Beware of Boundaries in Binominal Proportion Confidence Intervals


Dean Marchiori


March 13, 2023

Binomial proportion confidence intervals are often employed when attempting to perform tests for significance, or sample size calculations around sample measurements resulting from a Bernoulli process.

The typical choice when calculating binomial proportion confidence intervals is the asymptotic, or normally approximated ‘Wald’ interval where success probability is measured by:

\[ \hat{p} \pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]

In many settings, such as marketing analytics or manufacturing processes the sample proportion is close to 0 or 1. Evaluating asymptotic confidence intervals near these boundary conditions will lead to underestimation of the error, and in some cases producing an interval outside \([0, 1]\).

Fortunately other methods exist, such as Wilson’s score interval, exact methods and Bayesian approaches. The recommendation here is to examine the probability coverage and explore alternative methods for sample size and CI calculation, especially when the parameter is near the boundary conditions, or in cases of very small n. 


n <- 50
p <- c(0.01, 0.5, 0.99)
x <- purrr::map_df(p, .f =  ~binom.confint(x = n * .x, n = n, methods = 'all'))
ggplot(x, aes(colour = factor(x))) +
  geom_point(aes(mean, method), show.legend = F) +
  geom_errorbarh(aes(xmin = lower, xmax = upper, y = method), show.legend = F) +
  geom_vline(xintercept =  c(0, 1), lty = 2, col = "grey") +
  facet_wrap(~(x*2/100)) +
  theme_bw() +
  labs(title = "A variety of binomial confidence interval methods for p = 0.01, 0.5 & 0.99",
       subtitle = "Note unusual behaviour near 0.01 and 0.99")

cov <- purrr::map_df(p, ~binom.coverage(.x, n, conf.level = 0.95, method = "all"))
ggplot(cov, aes(colour = factor(p))) +
  geom_point(aes(coverage, method), show.legend = F) +
  geom_vline(xintercept =  0.95, lty = 2) +
  facet_wrap(~(p)) +
  theme_bw() +
  labs(title = "Probability coverage for a variety of binomial confidence interval methods",
       subtitle = "Reference line at 0.95 coverage")

A good discussion is contained in:

Wallis, Sean A. (2013). “Binomial confidence intervals and contingency tests: mathematical fundamentals and the evaluation of alternative methods” (PDF). Journal of Quantitative Linguistics. 20 (3): 178–208. doi:10.1080/09296174.2013.799918. S2CID 16741749.