Small-sample comparisons of confidence intervals for the difference of two independent binomial proportions

This paper compares the exact small-sample achieved coverage and expected lengths of five methods for computing the confidence interval of the difference of two independent binomial proportions. We strongly recommend that one of these be used in practice. The first method we compare is an asymptotic method based on the score statistic (AS) as proposed by Miettinen and Nurminen [1985. Comparative analysis of two rates. Statist. Med. 4, 213–226.]. Newcombe [1998. Interval estimation for the difference between independent proportions: comparison of seven methods. Statist. Med. 17, 873–890.] has shown that under a certain asymptotic set-up, confidence intervals formed from the score statistic perform better than those formed from the Wald statistic (see also [Farrington, C.P., Manning, G., 1990. Test statistics and sample size formulae for comparative binomial trials with null hypothesis of non-zero risk difference or non-unity relative risk. Statist. Med. 9, 1447–1454.]). The remaining four methods compared are the exact methods of Agresti and Min (AM), Chan and Zhang (CZ), Coe and Tamhane (CT), and Santner and Yamagami (SY). We find that the CT has the best small-sample performance, followed by AM and CZ. Although AS is claimed to perform reasonably well, it performs the worst in this study; about 50% of the time it fails to achieve nominal coverage even with moderately large sample sizes from each binomial treatment.