A multivariate version of the Benjamini–Hochberg method

We propose a multivariate method for combining results from independent studies about the same ‘large scale’ multiple testing problem. The method works asymptotically in the number of hypotheses and consists of applying the Benjamini–Hochberg procedure to the pp-values of each study separately by determining the ‘individual false discovery rates’ which maximize power subject to a restriction on the (global) false discovery rate. We show how to obtain solutions to the associated optimization problem, provide both theoretical and numerical examples, and compare the method with univariate ones.