We develop a new method for large-scale frequentist multiple testing with Bayesian prior information. We find optimal @math -value weights that maximize the average power of the weighted Bonferroni method. Due to the nonconvexity of the optimization problem, previous methods that account for uncertain prior information are suitable for only a small number of tests. For a Gaussian prior on the effect sizes, we give an efficient algorithm that is guaranteed to find the optimal weights nearly exactly. Our method can discover new loci in genome-wide association studies and compares favourably to competitors. An open-source implementation is available.
We maximize power in a replicated clinical trial involving multiple endpoints by adjusting the individual significance levels for each hypothesis, using preliminary data to obtain the optimal adjustments. The levels are constrained to control the familywise error rate. Power is defined as the expected number of significances, where expectations are taken with respect to the posterior distributions of the non-centrality parameters under non-informative priors. Sample size requirements for the replicate study are given. Intuitive principles such as downweighting insignificant variables from a preliminary study and giving primary endpoints more emphasis are justifiable within the conceptual framework. © 1998 John Wiley & Sons, Ltd.