FDR control with adaptive procedures and FDR monotonicity

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.

💡 Research Summary

The paper addresses the persistent challenge of controlling the false discovery rate (FDR) in large‑scale multiple testing, a problem that has become central with the advent of high‑throughput biological technologies. While the classic Benjamini–Hochberg (BH) procedure guarantees FDR ≤ q, it can be overly conservative because it treats the total number of hypotheses m as if all are true nulls. Adaptive procedures aim to improve power by estimating the number of true null hypotheses m₀ and substituting this estimate (\hat m_0) into the BH threshold.

The authors contribute two rigorous theoretical results under the assumption of independent p‑values. First, they prove a universal bound for any adaptive BH‑type procedure:
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