A Comprehensive Comparison of the Wald, Wilson, and adjusted Wilson Confidence Intervals for Proportions

TL;DR

This paper provides a comprehensive comparison of Wald, Wilson, and adjusted Wilson confidence intervals for proportions, demonstrating that adjusted Wilson intervals with added pseudo-observations outperform traditional methods across various sample sizes and confidence levels.

Contribution

It introduces novel visualization techniques to compare confidence intervals and identifies optimal pseudo-observations for improved coverage performance.

Findings

Adjusted Wilson intervals with 3-6 pseudo-observations outperform Wald and Wilson intervals.

Color-coded plots effectively illustrate coverage probabilities across parameters.

Optimal pseudo-observations depend on the desired confidence level.

Abstract

The standard confidence interval for a population proportion covered in the overwhelming majority of introductory and intermediate statistics textbooks surprisingly remains the Wald confidence interval despite having a poor coverage probability, especially for small sample sizes or when the unknown population proportion is close to either 0 or 1. Using the mean coverage probability, and for some sample sizes, Agresti and Coull showed not only that the 95\% Wilson confidence interval performs better, but also showed that 95\% adjusted Wilson of type 4 confidence interval, obtained by simply adding four pseudo-observations, outperforms both the Wald and the Wilson confidence intervals. In this paper, we introduce a rainbow color code and pixel-color plots as ways to comprehensively compare the Wald, Wilson, and adjusted-Wilson of type $\epsilon$ confidence intervals across all sample sizes $n=1, 2, \dots, 1000$, population proportion values $p=0.01, 0.02, \dots, 0.99$, and for the three typical confidence levels. We show not only that adding 3 (resp., 4 and 6) pseudo-observations is the best for the 90\% (resp., 95\% and 99\%) adjusted Wilson confidence interval, but it also performs better than both the 90\% (resp., 95\% and 99\%) Wald and Wilson confidence intervals.