Further results on controlling the false discovery proportion

TL;DR

This paper advances the theory of controlling the false discovery proportion (FDP) in multiple testing under dependency, proposing more powerful procedures and generalizations for scenarios with dependent p-values.

Contribution

It introduces a larger class of procedures controlling the $ ext{γ}$-FDP under positive dependence, and generalizes the $ ext{γ}$-FDP concept for high-dependency situations with new control methods.

Findings

New procedures controlling $ ext{γ}$-FDP under positive dependence.

Improved methods using pairwise joint distributions of null p-values.

Numerical studies support the theoretical results.

Abstract

The probability of false discovery proportion (FDP) exceeding $\gamma\in[0,1)$, defined as $\gamma$-FDP, has received much attention as a measure of false discoveries in multiple testing. Although this measure has received acceptance due to its relevance under dependency, not much progress has been made yet advancing its theory under such dependency in a nonasymptotic setting, which motivates our research in this article. We provide a larger class of procedures containing the stepup analog of, and hence more powerful than, the stepdown procedure in Lehmann and Romano [Ann. Statist. 33 (2005) 1138-1154] controlling the $\gamma$-FDP under similar positive dependence condition assumed in that paper. We offer better alternatives of the stepdown and stepup procedures in Romano and Shaikh [IMS Lecture Notes Monogr. Ser. 49 (2006a) 33-50, Ann. Statist. 34 (2006b) 1850-1873] using pairwise joint distributions of the null $p$-values. We generalize the notion of $\gamma$-FDP making it appropriate in situations where one is willing to tolerate a few false rejections or, due to high dependency, some false rejections are inevitable, and provide methods that control this generalized $\gamma$-FDP in two different scenarios: (i) only the marginal $p$-values are available and (ii) the marginal $p$-values as well as the common pairwise joint distributions of the null $p$-values are available, and assuming both positive dependence and arbitrary dependence conditions on the $p$-values in each scenario. Our theoretical findings are being supported through numerical studies.