# R-program for Supplementary Exercise 8.62 of IPS7e

succ <- c(63,43)
patients <- c(78,77)

# (b): confidence intervals
# note: built-in function prop.test uses different method
prop.test(succ, patients, conf.level=0.90, correct=F)

# function to compute classical CI for difference between two proportions
zprop2.ci = function(x, n, conf.level=0.95){
   p <- x/n
   lower <- p[1]-p[2] - qnorm(1-(1-conf.level)/2)*sqrt(sum(p*(1-p)/n))
   upper <- p[1]-p[2] + qnorm(1-(1-conf.level)/2)*sqrt(sum(p*(1-p)/n))
   return(as.vector(c(lower,upper)))
}
# 90% classical CI
zprop2.ci(succ, patients, conf.level=0.90)

# (c): test
# note: built-in function prop.test computes chi-square test (=z^2)

zprop2.test = function(x, n, pdiff=0, alternative = c("two.sided")){
   p <- x/n
   pc <- sum(x)/sum(n)
   z <- (p[1]-p[2]-pdiff)/sqrt(pc*(1-pc)*sum(1/n))
   if (alternative == "two.sided")
     pval <- 2*pnorm(-abs(z))
   if (alternative == "less")
     pval <- pnorm(z)
   if (alternative == "greater")
     pval <- pnorm(z, lower.tail=F)
   return(as.vector(c(z,pval)))
}
# z-test based on normal approximation
zprop2.test(succ, patients)