Two simple sufficient conditions for FDR control

TL;DR

This paper introduces two simple sufficient conditions that ensure FDR control in multiple testing procedures, unifying and extending existing results across various dependency structures and testing frameworks.

Contribution

It presents a unified framework with two conditions that guarantee FDR control, applicable to many procedures and dependency types, and extends to continuous hypothesis spaces.

Findings

Unified proof for FDR control under various dependencies.

Extension to weighted and adaptive procedures.

Application to continuous hypothesis spaces.

Abstract

We show that the control of the false discovery rate (FDR) for a multiple testing procedure is implied by two coupled simple sufficient conditions. The first one, which we call ``self-consistency condition'', concerns the algorithm itself, and the second, called ``dependency control condition'' is related to the dependency assumptions on the $p$-value family. Many standard multiple testing procedures are self-consistent (e.g. step-up, step-down or step-up-down procedures), and we prove that the dependency control condition can be fulfilled when choosing correspondingly appropriate rejection functions, in three classical types of dependency: independence, positive dependency (PRDS) and unspecified dependency. As a consequence, we recover earlier results through simple and unifying proofs while extending their scope to several regards: weighted FDR, $p$-value reweighting, new family of step-up procedures under unspecified $p$-value dependency and adaptive step-up procedures. We give additional examples of other possible applications. This framework also allows for defining and studying FDR control for multiple testing procedures over a continuous, uncountable space of hypotheses.