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Biometrika 1997 84(2):419-427; doi:10.1093/biomet/84.2.419
© 1997 by Biometrika Trust
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A Bayesian perspective on the Bonferroni adjustment

PETER H. WESTFALL, WESLEY O. JOHNSON and JESSICA M. UTTS

Department of Information Systems and Quantitative Sciences, Texas Tech University Lubbock, Texas 79409, U.S.A. e-mail: westfall{at}ttu.edu
Division of Statistics, University of California at Davis Davis, California 95616, U.S.A. e-mail: wojohnson{at}ucdavis.edu jmutts{at}ucdavis.edu

Bayes/frequentist correspondences between the p-value and the posterior probability of the null hypothesis have been studied in univariate hypothesis testing situations. This paper extends these comparisons to multiple testing and in particular to the Bonferroni multiple testing method, in which p-values are adjusted by multiplying by k, the number of tests considered. In the Bayesian setting, prior assessments may need to be adjusted to account for multiple hypotheses, resulting in corresponding adjustments to the posterior probabilities. Conditions are given for which the adjusted posterior probabilities roughly correspond to Bonferroni adjusted p-values.

Key Words: Adjusted p-value • Bayes factor • Multiple testing • Multiplicity adjustment • Simultaneous inference


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