False discovery rate control with multivariate $p$-values

TL;DR

This paper develops methods for controlling false discovery rate and positive FDR using multivariate p-values, providing explicit procedures with optimal power under known distributions and asymptotic guarantees when distributions are partially known.

Contribution

It introduces a novel approach using nested regions of multivariate p-values for FDR control, including explicit procedures for known distributions and asymptotic methods for partial knowledge.

Findings

Explicit FDR control with maximum power when distributions are known.

Asymptotic FDR control with partial distribution knowledge.

Comparison showing advantages over simpler or more straightforward methods.

Abstract

Multivariate statistics are often available as well as necessary in hypothesis tests. We study how to use such statistics to control not only false discovery rate (FDR) but also positive FDR (pFDR) with good power. We show that FDR can be controlled through nested regions of multivariate $p$-values of test statistics. If the distributions of the test statistics are known, then the regions can be constructed explicitly to achieve FDR control with maximum power among procedures satisfying certain conditions. On the other hand, our focus is where the distributions are only partially known. Under certain conditions, a type of nested regions are proposed and shown to attain (p)FDR control with asymptotically maximum power as the pFDR control level approaches its attainable limit. The procedure based on the nested regions is compared with those based on other nested regions that are easier to construct as well as those based on more straightforward combinations of the test statistics.