FDR control with adaptive procedures and FDR monotonicity

TL;DR

This paper establishes theoretical bounds and properties for adaptive FDR controlling procedures, introduces improved methods, and demonstrates their increased power over existing procedures in biological data analysis.

Contribution

It provides rigorous FDR bounds for adaptive procedures, proves a monotonicity property for BH-like methods, and proposes new procedures with better power in practice.

Findings

Proved FDR bounds for adaptive procedures using any estimator of true nulls.

Established monotonicity property of BH-like procedures under independence.

Demonstrated improved power of new procedures over BH in simulations and gene expression data.

Abstract

The steep rise in availability and usage of high-throughput technologies in biology brought with it a clear need for methods to control the False Discovery Rate (FDR) in multiple tests. Benjamini and Hochberg (BH) introduced in 1995 a simple procedure and proved that it provided a bound on the expected value, $\mathit{FDR}\leq q$. Since then, many authors tried to improve the BH bound, with one approach being designing adaptive procedures, which aim at estimating the number of true null hypothesis in order to get a better FDR bound. Our two main rigorous results are the following: (i) a theorem that provides a bound on the FDR for adaptive procedures that use any estimator for the number of true hypotheses ($m_0$), (ii) a theorem that proves a monotonicity property of general BH-like procedures, both for the case where the hypotheses are independent. We also propose two improved procedures for which we prove FDR control for the independent case, and demonstrate their advantages over several available bounds, on simulated data and on a large number of gene expression data sets. Both applications are simple and involve a similar amount of computation as the original BH procedure. We compare the performance of our proposed procedures with BH and other procedures and find that in most cases we get more power for the same level of statistical significance.