, observe that the partial pooling model with added child-level predictors (Model 12.2B)  is pulled substantially towards the complete pooling model (group average) compared to  Model 12.2.A

* If you take a look at children with less observations (incomplete data), you will notice that Model 12.2.B is pulled more towards the complete pooling model (group average) compared to  Model 12.2.A. See child 67, 79, 77


### Numerically
```{r results='asis', echo=FALSE, warning=FALSE}

anova_mixd <- anova(model_chap12_q2a,model_chap12_q2b)
xtable(anova_mixd, comment=FALSE)
```

Using ANOVA to check the changes between the 2 models, we find out that adding additional child-level predictors actually reduced the residual sum of squares . This reduction was  statistically significant. We also had lower AIC and BIC in the second model implying that the additional regressors actually improved the model



```{r results='asis', echo=FALSE, warning=FALSE}
#coef(model_chap12_q2a)
q2a_plot <- as.data.frame(lme4::ranef(model_chap12_q2a,condVar=T)) %>%
    ggplot(aes(y=grp,x=condval))+
    geom_point()+facet_wrap(~term,scales="free_x")+
    geom_errorbarh(aes(xmin=condval-2*condsd,xmax=condval+2*condsd),height=0)+
   labs(x="Random Effects", y="Children") +
      theme_classic()+ 
      guides(fill=FALSE) +
   ggtitle("Model 12.2 A")

q2b_plot <-  as.data.frame(lme4::ranef(model_chap12_q2b,condVar=T)) %>%
    ggplot(aes(y=grp,x=condval))+
    geom_point()+facet_wrap(~term,scales="free_x")+
    geom_errorbarh(aes(xmin=condval-2*condsd,xmax=condval+2*condsd),height=0)+
    labs(x="Random Effects", y="Children") +
      theme_classic()+ 
      guides(fill=FALSE)+
   ggtitle("Model 12.2 B")

grid.arrange(q2a_plot,q2b_plot, ncol = 2.)


```

From the above plot, notice the scale of each plot. Model 12.2.A  has higher random error than Model 12.2.B. This implies that adding child-level predictors reduces the variance of our random effect estimate. We noticed a substantial reduction in the variance of the intercept - 128 to 123. This is mainly because some of the variations had been explained by the addition of fixed effect predictors like base age and treatment.

## Part D:
Compare results in (b) to those obtained in part (c).

As seen in the previous question, We observed that by adding child-level predictors (i.e, group-level predictors) the model improved significantly hence increasing the precision of our estimates. The addition of the 2 fixed effect predictors like  base age and treatment made the variance of the random effect estimates to reduce by 5 units essentially by compromising between group level and individual level variation. 
However, the residual did not reduced by much  possibly because we did not add another random effect predictor that is due to child-specific idiosyncrasies e.g ARVS.

```{r results='asis', echo=FALSE, warning=FALSE}
df_zoom <- df_models %>% 
  filter(newpid %in% c("63", "67", "79", "77"))

model_comparison  %+% df_zoom
```

In summary, this shrinkage effect observed in model 12.2.B ensures that our model compromised between  individual-level effects and group level effects at the same regularizing extreme estimates and taming children with less  visits/observations as depicted by child 67,79 and 77.

\onecolumn

# Chapter 12 Question 3:

`Predictions for new observations and new groups:`

## Part A:
``Use the model fit from Exercise 12.2(b) to generate simulation of predicted CD4 percentages for each child in the dataset at a hypothetical next time point.``

```{r results='asis', echo=FALSE, warning=FALSE}


set.seed(111)
n.sims = 1000

# create hypothetical next time point
cd4.df3 = cd4.df %>% 
            na.omit() %>% # remove missing values
            group_by(newpid) %>% # group by PID
            mutate(current.time = max(time)) %>% # get maximum time for each child
            mutate(future.time = current.time + .25) # add .25 year beacause appintments are done quarterly

# slelect only relevant time
cd4.df4 = cd4.df3 %>%
          dplyr::select(newpid,treatmnt,baseage,future.time) %>% 
          rename(time=future.time) %>%
          unique() # make sure it is a one row per patient dataset

# Simulate 1000 times using sim fx
model_chap12_q2b.sim = sim(model_chap12_q2b,n.sims)
# fetch array offixed effect for intercept for each simulation
fixed.intercept = coef(model_chap12_q2b.sim)$fixef[,1]
# get random effect array for intercept for each simulation
rand.eff = coef(model_chap12_q2b.sim)$ranef$newpid
# get the intercept for each person by summing fixed and random effects
sim.intercept = fixed.intercept + rand.eff
# create matrix
sim.intercept = matrix(sim.intercept,nrow=n.sims,ncol=length(unique(cd4.df$newpid)))

# generate x matrix
X = cbind(1,cd4.df4$time,as.numeric(cd4.df4$treatmnt)-1,cd4.df4$baseage)

cd4.df.pred.future.sim = list()
for (simnum in 1:n.sims){
  b.hat = rbind(sim.intercept[simnum,],coef(model_chap12_q2b.sim)$fixef[simnum,2],coef(model_chap12_q2b.sim)$fixef[simnum,3], coef(model_chap12_q2b.sim)$fixef[simnum,4]) 
  rownames(b.hat) = c("intercept","time","treatmnt","baseage")
  
  sigma.y.hat <- sigma.hat(model_chap12_q2b.sim)
  
  cd4.future.sim = rep(NA,nrow(cd4.df4))
  # actual simulation 
  for (idnum in 1:length(cd4.future.sim)){
    cd4.future.sim[idnum] = rnorm (n.sims, X[idnum,] %*% b.hat[,idnum], sigma.y.hat)
  }
  
  cd4.df.pred.future.sim[[simnum]] = cd4.df4 %>%
                  cbind(cd4.future.sim) %>%
                  rename(future.time = time)
}

  cd4.df.pred.future.sim.all = bind_rows(cd4.df.pred.future.sim) %>%
                                      group_by(newpid) %>% 
                                      mutate(mean.future.cd4 = mean(cd4.future.sim))
  
  cd4.df.pred.future.sim.all$index = as.numeric(cd4.df.pred.future.sim.all$newpid)
  cd4.df.set1 = cd4.df.pred.future.sim.all%>%
                filter(index <= 25) 

  simulation_plot_12_3 <- ggplot(cd4.df.set1, aes(x=newpid, y =cd4.future.sim, color=factor(index))) +
  geom_jitter(alpha = 0.1) +
    #stat_boxplot(geom="errorbar", width=.5,  color = "black")+
  geom_boxplot(alpha = 0, color = "black") +
  labs(y = "Predicted CD4 %", x="Children")+
  theme_classic()+ guides(fill=FALSE)+
  theme(legend.position="none") 

  simulation_plot_12_3

  plot1.df = cd4.df.pred.future.sim.all %>% dplyr::filter(index > 25 & index <=52)
  plot2.df = cd4.df.pred.future.sim.all %>% dplyr::filter(index > 52 & index <=78)
  plot3.df =  cd4.df.pred.future.sim.all %>% dplyr::filter(index > 78 & index <=104)
  plot4.df =  cd4.df.pred.future.sim.all %>% dplyr::filter(index > 104 & index <=129) 
  plot5.df =  cd4.df.pred.future.sim.all %>% dplyr::filter(index > 129 & index <=154) 
  plot6.df =  cd4.df.pred.future.sim.all %>% dplyr::filter(index > 154 & index <=179)
  plot7.df =  cd4.df.pred.future.sim.all %>% dplyr::filter(index > 179 & index <=204)
  plot8.df =  cd4.df.pred.future.sim.all %>% dplyr::filter(index > 204)
  
  


```

Since most appointments/visits occurred every quarterly as shown in Chapter 11 Question 4, we made our  hypothetical next time point as the sum of maximum visit time for each child + 0.25 year. We then generated 1000 predictive simulation for each child. Each box plot represent each child and each  geometric point represent each predicted simulated cd4% (1000). Notice the wide variation around the mean. This is because We did not use the classical predict function, instead we simulated from the distribution of regression coefficients. This in turn ensured we captured the uncertainity in regression coefficients that is necessary for replicating new data.

In general most children with multiple visits had much more precise prediction interval while children with less observation had less precise prediction interval. Additionally children who had visits/observations greater 1.75 years had less precise prediction interval compared to children who had observations less than 1.75. This 2 aspects are also related to the shrinkage effects discussed in the previous question.




```{r results='asis', echo=FALSE, warning=FALSE, out.width="97%"}

grid.arrange( simulation_plot_12_3  %+% plot1.df,
                simulation_plot_12_3  %+% plot2.df,
               
               nrow =2., ncol = 1.)
  
   
```

```{r results='asis', echo=FALSE, warning=FALSE, out.width="97%"}

 grid.arrange(
                simulation_plot_12_3  %+% plot3.df,
                simulation_plot_12_3  %+% plot4.df,
               nrow =2., ncol = 1.)
  

```

```{r results='asis', echo=FALSE, warning=FALSE, out.width="97%"}

    grid.arrange( simulation_plot_12_3  %+% plot5.df,
                simulation_plot_12_3  %+% plot6.df,
               nrow =2., ncol = 1.)
```

```{r results='asis', echo=FALSE, warning=FALSE, out.width="97%"}

    grid.arrange( 
                simulation_plot_12_3  %+% plot7.df,
                simulation_plot_12_3  %+% plot8.df,
               nrow =2., ncol = 1.)
```


## Part B:
``Use the same model fit to generate simulations of CD4 percentages at each of the time periods for a new child who was 4 years old at baseline.``

```{r results='asis', echo=FALSE, warning=FALSE}

n.sims = 100

cd4.df4 = cd4.df %>%
  dplyr::select(newpid,treatmnt,baseage,time)

# Simulates the fit model 1000 times
model_chap12_q2b.sim = sim(model_chap12_q2b,n.sims)
# Extracts fixed effect for intercept for each simulation
fixed.intercept = coef(model_chap12_q2b.sim)$fixef[,1]
# Extracts random effect for intercept for each person for each simulation
rand.eff = coef(model_chap12_q2b.sim)$ranef$newpid
# Computes intercept for each person for each simulation
sim.intercept = fixed.intercept + rand.eff
sim.intercept = matrix(sim.intercept,nrow=n.sims,ncol=length(unique(cd4.df4$newpid)))

X = cbind(1,cd4.df4) %>% dplyr::select(1,time,treatmnt,baseage,newpid) %>% dplyr::mutate(treatmnt =as.numeric(treatmnt)-1)

cd4.df.4y.pred.sim = list()

for (simnum in 1:n.sims){
  b.hat = rbind(sim.intercept[simnum,],coef(model_chap12_q2b.sim)$fixef[simnum,2],coef(model_chap12_q2b.sim)$fixef[simnum,3], coef(model_chap12_q2b.sim)$fixef[simnum,4]) 
  rownames(b.hat) = c("intercept","time","treatmnt","baseage")
  colnames(b.hat) = unique(X$newpid)
  
  cd4.4y.pred.sim = rep(NA,nrow(cd4.df4))
  for (i in 1:length(cd4.4y.pred.sim)){
    cd4.4y.pred.sim[i] = as.vector(t(X[i,1:4])) %*% as.vector(t(b.hat[,as.character(X[i,"newpid"])]))
  }
  
  cd4.df.4y.pred.sim[[simnum]] = cd4.df4 %>% cbind(cd4.4y.pred.sim)
}

cd4.df.4y.pred.sim.all = bind_rows(cd4.df.4y.pred.sim) %>% 
                        arrange(newpid) %>% 
                        group_by(newpid) %>%
                        mutate(mean.4yo.cd4 = mean(cd4.4y.pred.sim)) %>%
                        filter(baseage>3.75 & baseage < 4.25)

ggplot(cd4.df.4y.pred.sim.all,
       aes(x=time,y=cd4.4y.pred.sim)) +
      geom_jitter(alpha = 0.3, aes(color=newpid)) +
    geom_smooth(se=T, colour="black", method = "lm", linetype=6) +
    #stat_boxplot(geom="errorbar", width=.5,  color = "black")+
  #geom_boxplot(alpha = 0, color = "black") +
  labs(y = "Predicted CD4 %", x="Time")+
  theme_classic()+ guides(fill=FALSE)
 # theme(legend.position="none") 


```

 We identified  16 children who had a baseline age of 4  years or  (+-.25). From the plot it is clear that children who get into the study with a baseline age of 4 years have higher cd4% decline rate. Most of their cd4 % declined by an average of 10 units
 
 
```{r results='asis', echo=FALSE, warning=FALSE}
cd4.df.4y.pred.plot<-ggplot(cd4.df.4y.pred.sim.all,
       aes(x=time,y=cd4.4y.pred.sim,color=factor(newpid))) +
      geom_jitter(alpha = 0.3) +
        geom_smooth(se=T, colour="black", method = "lm", linetype=6) +
        #stat_boxplot(geom="errorbar", width=.5,  color = "black")+
    
      labs(y = "Predicted CD4 %", x="Time")+
      facet_wrap(~newpid)+
      theme_classic()+ guides(fill=FALSE)+
      theme(legend.position="none") 

 plot1.df = cd4.df.4y.pred.sim.all[cd4.df.4y.pred.sim.all$newpid %in% c(1,103,234,130),]
cd4.df.4y.pred.plot  %+% plot1.df

```

\onecolumn

# Chapter 12 Question 4:
``Posterior predictive checking: continuing the previous exercise, use the fitted model from Exercise 12.2(b) to simulate a new dataset of CD4 percentages (with the same sample size and ages of the original dataset) for the final time point of the study, and record the average CD4 percentage in this sample. Repeat this process 1000 times and compare the simulated distribution to the observed CD4 percentage at the final time point for the actual data.``


```{r results='asis', echo=FALSE, warning=FALSE}
n.sims = 1000
# get uniqui child id
child_id = cd4.df%>%
  na.omit()%>%
  distinct(newpid, treatmnt, baseage)

# count num of id
num_id = dim(child_id)[1]

final_time_point.df = cd4.df%>%
  filter(visitno == 19)%>%
  dplyr::select(time, cd4pct)

final_time = mean(final_time_point.df$time)
y = array(NA, c(n.sims, num_id))
average = array(NA, n.sims)

for(s in 1:num_id){
  sigma.y.hat = sigma.hat(model_chap12_q2b)$sigma$data
  coef.hat = as.matrix(coef(model_chap12_q2b)$newpid)[s,]
  variable = child_id[s,]
  y[,s] = rnorm (n.sims, coef.hat %*% c(1, final_time, variable$treatmnt, variable$baseage), sigma.y.hat)
}

predicted = apply(y, 1, mean)
actual = cd4.df%>%
   na.omit()%>%
  filter(visitno == 19)%>%
  dplyr::select(cd4pct)

par(mfrow = c(1, 2)) 
hist(predicted, main = "simulated distribution", col="blue",  xlab="cd4%")
abline(v=mean(predicted),col="red")
hist(actual$cd4pct, main = "observed distribution", col="blue", xlab="cd4%")


```



A few observation to make. In general we notice the shrinkage effect depicted by the model when we try to extrapolate. The  simulated distribution of the predicted  cd4 % ranges from 21 to 24 with a mean of about 22.5. On the other hand the observed distribution ranges from 0 to 60. Comparing to the observed distribution, simulated prediction have been pulled substantially towards (group average). It is clear that the model can be  terrible at extrapolating beyond the 3rd IQR (the final time point).

In summary, this shrinkage effect observed in model 12.2.B ensures that our model compromised between  individual-level effects and group level effects at the same regularizing extreme estimates and taming children with less  visits/observations 


\onecolumn

# Source Code

```{r echo=T, warning=F}
#' ---	
#' title: "Homework 5"	
#' author: "Allan Kimaina"	
#' header-includes:	
#' - \usepackage{pdflscape}	
#' - \newcommand{\blandscape}{\begin{landscape}}	
#' - \newcommand{\elandscape}{\end{landscape}}	
#' output:	
#'   pdf_document: default	
#'   html_document: default	
#' ---	
#' 	
#' 	
#knitr::opts_chunk$set(echo = F)	
	
# loadl lm  package	
library(plyr)	
library(dplyr)	
library("tibble")	
library(car)	
library(sjPlot)	
library(sjmisc)	
library(sjlabelled)	
library(ggpubr)	
library(ggpmisc)	
library(gridExtra)	
library(stargazer)	
library(e1071)	
library(jtools)	
library(effects)	
library(multcompView)	
library(ggplot2)	
library(ggrepel)	
library(MASS)	
library(broom)	
library(ggcorrplot)	
library(leaps)	
library(relaimpo)	
library(olsrr)	
	
# aestetics	
library(xtable)# latex table	
library(texreg) # for latex table	
	
# load GLM packages	
library(ROCR)	
library(arm)	
library(foreign)	
library(nnet)	
library(VGAM)	
library(ordinal)	
library(ModelGood)	
library(InformationValue)	
library(rms)	
library(AER) # for overdispersion	
library(metRology)	
library(hett)	
library(lme4)	
	
	
	
	
# load required data	
riskyBehaviours.df=read.csv("data/risky_behaviors.csv")	
# Data wrangling	
riskyBehaviours.df$couples <- factor(riskyBehaviours.df$couples)	
riskyBehaviours.df$women_alone <- factor(riskyBehaviours.df$women_alone)	
	
# https://stackoverflow.com/questions/14510277/scale-back-linear-regression-coefficients-in-r-from-scaled-and-centered-data	
# scaling	
riskyBehaviours.df$cBupacts <- (riskyBehaviours.df$bupacts - mean(riskyBehaviours.df$bupacts)) / (2 * sd(riskyBehaviours.df$bupacts))	
	
#hist(riskyBehaviours.df$fupacts)	
	
#' 	
#' 	
#' \onecolumn	
#' # Chapter 7 Question 2:	
#' 	
#' Continuous probability simulation: the logarithms of weights (in pounds) of men in the United States are approximately normally distributed with mean 5.13 and standard deviation 0.17; women with mean 4.96 and standard deviation 0.20. Suppose 10 adults selected at random step on an elevator with a capacity of 1750 pounds.	
#' 	
#' ``What is the probability that the elevator cable breaks?``	
#' 	
#' 	
set.seed(111)	
men_weight <- 5.13	
men_sd <- .17	
prop_female <- .52	
women_weight <- 4.96	
women_sd <- .20	
n <- 10	
num_sims <- 1000	
capacity <- 1750	
elevatorCableBreaks <- rep(NA, num_sims)	
for (j in 1:num_sims) {	
  weights <- rep(NA, n)	
  is_female <- rbinom(n=n, size=1, prob=prop_female)	
  for (i in 1:n) {	
    log_weight <- ifelse(is_female[i]==1,	
                         rnorm(n=1, mean=women_weight, sd=women_sd),	
                         rnorm(n=1, mean=men_weight, sd=men_sd)	
                         )	
    weights[i]<- exp(log_weight)	
  }	
  elevatorCableBreaks[j] <- ifelse(capacity<sum(weights), 1, 0)	
}	
sum(elevatorCableBreaks) / num_sims	
	
	
#' 	
#' 	
#'  We assumed the proportion of women to men is .52 (provided in the chapter but not in this question)	
#' \onecolumn	
#' 	
#' # Chapter 7 Question 8:	
#' Inference for the ratio of parameters: a (hypothetical) study compares the costs and effectiveness of two different medical treatments. 	
#' 	
#'  * In the first part of the study, the difference in costs between treatments A and B is estimated at $600 per patient, with a standard error of $400, based on a regression with 50 degrees of freedom.	
#'  	
#'  *  In the second part of the study, the difference in effectiveness is estimated at 3.0 (on some relevant measure), with a standard error of 1.0, based on a regression with 100 degrees of freedom. 	
#'  	
#'  *   For simplicity, assume that the data from the two parts of the study were collected independently.	
#' 	
#' Inference is desired for the incremental cost-effectiveness ratio: the difference between the average costs of the two treatments, divided by the difference between their average effectiveness. (This problem is discussed further by heitjan, Moskowitz, and Whang, 1999.)	
#' 	
#' ## Part A:	
#' `` Create 1000 simulation draws of the cost difference and the effectiveness difference, and make a scatterplot of these draws. ``	
#' 	
#' 	
set.seed(111)	
num_sims <- 1000	
	
# initialize with NA	
cost_diff = rep(NA, num_sims)	
effect_diff = rep(NA, num_sims)	
	
n_cost = 52	
n_effect = 102	
	
for (i in 1:num_sims) {	
  # cost difference	
  treatment = rbinom(n = n_cost, size = 1, prob = 0.5)	
  X = cbind(1, treatment)	
  sigma = sqrt(400 ^ 2 / solve((t(X) %*% X))[2, 2])	
  error = rnorm(n_cost, mean = 0, sd = sigma)	
  cost = 100000 + 600 * treatment + error	
  Cost_model = lm(cost ~ treatment)	
  cost_diff[i] = coef(Cost_model)[2]	
  	
  # effect difference	
  treatment = rbinom(n = n_effect, size = 1, prob = 0.5)	
  X = cbind(1, treatment)	
  sigma = sqrt(1 / solve((t(X) %*% X))[2, 2])	
  error = rnorm(n_effect, mean = 0, sd = sigma)	
  effect = 100000 + 3 * treatment + error	
  effect_model = lm(effect ~ treatment)	
  effect_diff[i] = coef(effect_model)[2]	
  	
}	
	
	
	
	
costEffect.df <- data.frame(cost=cost_diff, effect=effect_diff)	
	
ggplot(data=costEffect.df, aes(x=cost, y=effect)) + 	
      geom_point() +	
      labs(x="Cost Difference", y="Effectiveness Difference") +	
      theme_classic()+ 	
      guides(fill=FALSE)	
	
	
	
#' 	
#' 	
#' 	
#' ## Part B: 	
#' ``Use simulation to come up with an estimate, 50% interval, and 95% interval for the incremental cost-effectiveness ratio. ``	
#' 	
#' 	
	
	
costEffect.df$ratio <- costEffect.df$cost/costEffect.df$effect	
	
hist(costEffect.df$ratio, 	
     col="blue",	
     main = "Cost Effectiveness", xlim = c(-2000,3000), xlab="ratio",	
     breaks=300)	
	
	
#' 	
#' 	
#' 	
quantile(costEffect.df$ratio, c(.25, .75))	
quantile(costEffect.df$ratio, c(.025, .975))	
	
#stargazer(summary(costEffect.df))	
	
	
#' 	
#' 	
#' ## Part C:	
#' ``Repeat this problem, changing the standard error on the difference in effectiveness to 2.0.``	
#' 	
#' 	
num_sims <- 1000	
effect_diff = rep(NA, num_sims)	
n_cost = 52	
n_effect = 102	
	
for (i in 1:num_sims) {	
  # effect difference	
  treatment = rbinom(n = n_effect, size = 1, prob = 0.5)	
  X = cbind(1, treatment)	
  sigma = sqrt(1 / solve((t(X) %*% X))[2, 2])	
  error = rnorm(n_effect, mean = 0, sd = sigma)	
  effect = 100000 + 2 * treatment + error	
  effect_model = lm(effect ~ treatment)	
  effect_diff[i] = coef(effect_model)[2]	
  	
}	
	
	
costEffect.df$ratio <- costEffect.df$cost/costEffect.df$effect	
hist(costEffect.df$ratio, 	
     col="blue",	
     main = "Cost Effectiveness", xlim = c(-2000,3000), xlab="ratio",	
     breaks=300)	
	
#' 	
#' 	
#' 	
#' 	
#' 	
quantile(costEffect.df$ratio, c(.25, .75))	
quantile(costEffect.df$ratio, c(.025, .975))	
	
#' 	
#' 	
#' 	
#' \onecolumn	
#' 	
#' # Chapter 8 Question 1:	
#' Fitting the wrong model: suppose you have 100 data points that arose from the following model: y =3+0 .1x1+0.5x2+error, with errors having a t distribution with mean 0, scale 5, and 4 degrees of freedom.We shall explore the implications of fitting a standard linear regression to these data.	
#' 	
#' 	
#' ## Part A:	
#' ``Simulate data from this model. For simplicity, suppose the values of x1 are simply the integers from 1 to 100, and that the values of x2 are random and equally likely to be 0 or 1.3 Fit a linear regression (with normal errors) to these data and see if the 68% confidence intervals for the regression coefficients (for each, the estimates ±1 standard error) cover the true values. ``	
#' 	
#' 	
set.seed(111)	
x1 <- 1:100	
x2 <- rbinom(100, 1, 0.5)	
error <- rnorm(100, 0, 1)	
	
y = 3 + .1*x1 + .5*x2 + error	
	
model_8.1.a <- lm(y ~ x1 + x2)	
texreg(list(model_8.1.a), 	
       custom.model.names = c("Model 8A"),	
       single.row=TRUE,  float.pos = "h")	
	
	
#' 	
#' 	
#' 	
coverage_test = c(3, 0.1, 0.5)	
regression_coef = as.data.frame(summary(model_8.1.a)$coefficients)	
int_coverage = cbind(regression_coef$Estimate-regression_coef$`Std. Error`,	
                   regression_coef$Estimate+regression_coef$`Std. Error`)	
	
int_coverage_test = cbind(coverage_test>=int_coverage[,1]) & (coverage_test<=int_coverage[,2])	
rownames(int_coverage) <- c("Intercept","X1","X2")	
rownames(int_coverage_test) <- c("Intercept","X1","X2")	
test_matrix <- merge(int_coverage, int_coverage_test, by = "row.names", all = TRUE)	
colnames(test_matrix) <- c("Coef","Lower","Upper","Coverage")	
xtable(test_matrix, comment=FALSE)	
#' 	
#' After generating 68% confidence intervals for the regression coefficient, all the point estimates except Intercept were not contained in the 68% confidence intervals. However the CI band was too wide making high uncertainty in the point estimate. Repeating this simulation without setting seed would definitely result in difference coverage conclusion due to this wide CI	
#' 	
#' ## Part B:	
#' `` Put the above step in a loop and repeat 1000 times. Calculate the confidence coverage for the 68% intervals for each of the three coefficients in the model. ``	
#' 	
#' 	
set.seed(111)	
coefs <- array(NA, c(3, 1000))	
se <- array(NA, c(3, 1000))	
	
for (i in 1:ncol(coefs)) {	
  x1 <- 1:100	
  x2 <- rbinom(100, 1, 0.5)	
  error <-rnorm(100, 0, 1)	
              	
  y = 3 + 0.1*x1 + 0.5*x2 + error	
  	
  lm.model <- summary(lm(y ~ x1 + x2))	
  coefs[1,i] <- tidy(lm.model)[1,2]	
  coefs[2,i] <- tidy(lm.model)[2,2]	
  coefs[3,i] <- tidy(lm.model)[3,2]	
  	
  se[1,i] <- tidy(lm.model)[1,3]	
  se[2,i] <- tidy(lm.model)[2,3]	
  se[3,i] <- tidy(lm.model)[3,3]	
}	
	
mean_coef <- rowMeans(coefs)	
mean_se <- rowMeans(se)	
	
int_coverage<- cbind(mean_coef + (-1 * mean_se), 	
                     mean_coef + (1 * mean_se))	
	
int_coverage_test = cbind(coverage_test>=int_coverage[,1]) & (coverage_test<=int_coverage[,2])	
rownames(int_coverage) <- c("Intercept","X1","X2")	
rownames(int_coverage_test) <- c("Intercept","X1","X2")	
test_matrix <- merge(int_coverage, int_coverage_test, by = "row.names", all = TRUE)	
colnames(test_matrix) <- c("Coef","Lower","Upper","Coverage")	
xtable(test_matrix, comment=FALSE)	
	
#' 	
#' 	
#' After simulating the previous step 1000 times, we calculated the mean of all the point estimates and standard error. All the 3 estimates were contained in the 68% confidence intervals.	
#' 	
#' ## Part C:	
#' ``Repeat this simulation, but instead fit the model using t errors (see Exercise 6.6).``	
#' 	
#' 	
set.seed(111)	
coefs <- array(NA, c(3, 1000))	
se <- array(NA, c(3, 1000))	
	
for (i in 1:ncol(coefs)) {	
  x1 <- 1:100	
  x2 <- rbinom(100, 1, 0.5)	
  error <- rt.scaled(100, df = 4, mean = 0, sd = 5)	
  y = 3 + 0.1*x1 + 0.5*x2 + error	
  	
  lm.model <-  summary(tlm(y ~ x1 + x2))	
 	
  coefs[1,i] <- lm.model$loc.summary$coefficients[1,1]	
  coefs[2,i] <- lm.model$loc.summary$coefficients[2,1]	
  coefs[3,i] <- lm.model$loc.summary$coefficients[3,1]	
  	
  se[1,i] <- lm.model$loc.summary$coefficients[1,2]	
  se[2,i] <- lm.model$loc.summary$coefficients[2,2]	
  se[3,i] <- lm.model$loc.summary$coefficients[3,2]	
}	
	
mean_coef <- rowMeans(coefs)	
mean_se <- rowMeans(se)	
	
int_coverage<- cbind(mean_coef + (-1 * mean_se), 	
                     mean_coef + (1 * mean_se))	
	
int_coverage_test = cbind(coverage_test>=int_coverage[,1]) & (coverage_test<=int_coverage[,2])	
rownames(int_coverage) <- c("Intercept","X1","X2")	
rownames(int_coverage_test) <- c("Intercept","X1","X2")	
test_matrix <- merge(int_coverage, int_coverage_test, by = "row.names", all = TRUE)	
colnames(test_matrix) <- c("Coef","Lower","Upper","Coverage")	
xtable(test_matrix, comment=FALSE)	
	
#' 	
#' After  the previous step 1000 times using t errors, we calculated the mean of all the point estimates and standard error. All the 3 estimates were contained in the 68% confidence intervals.	
#' However the standard error were inflated making the CI band  too wide compared to the previous model	
#' \onecolumn	
#' 	
#' # Chapter 8 Question 4:	
#' Model checking for count data: the folder risky.behavior contains data from a study of behavior of couples at risk for HIV; see Exercise 6.1.	
#' 	
#' ## Part A:	
#' ``Fit a Poisson regression model predicting number of unprotected sex acts from baseline hIV status. Perform predictive simulation to generate 1000 datasets and record both the percent of observations that are equal to 0 and the percent that are greater than 10 (the third quartile in the observed data) for each. Compare these values to the observed value in the original data. ``	
#' 	
#' ### Fit a Poisson regression model	
#' 	
riskyBehaviours.glm1 <- glm(fupacts ~ bs_hiv, family=poisson, data=riskyBehaviours.df)	
texreg(list(riskyBehaviours.glm1), 	
       custom.model.names = c("Model 1: Poisson"),	
       single.row=TRUE,  float.pos = "h")	
	
#' 	
#' This model does not fit well, the rule of thumb dictates that the ratio of residual deviance to degree of freedom should be close to 1, however we get a value which is way far from 1.	
#' 	
#' ### 1000 Simulations	
#' 	
# data wrangling and cleaning	
riskyBehaviours.df$bs_hiv_bin <- ifelse(riskyBehaviours.df$bs_hiv=="negative",0,1)	
X = cbind(1,as.numeric(riskyBehaviours.df$bs_hiv_bin))	
	
# simulate 1000	
n.sims <- 1000	
riskyBehaviours.sims1 <- arm::sim(riskyBehaviours.glm1, n.sims)	
n<- length(riskyBehaviours.df$fupacts)	
y.rep <- array(NA, c(n.sims,n))	
beta <- coef(riskyBehaviours.sims1)	
	
# do 1000 simulations	
for(i in 1:n.sims){	
  y.hat <- exp(X%*%beta[i,])	
  y.rep[i,]<-rpois(n,y.hat)	
}	
	
# test statistics	
test.rep <- rep(NA, n.sims)	
test.rep.gt10 <- rep(NA, n.sims)	
for (i in 1:n.sims){	
  test.rep[i]<- mean(y.rep[i,]==0)	
  test.rep.gt10[i]<-mean(y.rep[i,]>10)	
}	
	
#summary(rowSums(y.hat == 0)/ncol(y.hat))	
#summary(rowSums(y.hat > 10)/ncol(y.hat))	
	
real.gt.0= mean(riskyBehaviours.df$fupacts == 0)	
real.gt.10= mean(riskyBehaviours.df$fupacts > 10)	
	
# summary(test.rep)	
# summary(test.rep.gt10)	
par(mfrow = c(1, 2)) 	
hist(test.rep, main="# Plot A ( 1K Simulations)", xlab="proportion of sex acts = 0",  col="blue")	
hist(test.rep.gt10, main="# Plot B (1K Simulations)", xlab="proportion of of sex acts > 10",  col="blue")	
	
#' 	
#' 	
#' ### Plot A:  Frequency of number of unprotected sex acts at followup = 0	
#' 	
#' ```	
#' ##     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 	
#' ## 0.000e+00 0.000e+00 0.000e+00 1.843e-05 0.000e+00 2.304e-03 	
#' ```	
#' 	
#' The actual dataset had 29% of the observation having  0 number of unprotected sex acts at followup. However, after performing predictive simulation to generate 1000 datasets, we found out that the percentage of hypothesized observations with  number of unprotected sex acts at followup equal to 0 ranged from 0% to 0.2% with most of replication at 0% - all of which were much much lower than the observed test statistic of 29%.	
#' 	
#' ### Plot B:  Frequency of  number of unprotected sex acts at followup > 10	
#' 	
#' ```	
#' ##  Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 	
#' ## 0.7995  0.8364  0.8479  0.8485  0.8594  0.9101 	
#' ```	
#'  	
#' The actual dataset had 36% of the observations having number of unprotected sex acts at followup that is greater than 10. However, after performing predictive simulation to generate 1000 datasets, we found out that the percentage of hypothesized observations with  number of unprotected sex acts at followup greater than 10 ranged from 84% `(1stIQR)` to 86% `(3rdIQR)` with an average of 85% - all of which were much much higher than the observed test statistic of 36%. 	
#'  	
#' ## Part B:	
#' ``Repeat (a) using an overdispersed Poisson regression model.``	
#' 	
#' ### Fit an overdispersed Poisson regression model	
#' 	
riskyBehaviours.glm2 <- glm(fupacts ~ bs_hiv, family=quasipoisson, data=riskyBehaviours.df)	
texreg(list(riskyBehaviours.glm2), 	
       custom.model.names = c("Model 2: overdispersed Poisson"),	
       single.row=TRUE,  float.pos = "h")	
	
	
	
#' 	
#' Much better model but it does not fit as well, the ratio of residual deviance to degree of freedom is still far from 1.	
#' 	
#' ### 1000 Simulations	
#' 	
	
# simulate 1000	
n.sims <- 1000	
riskyBehaviours.sims2 <- arm::sim(riskyBehaviours.glm2, n.sims)	
n<- length(riskyBehaviours.df$fupacts)	
y.rep <- array(NA, c(n.sims,n))	
beta <- coef(riskyBehaviours.sims2)	
	
# do 1000 simulations	
overdisp <- summary(riskyBehaviours.glm2)$dispersion	
for(i in 1:n.sims){	
  y.hat <- exp(X%*%beta[i,])	
  a <- y.hat/(overdisp-1) # dispersion param	
  y.rep[i,]<-rnegbin(n,y.hat, a)	
}	
	
# test statistics	
test.rep <- rep(NA, n.sims)	
test.rep.gt10 <- rep(NA, n.sims)	
for (i in 1:n.sims){	
  test.rep[i]<- mean(y.rep[i,]==0)	
  test.rep.gt10[i]<-mean(y.rep[i,]>10)	
}	
	
#summary(rowSums(y.hat == 0)/ncol(y.hat))	
#summary(rowSums(y.hat > 10)/ncol(y.hat))	
	
real.gt.0= mean(riskyBehaviours.df$fupacts == 0)	
real.gt.10= mean(riskyBehaviours.df$fupacts > 10)	
	
# summary(test.rep)	
# summary(test.rep.gt10)	
par(mfrow = c(1, 2)) 	
hist(test.rep, main="# Plot C ( 1K Simulations)", xlab="proportion of sex acts = 0",  col="blue")	
hist(test.rep.gt10, main="# Plot D (1K Simulations)", xlab="proportion of of sex acts > 10",  col="blue")	
	
#' 	
#' 	
#' ### Plot C: Frequency of number of unprotected sex acts at followup = 0	
#' 	
#' ```	
#' ##  Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 	
#' ## 0.1290  0.2166  0.2373  0.2388  0.2604  0.3364 	
#' ```	
#' 	
#' After performing predictive simulation based on overdispersed model, we found out that the percentage of hypothesized observations with  number of unprotected sex acts at followup equal to 0 varied from 21% to 26%, with most of replication averaging at 24% - all of which were reasonably much closer to the observed test statistic of 29% compared to the previous model. In summary, this tells us that this model reasonably fits this aspect of data  well, however other aspects of this data might not be well fit by this model.	
#' 	
#' ### Plot D: Frequency of  number of unprotected sex acts at followup > 10	
#' 	
#' ```	
#' ##  Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 	
#' ## 0.2834  0.3664  0.3871  0.3881  0.4124  0.4862  	
#' ```	
#'  	
#' The actual dataset had 36% of the observations having number of unprotected sex acts at followup that is greater than 10. After the predictive simulation, we found out that the percentage of hypothesized observations with  number of unprotected sex acts at followup greater than 10 varied from 37% `(1stIQR)` to 41% `(3rdIQR)` with an average of 39% - all of which were reasonably much closer to the observed test statistic of 36% than the previous model. This tells us that this model reasonably fits this aspect of data  well, however other aspects of this data might not be well fit by this model.	
#' 	
#' 	
#' ## Part C:	
#' ``Repeat (b), also including ethnicity and baseline number of unprotected sex acts as input variables.``	
#' 	
#' ### Fit an overdispersed Poisson regression model	
#' 	
riskyBehaviours.glm3 <- glm(fupacts ~ bs_hiv + bupacts, family=quasipoisson, data=riskyBehaviours.df)	
texreg(list(riskyBehaviours.glm3), 	
       custom.model.names = c("Model 3: overdispersed Poisson"),	
       single.row=TRUE,  float.pos = "h")	
	
	
	
#' 	
#' Much  better model than the previous model but it does not meet the standard, the ratio of residual deviance to degree of freedom is 24 which is still far from 1.	
#' 	
#' ### 1000 Simulations	
#' 	
	
# simulate 1000	
n.sims <- 1000	
 riskyBehaviours.sims3<- arm::sim(riskyBehaviours.glm3, n.sims)	
n<- length(riskyBehaviours.df$fupacts)	
y.rep <- array(NA, c(n.sims,n))	
beta <- coef(riskyBehaviours.sims3)	
X = cbind(1,as.numeric(riskyBehaviours.df$bs_hiv_bin),riskyBehaviours.df$bupacts)	
	
# do 1000 simulations	
overdisp <- summary(riskyBehaviours.glm3)$dispersion	
for(i in 1:n.sims){	
  y.hat <- exp(X%*%beta[i,])	
  a <- y.hat/(overdisp-1) # dispersion param	
  y.rep[i,]<-rnegbin(n,y.hat, a)	
}	
	
# test statistics	
test.rep <- rep(NA, n.sims)	
test.rep.gt10 <- rep(NA, n.sims)	
for (i in 1:n.sims){	
  test.rep[i]<- mean(y.rep[i,]==0)	
  test.rep.gt10[i]<-mean(y.rep[i,]>10)	
}	
	
#summary(rowSums(y.hat == 0)/ncol(y.hat))	
#summary(rowSums(y.hat > 10)/ncol(y.hat))	
	
real.gt.0= mean(riskyBehaviours.df$fupacts == 0)	
real.gt.10= mean(riskyBehaviours.df$fupacts > 10)	
	
# summary(test.rep)	
# summary(test.rep.gt10)	
par(mfrow = c(1, 2)) 	
hist(test.rep, main="# Plot E ( 1K Simulations)", xlab="proportion of sex acts = 0",  col="blue")	
hist(test.rep.gt10, main="# Plot F (1K Simulations)", xlab="proportion of of sex acts > 10",  col="blue")	
	
#' 	
#' 	
#' ### Plot E: Frequency of number of unprotected sex acts at followup = 0	
#' 	
#' ```	
#' ##    Min.  1st Qu. Median  Mean    3rd Qu. Max 	
#' ##  0.1198  0.1843  0.2028  0.2041  0.2235  0.2857 	
#' ```	
#' 	
#' After performing predictive simulation based on overdispersed model with added predictor (bupacts), we found out that the percentage of hypothesized observations with  number of unprotected sex acts at followup equal to 0 varied from 18% to 22%, with most of replication averaging at 20% - most of which were closer to the observed test statistic of 29%	
#' 	
#' ### Plot F: Frequency of  number of unprotected sex acts at followup > 10	
#' 	
#' ```	
#' ##   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 	
#' ## 0.3111  0.3802  0.4032  0.4033  0.4263  0.4908  	
#' ```	
#'  	
#' The actual dataset had 36% of the observations having number of unprotected sex acts at followup that is greater than 10. After the predictive simulation, we found out that the percentage of hypothesized observations with  number of unprotected sex acts at followup greater than 10 varied from 38% `(1stIQR)` to 42% `(3rdIQR)` with an average of 40% - all of which were closer to the observed test statistic of 36%.	
#' 	
#' 	
#' In summary this model really does well in compromising and balancing between the 2 aspect of data we have accessed. The upshot  is that this model fit the data much much better compared to the previous 2 model	
#' 	
#' 	
#' \onecolumn	
#' 	
#' # Chapter 11 Question 4:	
#' The folder cd4 has CD4 percentages for a set of young children with HIV who were measured several times over a period of two years. The dataset also includes the ages of the children at each measurement.	
#' 	
#' ### Time Function	
#' 	
# load required data	
cd4.df=read.csv("data/cd4.csv")	
cd4.df$newpid= as.character(cd4.df$newpid)	
cd4.df$treatmnt= as.factor(cd4.df$treatmnt)	
	
# transfrom data	
time<-cd4.df$visage-cd4.df$baseage	
cd4.df<-cbind(cd4.df,time)	
	
# define ggplot alpha level	
cd4.df = ddply(cd4.df, .(newpid), function(x){	
  x$alpha = ifelse(runif(n = 1) > 0.9, 1, 0.1)	
  x$is_test = factor(ifelse(x$newpid<5, 0, 1))	
  x	
})	
	
# define timne function	
time_function_plot <- ggplot(cd4.df, aes(x = time, y = cd4pct) )+	
                    geom_point()+	
                    labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+	
                    aes(alpha=alpha, color=factor(newpid)) +	
                    theme_classic()+	
                    theme(legend.position="none") 	
time_function_plot	
#' 	
#' 	
#' 	
#' We define our time function as visit_age - base_Age. This ensured that all children visit started from day 0 and had a serial/sequential progression of events as shown in the graph above. Some data points (children) have been grey scaled for interchangeability.	
#' 	
#' It is also interesting to note that most visit occurred quarterly	
#' 	
#' 	
#' ## Part A: 	
#' ``Graph the outcome (the CD4 percentage, on the square root scale) for each child as a function of time.``	
#' 	
#' 	
#' 	
#' ### Overall Plot - all 254 children	
#' 	
	
	
cd4_c11_q4.plot<- ggplot(cd4.df, aes(x = time, y = cd4pct)) +	
    guides(colour=FALSE) +	
    labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+	
    theme_classic()+ guides(fill=FALSE)+	
    theme(legend.position="none") +	
    scale_y_sqrt()	
    #geom_smooth(se=FALSE, colour="black", method = "lm", size=2)	
    #facet_wrap(~person_id)	
	
 cd4_c11_q4.plot +	
       geom_line()+	
   geom_point()+	
 aes(color  = factor(newpid)) 	
	
	
#' 	
#' 	
#' The plot above is difficult synthesize and inter prate, let's try plotting 5% of the children.	
#' The below shows 5% of the children (colored) and the remaining 90% of children (grey scale).	
#' 	
#' ### Plot for 5% of all children	
#' 	
	
	
cd4_c11_q4a.plot.2<-  cd4_c11_q4.plot+	
      geom_line(aes(alpha=alpha, color=factor(newpid)) )+	
   geom_point(aes(alpha=alpha, color=factor(newpid)) )+	
  guides(alpha=FALSE)  	
  #geom_smooth(se=T, colour="black", method = "lm", size=1, linetype=6)	
	
subset <- c(3,4,6,7,10,11,12,15,17,19,62,63)	
cd4_c11_q4a.plot.3<- ggplot(cd4.df[cd4.df$newpid %in% subset,],	
                            aes(x = time, y = cd4pct)) +	
    guides(colour=FALSE) +	
    labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+	
    theme_bw()+	
    theme(legend.position="none") +	
    geom_line()+	
    geom_point()+	
    aes(color  = factor(newpid)) +	
  # geom_smooth(se=T, colour="red", method = "lm",  linetype=6)+	
    facet_wrap(~factor(newpid)) 	
	
 #grid.arrange(cd4_c11_q4a.plot.2, cd4_c11_q4a.plot.3, nrow = 2.)   	
cd4_c11_q4a.plot.2	
cd4_c11_q4a.plot.3	
#' 	
#' 	
#' 	
#' 	
#' 	
#' 	
#' ## Part B:	
#' ``Each child’s data has a time course that can be summarized by a linear fit. Estimate these lines and plot them for all the children.``	
#' 	
#' 	
#' 	
	
cd4_c11_q4b.plot.3<- ggplot(cd4.df, aes(x = time, y = cd4pct, color=factor(newpid)) )+	
                    #  geom_point()+	
                    geom_smooth(se=F,method = "lm", linetype=1) +	
                    labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+	
                    theme_classic()+	
                    theme(legend.position="none") 	
	
cd4_c11_q4b.plot.3	
#' 	
#' 	
#' From the above plot, it is difficult to decipher the magnitude and direction of each regression line. The next page provides a plot for only a subset of these children which is more interpretable.	
#' 	
#' 	
	
cd4_c11_q4b.plot.4 <- ggplot(cd4.df)+	
                    #geom_point(	
                    #  aes(x = time, y = cd4pct, alpha=alpha, color=factor(newpid) ) )+	
                    geom_line(stat = "smooth", 	
                              method = lm, 	
                              aes(x = time, y = cd4pct, alpha=alpha, color=factor(newpid))	
                              ) +	
                    labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+	
                     coord_cartesian(ylim = c(0,60), xlim = c(0, 2))+	
                    theme_classic()+	
                    theme(legend.position="none") 	
	
cd4_c11_q4b.plot.4	
#' 	
#' 	
#' 	
#' 	
	
# selective subsetting	
subset <- c(3,4,6,7,10,11,12,15,17,19,62,63)	
cd4_c11_q4b.plot.2<- ggplot(cd4.df[cd4.df$newpid %in% subset,],	
                            aes(x = time, y = cd4pct)) +	
    guides(colour=FALSE) +	
    labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+	
    theme_bw()+	
    theme(legend.position="none") +	
    geom_line()+	
    geom_point()+	
    aes(color  = factor(newpid)) +	
  # geom_smooth(se=T, colour="red", method = "lm",  linetype=6)+	
    facet_wrap(~factor(newpid)) +	
    geom_smooth(se=F, colour="black", method = "lm",  linetype=6)	
cd4_c11_q4b.plot.2	
#' 	
#' 	
#' For some children, cd4 percent increases over time. However, for most children, cd4 percent decreases with time  indicating that time is negatively associated with  the outcome.	
#' 	
#' This is rather odd! we expect that most of the children to have increasing cd4 percent over time. Lets take a look at the general  trend of all children in general (Group level regression) as shown below.	
#' 	
#' 	
cd4_c11_q4b.plot<- ggplot(cd4.df, aes(x = time, y = cd4pct)) +	
 geom_line() +	
 # geom_point()+	
    guides(colour=FALSE) +	
    labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+	
    theme_classic()+ guides(fill=FALSE)+	
    theme(legend.position="none") +	
    aes(alpha=alpha, color=factor(newpid)) +	
    geom_smooth(se=T, colour="black", method = "lm", linetype=6)	
	
cd4_c11_q4b.plot	
#' 	
#' 	
#' The dotted line is the population level regression and apparently it has a negative slope. It's likely that  some treatment/ARVs works better than others or maybe other child level factors might be confounded with the design variables. This is beyond the scope of this analysis, therefore we shall not make any adjustment or include other predictors.	
#' 	
#' 	
#' ## Part C:	
#' ``Set up a model for the children’s slopes and intercepts as a function of the treatment and age at baseline. Estimate this model using the two-step procedure–first estimate the intercept and slope separately for each child, then fit the between-child models using the point estimates from the first step.``	
#' 	
#' 	
# step 1	
df_1 = cd4.df %>% na.omit()  %>% 	
  group_by(newpid) %>%	
  dplyr::select(newpid,cd4pct,time) %>% 	
  mutate(intercept = coef(lm(cd4pct~time))[1]) %>% 	
  mutate(slope = coef(lm(cd4pct~time))[2]) %>% 	
  dplyr::select(newpid, intercept,slope)	
	
# step 2	
df_2 = inner_join(cd4.df,df_1,by="newpid")	
	
model.intercept = lm(intercept~baseage+treatmnt, data =df_2 )	
model.slope = lm(slope~baseage+treatmnt, data =df_2 )	
	
texreg(list(model.intercept,model.slope), 	
       custom.model.names = c("Intercept Model", "Slope Model"),	
       single.row=TRUE,  float.pos = "h")	
	
#' 	
#' 	
#' \onecolumn	
#' 	
#' # Chapter 12 Question 2: 	
#' Continuing with the analysis of the CD4 data from Exercise 11.4:	
#' 	
#' ## Part A:	
#' ``Write a model predicting CD4 percentage as a function of time with varying intercepts across children. Fit using lmer() and interpret the coefficient for time.``	
#' 	
#' 	
model_chap12_q2a = lmer(cd4pct~1+time+(1|newpid),data=cd4.df,REML=F) # varying intercepts	
	
texreg(list(model_chap12_q2a), 	
       custom.model.names = c("Model 12.2 A"),	
       single.row=TRUE,  float.pos = "h")	
#summary(model_chap12_q2a)	
#coef(model_chap12_q2a)	
	
#' 	
#' 	
#' Given that this model is a random intercept model, it only accounts for "idiosyncratic" variation that is due to individual level differences in baseline cd4 percent. It also assumes that time effects on cd4 percent is fixed meaning that every child cd4 percent varies systematically with time. 	
#' 	
#' * **Time** -  Given that time is a fixed effect on every child, each year on the study is associated with a decreased in cd4 percentage by 3 % on average. In this model, time was statistically significant and had a substantial effect on cd4 percentage.	
#' 	
#' ## Part B:	
#' ``Extend the model in (a) to include child-level predictors (that is, group-level predictors) for treatment and age at baseline. Fit using lmer() and interpret the coefficients on time, treatment, and age at baseline.``	
#' 	
#' 	
#' 	
#' 	
#model_chap12_q2b=lmer(cd4pct~1+time+(1+treatmnt+baseage|newpid),data=cd4.df,REML=F) 	
model_chap12_q2b=lmer(cd4pct~1+time+treatmnt+baseage+(1|newpid),data=cd4.df,REML=F) 	
	
	
	
texreg(list(model_chap12_q2b), 	
       custom.model.names = c("Model 12.2 B"),	
       single.row=TRUE,  float.pos = "h")	
#summary(model_chap12_q2b)	
#coef(model_chap12_q2b)	
#toLatex(mtable(model_chap12_q2b))	
#' 	
#' 	
#' In general, adding child-level predictors (i.e, group-level predictors) definitely improves the model. In fact it increased the precision of our estimates. Note that the variance of our random effect estimate has reduced from 128 to 123 meaning some of the variations has been explained by addition of fixed effect predictors like base age and treatment. Conversely, residual has not reduced by much  possibly because we haven't added another random effect predictor that is due to child-specific idiosyncrasies.	
#' 	
#' ### Interpretation	
#' 	
#' * **Treatment** - Children who were on second treatment had 1.2125% higher cd4 percent on average  than children on first treatment. This predictor had substantial effect on cd4 percent however it was insignificant.	
#' 	
#' *  **Baseline Age** - 1 year difference in baseline age is associate with changing cd4 percentage by 0.9485 % on average. Meaning children who were enrolled into the study early had higher cd4 percentage. This predictor was highly significant.	
#' 	
#' * **Time** -  Each year on treatment was associated with a decreased in cd4 percentage by 2.9615 % on average. This predictor was highly significant.	
#' 	
#' ## Part C:	
#' ``Investigate the change in partial pooling from (a) to (b) both graphically and numerically.``	
#' 	
#' 	
#' 	
#' 	
#' ### Graphically	
#' 	
#' 	
	
# complete pooling	
complete_pooling.fit<-lm(cd4pct~time, data=cd4.df)	
	
df_complete_pooling <- data_frame(	
  model = "Complete pooling",	
  newpid = unique(cd4.df$newpid),	
  intercept = coef(complete_pooling.fit)[1], 	
  slope_time = coef(complete_pooling.fit)[2])	
	
	
	
# model for question 2 A	
df = coef(model_chap12_q2a)[["newpid"]]	
df$intercept = df$'(Intercept)'	
df_chap12_q2a <- df %>% 	
   dplyr::select(intercept, slope_time = time)  %>% 	
  as_tibble() %>% 	
   rownames_to_column("newpid") %>% 	
  add_column(model = "Model 12.2 A")	
	
	
# model for question 2 B	
df = coef(model_chap12_q2b)[["newpid"]]	
df$intercept = df$'(Intercept)'	
df_chap12_q2b <- df %>% 	
   dplyr::select(intercept, 	
         slope_time = time,	
         treatmnt_slope= treatmnt2, baseage_slope= baseage )  %>% 	
  as_tibble() %>% 	
   rownames_to_column("newpid") %>% 	
  add_column(model = "Model 12.2 B")	
	
	
# bind data	
df_models <- bind_rows(df_chap12_q2a, df_chap12_q2b, df_complete_pooling) %>% 	
  left_join(cd4.df, by = "newpid")	
	
# create subset	
subset <- c(67,72,77,78,79,11,12,19,27,23,62,63)	
	
# construct plot	
model_comparison <- ggplot(df_models[df_models$newpid %in% subset,]) + 	
  aes(x = time, y = cd4pct) + 	
  geom_abline(aes(intercept = intercept, slope = slope_time, color = model),	
              size = .75) + 	
   #theme_bw()+	
    #geom_line( linetype=6)+	
    geom_point()+	
  facet_wrap("newpid") +	
  labs(y = "CD4 Percentage", x="Time (visit_age - base_Age) ")+	
  scale_x_continuous(breaks = 0:4 * 2) + 	
  scale_color_brewer(palette = "Dark2") + 	
  theme(legend.position = "top")	
	
model_comparison	
	
#' 	
#' 	
#' 	
#' A few observation to make.	
#' 	
#' * From the plot, observe that the partial pooling model with added child-level predictors (Model 12.2B)  is pulled substantially towards the complete pooling model (group average) compared to  Model 12.2.A	
#' 	
#' * If you take a look at children with less observations (incomplete data), you will notice that Model 12.2.B is pulled more towards the complete pooling model (group average) compared to  Model 12.2.A. See child 67, 79, 77	
#' 	
#' 	
#' ### Numerically	
#' 	
	
anova_mixd <- anova(model_chap12_q2a,model_chap12_q2b)	
xtable(anova_mixd, comment=FALSE)	
#' 	
#' 	
#' Using ANOVA to check the changes between the 2 models, we find out that adding additional child-level predictors actually reduced the residual sum of squares . This reduction was  statistically significant. We also had lower AIC and BIC in the second model implying that the additional regressors actually improved the model	
#' 	
#' 	
#' 	
#' 	
#coef(model_chap12_q2a)	
q2a_plot <- as.data.frame(lme4::ranef(model_chap12_q2a,condVar=T)) %>%	
    ggplot(aes(y=grp,x=condval))+	
    geom_point()+facet_wrap(~term,scales="free_x")+	
    geom_errorbarh(aes(xmin=condval-2*condsd,xmax=condval+2*condsd),height=0)+	
   labs(x="Random Effects", y="Children") +	
      theme_classic()+ 	
      guides(fill=FALSE) +	
   ggtitle("Model 12.2 A")	
	
q2b_plot <-  as.data.frame(lme4::ranef(model_chap12_q2b,condVar=T)) %>%	
    ggplot(aes(y=grp,x=condval))+	
    geom_point()+facet_wrap(~term,scales="free_x")+	
    geom_errorbarh(aes(xmin=condval-2*condsd,xmax=condval+2*condsd),height=0)+	
    labs(x="Random Effects", y="Children") +	
      theme_classic()+ 	
      guides(fill=FALSE)+	
   ggtitle("Model 12.2 B")	
	
grid.arrange(q2a_plot,q2b_plot, ncol = 2.)	
	
	
#' 	
#' 	
#' From the above plot, notice the scale of each plot. Model 12.2.A  has higher random error than Model 12.2.B. This implies that adding child-level predictors reduces the variance of our random effect estimate. We noticed a substantial reduction in the variance of the intercept - 128 to 123. This is mainly because some of the variations had been explained by the addition of fixed effect predictors like base age and treatment.	
#' 	
#' ## Part D:	
#' Compare results in (b) to those obtained in part (c).	
#' 	
#' As seen in the previous question, We observed that by adding child-level predictors (i.e, group-level predictors) the model improved significantly hence increasing the precision of our estimates. The addition of the 2 fixed effect predictors like  base age and treatment made the variance of the random effect estimates to reduce by 5 units essentially by compromising between group level and individual level variation. 	
#' However, the residual did not reduced by much  possibly because we did not add another random effect predictor that is due to child-specific idiosyncrasies e.g ARVS.	
#' 	
#' 	
df_zoom <- df_models %>% 	
  filter(newpid %in% c("63", "67", "79", "77"))	
	
model_comparison  %+% df_zoom	
#' 	
#' 	
#' In summary, this shrinkage effect observed in model 12.2.B ensures that our model compromised between  individual-level effects and group level effects at the same regularizing extreme estimates and taming children with less  visits/observations as depicted by child 67,79 and 77.	
#' 	
#' \onecolumn	
#' 	
#' # Chapter 12 Question 3:	
#' 	
#' `Predictions for new observations and new groups:`	
#' 	
#' ## Part A:	
#' ``Use the model fit from Exercise 12.2(b) to generate simulation of predicted CD4 percentages for each child in the dataset at a hypothetical next time point.``	
#' 	
#' 	
	
	
set.seed(111)	
n.sims = 1000	
	
# create hypothetical next time point	
cd4.df3 = cd4.df %>% 	
            na.omit() %>% # remove missing values	
            group_by(newpid) %>% # group by PID	
            mutate(current.time = max(time)) %>% # get maximum time for each child	
            mutate(future.time = current.time + .25) # add .25 year beacause appintments are done quarterly	
	
# slelect only relevant time	
cd4.df4 = cd4.df3 %>%	
          dplyr::select(newpid,treatmnt,baseage,future.time) %>% 	
          rename(time=future.time) %>%	
          unique() # make sure it is a one row per patient dataset	
	
# Simulate 1000 times using sim fx	
model_chap12_q2b.sim = sim(model_chap12_q2b,n.sims)	
# fetch array offixed effect for intercept for each simulation	
fixed.intercept = coef(model_chap12_q2b.sim)$fixef[,1]	
# get random effect array for intercept for each simulation	
rand.eff = coef(model_chap12_q2b.sim)$ranef$newpid	
# get the intercept for each person by summing fixed and random effects	
sim.intercept = fixed.intercept + rand.eff	
# create matrix	
sim.intercept = matrix(sim.intercept,nrow=n.sims,ncol=length(unique(cd4.df$newpid)))	
	
# generate x matrix	
X = cbind(1,cd4.df4$time,as.numeric(cd4.df4$treatmnt)-1,cd4.df4$baseage)	
	
cd4.df.pred.future.sim = list()	
for (simnum in 1:n.sims){	
  b.hat = rbind(sim.intercept[simnum,],coef(model_chap12_q2b.sim)$fixef[simnum,2],coef(model_chap12_q2b.sim)$fixef[simnum,3], coef(model_chap12_q2b.sim)$fixef[simnum,4]) 	
  rownames(b.hat) = c("intercept","time","treatmnt","baseage")	
  	
  sigma.y.hat <- sigma.hat(model_chap12_q2b.sim)	
  	
  cd4.future.sim = rep(NA,nrow(cd4.df4))	
  # actual simulation 	
  for (idnum in 1:length(cd4.future.sim)){	
    cd4.future.sim[idnum] = rnorm (n.sims, X[idnum,] %*% b.hat[,idnum], sigma.y.hat)	
  }	
  	
  cd4.df.pred.future.sim[[simnum]] = cd4.df4 %>%	
                  cbind(cd4.future.sim) %>%	
                  rename(future.time = time)	
}	
	
  cd4.df.pred.future.sim.all = bind_rows(cd4.df.pred.future.sim) %>%	
                                      group_by(newpid) %>% 	
                                      mutate(mean.future.cd4 = mean(cd4.future.sim))	
  	
  cd4.df.pred.future.sim.all$index = as.numeric(cd4.df.pred.future.sim.all$newpid)	
  cd4.df.set1 = cd4.df.pred.future.sim.all%>%	
                filter(index <= 25) 	
	
  simulation_plot_12_3 <- ggplot(cd4.df.set1, aes(x=newpid, y =cd4.future.sim, color=factor(index))) +	
  geom_jitter(alpha = 0.1) +	
    #stat_boxplot(geom="errorbar", width=.5,  color = "black")+	
  geom_boxplot(alpha = 0, color = "black") +	
  labs(y = "Predicted CD4 %", x="Children")+	
  theme_classic()+ guides(fill=FALSE)+	
  theme(legend.position="none") 	
	
  simulation_plot_12_3	
	
  plot1.df = cd4.df.pred.future.sim.all %>% dplyr::filter(index > 25 & index <=52)	
  plot2.df = cd4.df.pred.future.sim.all %>% dplyr::filter(index > 52 & index <=78)	
  plot3.df =  cd4.df.pred.future.sim.all %>% dplyr::filter(index > 78 & index <=104)	
  plot4.df =  cd4.df.pred.future.sim.all %>% dplyr::filter(index > 104 & index <=129) 	
  plot5.df =  cd4.df.pred.future.sim.all %>% dplyr::filter(index > 129 & index <=154) 	
  plot6.df =  cd4.df.pred.future.sim.all %>% dplyr::filter(index > 154 & index <=179)	
  plot7.df =  cd4.df.pred.future.sim.all %>% dplyr::filter(index > 179 & index <=204)	
  plot8.df =  cd4.df.pred.future.sim.all %>% dplyr::filter(index > 204)	
  	
  	
	
	
#' 	
#' 	
#' Since most appointments/visits occurred every quarterly as shown in Chapter 11 Question 4, we made our  hypothetical next time point as the sum of maximum visit time for each child + 0.25 year. We then generated 1000 predictive simulation for each child. Each box plot represent each child and each  geometric point represent each predicted simulated cd4% (1000). Notice the wide variation around the mean. This is because We did not use the classical predict function, instead we simulated from the distribution of regression coefficients. This in turn ensured we captured the uncertainity in regression coefficients that is necessary for replicating new data.	
#' 	
#' In general most children with multiple visits had much more precise prediction interval while children with less observation had less precise prediction interval. Additionally children who had visits/observations greater 1.75 years had less precise prediction interval compared to children who had observations less than 1.75. This 2 aspects are also related to the shrinkage effects discussed in the previous question.	
#' 	
#' 	
#' 	
#' 	
#' 	
	
grid.arrange( simulation_plot_12_3  %+% plot1.df,	
                simulation_plot_12_3  %+% plot2.df,	
               	
               nrow =2., ncol = 1.)	
  	
   	
#' 	
#' 	
#' 	
	
 grid.arrange(	
                simulation_plot_12_3  %+% plot3.df,	
                simulation_plot_12_3  %+% plot4.df,	
               nrow =2., ncol = 1.)	
  	
	
#' 	
#' 	
#' 	
	
    grid.arrange( simulation_plot_12_3  %+% plot5.df,	
                simulation_plot_12_3  %+% plot6.df,	
               nrow =2., ncol = 1.)	
#' 	
#' 	
#' 	
	
    grid.arrange( 	
                simulation_plot_12_3  %+% plot7.df,	
                simulation_plot_12_3  %+% plot8.df,	
               nrow =2., ncol = 1.)	
#' 	
#' 	
#' 	
#' ## Part B:	
#' ``Use the same model fit to generate simulations of CD4 percentages at each of the time periods for a new child who was 4 years old at baseline.``	
#' 	
#' 	
	
n.sims = 100	
	
cd4.df4 = cd4.df %>%	
  dplyr::select(newpid,treatmnt,baseage,time)	
	
# Simulates the fit model 1000 times	
model_chap12_q2b.sim = sim(model_chap12_q2b,n.sims)	
# Extracts fixed effect for intercept for each simulation	
fixed.intercept = coef(model_chap12_q2b.sim)$fixef[,1]	
# Extracts random effect for intercept for each person for each simulation	
rand.eff = coef(model_chap12_q2b.sim)$ranef$newpid	
# Computes intercept for each person for each simulation	
sim.intercept = fixed.intercept + rand.eff	
sim.intercept = matrix(sim.intercept,nrow=n.sims,ncol=length(unique(cd4.df4$newpid)))	
	
X = cbind(1,cd4.df4) %>% dplyr::select(1,time,treatmnt,baseage,newpid) %>% dplyr::mutate(treatmnt =as.numeric(treatmnt)-1)	
	
cd4.df.4y.pred.sim = list()	
	
for (simnum in 1:n.sims){	
  b.hat = rbind(sim.intercept[simnum,],coef(model_chap12_q2b.sim)$fixef[simnum,2],coef(model_chap12_q2b.sim)$fixef[simnum,3], coef(model_chap12_q2b.sim)$fixef[simnum,4]) 	
  rownames(b.hat) = c("intercept","time","treatmnt","baseage")	
  colnames(b.hat) = unique(X$newpid)	
  	
  cd4.4y.pred.sim = rep(NA,nrow(cd4.df4))	
  for (i in 1:length(cd4.4y.pred.sim)){	
    cd4.4y.pred.sim[i] = as.vector(t(X[i,1:4])) %*% as.vector(t(b.hat[,as.character(X[i,"newpid"])]))	
  }	
  	
  cd4.df.4y.pred.sim[[simnum]] = cd4.df4 %>% cbind(cd4.4y.pred.sim)	
}	
	
cd4.df.4y.pred.sim.all = bind_rows(cd4.df.4y.pred.sim) %>% 	
                        arrange(newpid) %>% 	
                        group_by(newpid) %>%	
                        mutate(mean.4yo.cd4 = mean(cd4.4y.pred.sim)) %>%	
                        filter(baseage>3.75 & baseage < 4.25)	
	
ggplot(cd4.df.4y.pred.sim.all,	
       aes(x=time,y=cd4.4y.pred.sim)) +	
      geom_jitter(alpha = 0.3, aes(color=newpid)) +	
    geom_smooth(se=T, colour="black", method = "lm", linetype=6) +	
    #stat_boxplot(geom="errorbar", width=.5,  color = "black")+	
  #geom_boxplot(alpha = 0, color = "black") +	
  labs(y = "Predicted CD4 %", x="Time")+	
  theme_classic()+ guides(fill=FALSE)	
 # theme(legend.position="none") 	
	
	
#' 	
#' 	
#'  We identified  16 children who had a baseline age of 4  years or  (+-.25). From the plot it is clear that children who get into the study with a baseline age of 4 years have higher cd4% decline rate. Most of their cd4 % declined by an average of 10 units	
#'  	
#'  	
#' 	
cd4.df.4y.pred.plot<-ggplot(cd4.df.4y.pred.sim.all,	
       aes(x=time,y=cd4.4y.pred.sim,color=factor(newpid))) +	
      geom_jitter(alpha = 0.3) +	
        geom_smooth(se=T, colour="black", method = "lm", linetype=6) +	
        #stat_boxplot(geom="errorbar", width=.5,  color = "black")+	
    	
      labs(y = "Predicted CD4 %", x="Time")+	
      facet_wrap(~newpid)+	
      theme_classic()+ guides(fill=FALSE)+	
      theme(legend.position="none") 	
	
 plot1.df = cd4.df.4y.pred.sim.all[cd4.df.4y.pred.sim.all$newpid %in% c(1,103,234,130),]	
cd4.df.4y.pred.plot  %+% plot1.df	
	
#' 	
#' 	
#' \onecolumn	
#' 	
#' # Chapter 12 Question 4:	
#' ``Posterior predictive checking: continuing the previous exercise, use the fitted model from Exercise 12.2(b) to simulate a new dataset of CD4 percentages (with the same sample size and ages of the original dataset) for the final time point of the study, and record the average CD4 percentage in this sample. Repeat this process 1000 times and compare the simulated distribution to the observed CD4 percentage at the final time point for the actual data.``	
#' 	
#' 	
#' 	
n.sims = 1000	
# get uniqui child id	
child_id = cd4.df%>%	
  na.omit()%>%	
  distinct(newpid, treatmnt, baseage)	
	
# count num of id	
num_id = dim(child_id)[1]	
	
final_time_point.df = cd4.df%>%	
  filter(visitno == 19)%>%	
  dplyr::select(time, cd4pct)	
	
final_time = mean(final_time_point.df$time)	
y = array(NA, c(n.sims, num_id))	
average = array(NA, n.sims)	
	
for(s in 1:num_id){	
  sigma.y.hat = sigma.hat(model_chap12_q2b)$sigma$data	
  coef.hat = as.matrix(coef(model_chap12_q2b)$newpid)[s,]	
  variable = child_id[s,]	
  y[,s] = rnorm (n.sims, coef.hat %*% c(1, final_time, variable$treatmnt, variable$baseage), sigma.y.hat)	
}	
	
predicted = apply(y, 1, mean)	
actual = cd4.df%>%	
   na.omit()%>%	
  filter(visitno == 19)%>%	
  dplyr::select(cd4pct)	
	
par(mfrow = c(1, 2)) 	
hist(predicted, main = "simulated distribution", col="blue",  xlab="cd4%")	
abline(v=mean(predicted),col="red")	
hist(actual$cd4pct, main = "observed distribution", col="blue", xlab="cd4%")	
	
	
#' 	
#' 	
#' 	
#' 	
#' A few observation to make. In general we notice the shrinkage effect depicted by the model when we try to extrapolate. The  simulated distribution of the predicted  cd4 % ranges from 21 to 24 with a mean of about 22.5. On the other hand the observed distribution ranges from 0 to 60. Comparing to the observed distribution, simulated prediction have been pulled substantially towards (group average). It is clear that the model can be  terrible at extrapolating beyond the 3rd IQR (the final time point).	
#' 	
#' In summary, this shrinkage effect observed in model 12.2.B ensures that our model compromised between  individual-level effects and group level effects at the same regularizing extreme estimates and taming children with less  visits/observations 	
#' 	
#' 	
#' \onecolumn	
#' 	
#' # Source Code	
#' 	
#' 	
	
	
#' 	


```