The cost of using exact confidence intervals for a binomial proportion

TL;DR

This paper examines the trade-offs of using exact versus approximate confidence intervals for binomial proportions, focusing on increased length and sample size, and explores their asymptotic properties and Bayesian connections.

Contribution

It provides a detailed analysis of the costs associated with exact Clopper--Pearson intervals, including a closed-form sample size formula and insights into their relation with Bayesian methods.

Findings

Exact intervals have longer expected length than approximate ones.

Using exact intervals increases required sample size for desired precision.

Asymptotic formulas help determine sample size for exact methods.

Abstract

When computing a confidence interval for a binomial proportion p one must choose between using an exact interval, which has a coverage probability of at least 1-{\alpha} for all values of p, and a shorter approximate interval, which may have lower coverage for some p but that on average has coverage equal to 1-\alpha. We investigate the cost of using the exact one and two-sided Clopper--Pearson confidence intervals rather than shorter approximate intervals, first in terms of increased expected length and then in terms of the increase in sample size required to obtain a desired expected length. Using asymptotic expansions, we also give a closed-form formula for determining the sample size for the exact Clopper--Pearson methods. For two-sided intervals, our investigation reveals an interesting connection between the frequentist Clopper--Pearson interval and Bayesian intervals based on noninformative priors.