Online Rules for Control of False Discovery Rate and False Discovery Exceedance

TL;DR

This paper develops and analyzes online procedures for controlling the false discovery rate (FDR) and false discovery exceedance in sequential hypothesis testing, extending existing methods to dependent p-values and demonstrating their effectiveness.

Contribution

It introduces generalized alpha-investing procedures for online FDR control, including conditions for dependency and methods for controlling false discovery exceedance.

Findings

Proves online FDR control under independence of true null p-values.

Extends control to dependent p-values with additional conditions.

Shows online procedures maintain power comparable to offline methods.

Abstract

Multiple hypothesis testing is a core problem in statistical inference and arises in almost every scientific field. Given a set of null hypotheses $\mathcal{H}(n) = (H_1,\dotsc, H_n)$, Benjamini and Hochberg introduced the false discovery rate (FDR), which is the expected proportion of false positives among rejected null hypotheses, and proposed a testing procedure that controls FDR below a pre-assigned significance level. Nowadays FDR is the criterion of choice for large scale multiple hypothesis testing. In this paper we consider the problem of controlling FDR in an "online manner". Concretely, we consider an ordered --possibly infinite-- sequence of null hypotheses $\mathcal{H} = (H_1,H_2,H_3,\dots )$ where, at each step $i$, the statistician must decide whether to reject hypothesis $H_i$ having access only to the previous decisions. This model was introduced by Foster and Stine. We study a class of "generalized alpha-investing" procedures and prove that any rule in this class controls online FDR, provided $p$-values corresponding to true nulls are independent from the other $p$-values. (Earlier work only established mFDR control.) Next, we obtain conditions under which generalized alpha-investing controls FDR in the presence of general $p$-values dependencies. Finally, we develop a modified set of procedures that also allow to control the false discovery exceedance (the tail of the proportion of false discoveries). Numerical simulations and analytical results indicate that online procedures do not incur a large loss in statistical power with respect to offline approaches, such as Benjamini-Hochberg.