# Reference Chart for Precision of Wilson Binomial Proportion Confidence Interval

**BioStatMatt » R**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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I am often asked about the number of subjects needed to study a binary outcome, which usually leads to a discussion of confidence intervals for binary proportions, and the associated precision. Sometimes the precision is quantified as the width or half-width of a 95% confidence interval. For proportions, I like the Wilson score interval because it’s simple to calculate and does not violate the constraints of probability estimates (i.e., estimates must be between 0 and 1). Below is a function that computes the Wilson interval, given the number of trials (`n`

) and the fraction of events (`p`

).

## Level (1-a) Wilson confidence interval for proportion ## WILSON, E. B. 1927. Probable inference, the law of succession, ## and statistical inference. Journal of the American Statistical ## Association 22: 209-212. WilsonBinCI <- function(n, p, a=0.05) { z <- qnorm(1-a/2,lower.tail=FALSE) l <- 1/(1+1/n*z^2)*(p + 1/2/n*z^2 + z*sqrt(1/n*p*(1-p) + 1/4/n^2*z^2)) u <- 1/(1+1/n*z^2)*(p + 1/2/n*z^2 - z*sqrt(1/n*p*(1-p) + 1/4/n^2*z^2)) list(lower=l, upper=u) }

The code below generates a figure that illustrates the 95% confidence bounds as a function of the probability estimate, and for a sequence of trial sizes. I’m posting this here for my future easy access, but I hope some readers will find it useful as well, or might suggest improvements.

pseq <- seq(0, 1, length.out=200) nseq <- c(10,20,50,200,1000) gcol <- gray.colors(length(nseq), start = 0.3, end = 0.7) par(mar=c(5,5,4,2)+0.1) plot(pseq, pseq, type="n", ylim=c(-0.3,0.3), main="Wilson Binomial 95% CI", xlab="Probability Estimate (P)", ylab="", yaxt="n") pbnd <- -3:3/10 axis(2, at=pbnd, labels= ifelse(pbnd<0, paste0("P",pbnd), ifelse(pbnd==0, rep("P",length(pbnd)), paste0("P+",pbnd))),las=2) mtext("95% CI", 2, line = 4) legend("topright", paste("N =", nseq), fill=gcol, border=NA, bty="n", cex=0.8) for(i in 1:length(nseq)) { bnds <- t(sapply(pseq, function(p) unlist(WilsonBinCI(nseq[i], p)))) bnds <- bnds - pseq polygon(x=c(pseq,rev(pseq)), y=c(bnds[,2],rev(bnds[,1])), border=NA, col=gcol[i]) pmin <- pseq[which.min(bnds[,1])] bmin <- min(bnds[,1]) lines(x=c(-1,rep(pmin,2)), y=c(rep(bmin,2),-1), lty=3) } abline(h=0, lty=3)

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