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zwilson <- function(dat, conf.level){
# Wilson, E.B. (1927) Probable inference, the law of succession, and statistical inference J. Amer. Stat. Assoc 22, 209-212. Changed 28 Oct 2021
N. <- 1 - ((1 - conf.level) / 2)
z <- qnorm(N., mean = 0, sd = 1)
a <- dat[,1]
n <- dat[,2]
p <- a / n
limits <- (z * ((p * (1 - p) + (z^2) / (4 * n)) / n)^(1/2)) / (1 + (z^2) / n)
pest <- (p + (z^2) / (2 * n)) / (1 + (z^2) / n)
upp <- pest + limits
low <- pest - limits
d. <- (a * (n - a)) / n^3
e. <- z^2 / (4 * n^2)
var.wil <- sqrt(d. + e.)
# Design effect equals [var.obs] / [var.srs].
# var.wil has been computed assuming simple random sampling so if an argument for design effect is provided adjust se.wil accordingly. Assume design = 1:
design <- 1
se.wil <- sqrt(design * var.wil)
rval <- data.frame(est = p, se = se.wil, lower = low, upper = upp)
rval
}
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