A Generalized Benjamini-Hochberg Procedure for Multivariate Hypothesis Testing

The introduction of the false discovery rate (FDR) by Benjamini and Hochberg has spurred a great interest in developing methodologies to control the FDR in various settings. The majority of existing approaches, however, address the FDR control for the case where an appropriate univariate test statistic is available. Modern hypothesis testing and data integration applications, on the other hand, routinely involve multivariate test statistics. The goal, in such settings, is to combine the evidence for each hypothesis and achieve greater power, while controlling the number of false discoveries. This paper considers data-adaptive methods for constructing nested rejection regions based on multivariate test statistics (z-values). It is proved that the FDR can be controlled for appropriately constructed rejection regions, even when the regions depend on data and are hence random. This flexibility is then exploited to develop optimal multiple comparison procedures in higher dimensions, where the distribution of non-null z-values is unknown. Results are illustrated using simulated and real data.