### Bootstrap: Relative Risk
###
### Consider a prospective cohort study to investigate the effect of apsirin on heart attacks.
### A group of patients who are at risk of heart attack are randomly assigned to take either 
### a placebo or an aspirin.  At the end of one year the number of patients suffering a heart
### attack is recorded.  The results were

### Aspirin: Heart Attack	104 	No 10933	No. Subjects 11037
### Placebo: Heart Attack	189		No 10845	No. Subjects 11034

### Get a 95% CI for the Relative Risk = p1/p2 of aspirin to placebo using the traditional 
### Mantel-Haenszel test.  Using SAS run Proc Freq.

### Use the Bootstrap to plot the Bootstrap distribution of the RR and give a 95% Bootstrap
### Empirical Confidence Interval for RR.

# create the data 

asp.ht <- 104
n1 <- 11037

x.aspirin <- numeric(n1)
x.aspirin[1:asp.ht] <- 1

sum(x.aspirin)
mean(x.aspirin)

pla.ht <- 189
n2 <- 11034

x.placebo <- numeric(n2)
x.placebo[1:pla.ht] <- 1

sum(x.placebo)
mean(x.placebo)

mean(x.aspirin)/mean(x.placebo)	# RR

### Implement the bootstrap directly.

B <- 1000

theta <- numeric(B)

for(i in 1:B){
	x.aspirin.sample <- sample(x.aspirin, size = n1, replace = T)
	x.placebo.sample <- sample(x.placebo, size = n2, replace = T)
	theta[i] <- mean(x.aspirin.sample)/mean(x.placebo.sample)
}

hist(theta)		# histogram of the bootstrap samples

plot(density(theta),type="l")  	# density plot

quantile(theta, c(0.025,0.975))	# empirical bootstrap CI.