Bayesian Decision Theory for Multiple Comparisons

Applying a decision theoretic approach to multiple comparisons very similar to that described by Lehmann [Ann. Math. Statist. 21 (1950) 126; Ann. Math. Statist. 28 (1975a) 1-25; Ann. Math. Statist. 28 (1975b) 5475721, we introduce a loss function based on the concept of the false discovery rate (FDR). We derive a Bayes rule for this loss function and show that it is very closely related to a Bayesian version of the original multiple comparisons procedure proposed by Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300] to control the sampling theory FDR. We provide the results of a Monte Carlo simulation that illustrates the very similar sampling behavior of our Bayes rule and Benjamini and Hochberg's procedure when applied to making all pair-wise comparisons in a one-way fixed effects analysis of variance setup with 10 and with 20 means.