Confidence interval from binomial distribution: different results using `qbinom` and `binom.test`

Question

I expect identical results calculating a CI for a proportion using qbinom and binom.test, but they are actually slightly different:

success <- 360
n <- 1226
lci <- qbinom(0.025, n, success/n)/n
uci <- qbinom(0.975, n, success/n)/n
c(lci, uci)

[1] 0.2683524 0.3189233

binom.test(success, n, success/n)$conf.int

[1] 0.2682571 0.3200123

What am I missing here?

Answer 1

The source code for the binom.test function uses qbeta rather than qbinom, as this is the accepted formula for exact binomial confidence intervals. Where x is the number of successes, the lci and uci as given by binom.test are:

p.L <- function(x, alpha) {
    if (x == 0) 
        0
    else qbeta(alpha, x, n - x + 1)
}
p.U <- function(x, alpha) {
    if (x == n) 
        1
    else qbeta(1 - alpha, x + 1, n - x)
}

alpha <- (1 - 0.95)/2
lci <- p.L(x, alpha)
uci <- p.U(x, alpha)