On the performance of FDR control: Constraints and a partial solution

TL;DR

This paper investigates the limitations of FDR control using the BH procedure under a random effects model, revealing a criticality phenomenon that restricts power and pFDR control, and proposes a multi-location BH-type method to improve performance.

Contribution

It identifies the criticality phenomenon limiting FDR control and proposes a novel multi-location BH-type procedure to enhance power and pFDR control.

Findings

Criticality phenomenon causes power to drop to zero below a threshold FDR level.

Proposed multi-location BH procedure improves power and pFDR control.

Simulations confirm the effectiveness of the new method.

Abstract

The False Discovery Rate (FDR) paradigm aims to attain certain control on Type I errors with relatively high power for multiple hypothesis testing. The Benjamini--Hochberg (BH) procedure is a well-known FDR controlling procedure. Under a random effects model, we show that, in general, unlike the FDR, the positive FDR (pFDR) of the BH procedure cannot be controlled at an arbitrarily low level due to the limited evidence provided by the observations to separate false and true nulls. This results in a criticality phenomenon, which is characterized by a transition of the procedure's power from being positive to asymptotically 0 without any reduction in the pFDR, once the target FDR control level is below a positive critical value. To address the constraints on the power and pFDR control imposed by the criticality phenomenon, we propose a procedure which applies BH-type procedures at multiple locations in the domain of $p$-values. Both analysis and simulations show that the proposed procedure can attain substantially improved power and pFDR control.