Controlling false discovery exceedance for heterogeneous tests

TL;DR

This paper introduces three new procedures for controlling the false discovery exceedance (FDX) in large-scale multiple testing with heterogeneous data, improving upon classical methods designed for homogeneous settings.

Contribution

The authors develop and compare three novel FDX control procedures that explicitly incorporate heterogeneity in null distributions, enhancing accuracy over traditional methods.

Findings

The [HLR] procedure uses the arithmetic average of null distribution functions.

The [HGR] procedure employs the geometric average of null distribution functions.

The [PB] procedure, based on Poisson-binomial distribution, outperforms others but is computationally intensive.

Abstract

Several classical methods exist for controlling the false discovery exceedance (FDX) for large scale multiple testing problems, among them the Lehmann-Romano procedure ([LR] below) and the Guo-Romano procedure ([GR] below). While these two procedures are the most prominent, they were originally designed for homogeneous test statistics, that is, when the null distribution functions of the $p$-values $F_i$, $1\leq i\leq m$, are all equal. In many applications, however, the data are heterogeneous which leads to heterogeneous null distribution functions. Ignoring this heterogeneity usually induces a conservativeness for the aforementioned procedures. In this paper, we develop three new procedures that incorporate the $F_i$'s, while ensuring the FDX control. The heterogeneous version of [LR], denoted [HLR], is based on the arithmetic average of the $F_i$'s, while the heterogeneous version of [GR], denoted [HGR], is based on the geometric average of the $F_i$'s. We also introduce a procedure [PB], that is based on the Poisson-binomial distribution and that uniformly improves [HLR] and [HGR], at the price of a higher computational complexity. Perhaps surprisingly, this shows that, contrary to the known theory of false discovery rate (FDR) control under heterogeneity, the way to incorporate the $F_i$'s can be particularly simple in the case of FDX control, and does not require any further correction term. The performances of the new proposed procedures are illustrated by real and simulated data in two important heterogeneous settings: first, when the test statistics are continuous but the $p$-values are weighted by some known independent weight vector, e.g., coming from co-data sets; second, when the test statistics are discretely distributed, as is the case for data representing frequencies or counts.