Statistics and Its Interface

Volume 16 (2023)

Number 4

A default Bayesian multiple comparison of two binomial proportions

Pages: 517 – 529



Emrah Gecili (Cincinnati Children’s Hospital Medical Center, Cincinnati, Ohio, U.S.A.)

Siva Sivaganesan (University of Cincinnati, Cincinnati, Ohio, U.S.A.)


We consider a default Bayesian approach to multiple testing of equality of two binomial proportions. While our approach is motivated by a scenario where one proportion corresponds to an experimental condition and the other to a control, we find it is also reasonable for comparing two proportions in general. We consider a selection of priors under the alternative(s) including the intrinsic prior and a newly proposed “mode-based” Beta prior, and investigate their properties in terms of certain desirable characteristics that we specify for default priors. We also develop priors for the hyperparameters based on the conventional hyperprior used for normal means multiple testing. We also consider a computationally more efficient empirical Bayes approach using the intrinsic prior and the proposed Beta prior. We use repeated simulation and real data sets to evaluate and illustrate the approach, and compare certain frequentist characteristics of the results based on intrinsic and mode based Beta prior using full Bayes and empirical Bayes approaches. Additionally, the results from the Bayesian approach are compared with a commonly used frequentist procedure using conventional thresholds in the respective settings. Overall, we find that the proposed mode-based Beta prior is a suitable default prior for multiple testing of equality of two proportions.


binomial proportions, multiple comparison, objective Bayesian, prior distribution

2010 Mathematics Subject Classification

62C10, 62F15

Received 1 February 2022

Accepted 7 May 2022

Published 14 April 2023