# Binomial Confidence Intervals

This stems from a couple of binomial distribution projects I have been working on recently.  It’s widely known that there are many different flavors of confidence intervals for the binomial distribution.  The reason for this is that there is a coverage problem with these intervals (see Coverage Probability).  A 95% confidence interval isn’t always (actually rarely) 95%.  So I got curious what would happen if I generated random binomial data to find out what percent of the simulated data actually fell within the confidence interval.  For simplicity I used $p=0.5$.  I wrote up some quick code that generates binomial data for varying sample sizes and runs a comparison for each of the nine confidence interval methods using the binom.confint() function.

library(binom)
set.seed(0)
nsims <- 10000 maxn <- 500 n <- seq(2,maxn, by=2) my.method <- c("exact", "ac", "asymptotic", "wilson", "prop.test", "bayes", "logit", "cloglog", "probit") my.method <- my.method[sort.list(my.method)] coverage <- matrix(NA, nrow=length(n), ncol=length(my.method)) ci.lower <- ci.upper <- matrix(NA, ncol=length(my.method), nrow=length(n)) for(i in 1:length(n)){ m <- n[i]/2 y <- rbinom(nsims, n[i], m/n[i]) ll <- binom.confint(m,n[i], conf.level=.95, method=my.method)\$lower ul <- binom.confint(m,n[i], conf.level=.95, method=my.method)\$upper ci.lower[i,] <- ll ci.upper[i,] <- ul for(j in 1:length(my.method)){ sig <- length(y[y/n[i]<=ul[j] &amp;amp;amp;amp; y/n[i]>=ll[j]])
coverage[i,j] <- sig/nsims } } plot(n,NULL, xlim=c(1,nrow(coverage)+1), ylim=c(.83,1), col=1, pch=16, ylab="Percent", xlab="N", main="95% Confidence Intervals for p=.5") points(replicate(ncol(coverage),n),coverage, col=c(1:9), pch=16, cex=.5) abline(h=seq(.93,.97, by=.01), col="#EEEEEE") abline(h=.95, col="#000000", lwd=2) abline(v=seq(2,maxn, by=20), col="#EEEEEE") legend("bottomright", my.method, col=c(1:9), pch=16, title="Interval Types", bg="#FFFFFF") plot(n,NULL, xlim=c(1,100), ylim=c(0,1), col=1, pch=16, ylab="Percent", xlab="N", main="95% Confidence Interval for p=.5") for(k in 1:ncol(coverage)){ lines(n, ci.lower[,k], col=k, lwd=1) lines(n, ci.upper[,k], col=k, lwd=1) } legend("bottomright", my.method, col=c(1:9), ncol=2, lwd=1, title="Interval Types", bg="#FFFFFF") [/sourcecode]

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## One reply on “Binomial Confidence Intervals”

1. Thanks for this great post! This prompted me to play around with the binom package. I knew the methods differed but I was surprised by how much! Time to read more journal articles.