Stepup procedures controlling generalized FWER and generalized FDR

TL;DR

This paper introduces and analyzes stepup procedures for controlling generalized error rates, specifically the $k$-FWER and $k$-FDR, in multiple hypothesis testing, accommodating various dependence structures among p-values.

Contribution

It extends existing error rate control methods to the generalized $k$-FWER and $k$-FDR, providing procedures that do not rely on specific dependence assumptions among p-values.

Findings

Procedures controlling $k$-FWER using $k$th order joint null distributions.

Development of $k$-FDR controlling procedures under independence and positive dependence.

Procedures applicable without assuming specific dependence among p-values.

Abstract

In many applications of multiple hypothesis testing where more than one false rejection can be tolerated, procedures controlling error rates measuring at least $k$ false rejections, instead of at least one, for some fixed $k\ge 1$ can potentially increase the ability of a procedure to detect false null hypotheses. The $k$-FWER, a generalized version of the usual familywise error rate (FWER), is such an error rate that has recently been introduced in the literature and procedures controlling it have been proposed. A further generalization of a result on the $k$-FWER is provided in this article. In addition, an alternative and less conservative notion of error rate, the $k$-FDR, is introduced in the same spirit as the $k$-FWER by generalizing the usual false discovery rate (FDR). A $k$-FWER procedure is constructed given any set of increasing constants by utilizing the $k$th order joint null distributions of the $p$-values without assuming any specific form of dependence among all the $p$-values. Procedures controlling the $k$-FDR are also developed by using the $k$th order joint null distributions of the $p$-values, first assuming that the sets of null and nonnull $p$-values are mutually independent or they are jointly positively dependent in the sense of being multivariate totally positive of order two (MTP$_2$) and then discarding that assumption about the overall dependence among the $p$-values.