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. 2026 Mar;45(6-7):e70479.
doi: 10.1002/sim.70479.

Pairwise Comparisons of k $$ k $$ Binomial Responses

Dennis D Boos  1 , James Schmidt  1
Affiliations

Pairwise Comparisons of k $$ k $$ Binomial Responses

Dennis D Boos et al. Stat Med. 2026 Mar.

Abstract

Independent binomially distributed data arise in many contexts, such as clinical trials, quality control monitoring, and stratified sampling. Moreover, the scope is much larger because multinomially distributed data from a 2 by k $$ k $$ contingency table can be viewed as k $$ k $$ conditionally independent binomial random variables. A standard approach is to use Fisher's conditional test to test equality of the k $$ k $$ underlying success probabilities. However, researchers often want to know where the important pairwise differences are. Thus, the closed method of pairwise comparisons is here combined with unconditional exact tests for 2 by 2 tables and Fisher's conditional test for larger tables to get p $$ p $$ values exhibiting strong control of the Family-Wise Error Rate and excellent power properties. In clinical trials, studies are often multicenter, but the results here pertain only to single-site studies.

Keywords: closed method; family‐wise error rate; multiple comparisons.

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