Optimal multiple testing under family-wise error control: elementary symmetric polynomials and a scalable algorithm

2026-04-13

AI Summary

This study addresses the loss of statistical power in multiple hypothesis testing due to reliance solely on p-value ordering. Focusing on K exchangeable hypotheses, the authors propose an optimal family-wise error rate (FWER)–controlling procedure based on elementary symmetric polynomials of likelihood ratios. The method maximizes statistical power while strictly controlling the FWER. Key contributions include the first computable closed-form solution for the dual vector, a proof of global monotonicity of the objective function ensuring a unique coordinate root, and an efficient, scalable algorithm combining coordinate descent with bisection search. Empirical results demonstrate substantial power gains over Hommel’s method—15% at K=3 and 84% at K=12—with superior performance validated in replication studies and clinical trials.

Technology Category

Search and Optimization: Mixed Discrete/Continuous SearchReasoning under Uncertainty: Stochastic OptimizationMachine Learning: Calibration & Uncertainty Quantification

Application Category

Web Mining and Content Analysis: Robustness and generalizability of Web mining methodsGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSecurity and Privacy: Large-scale security measurements

Abstract

Simultaneously testing $K$ hypotheses while controlling the family-wise error rate is a fundamental problem in statistics. Existing procedures (Bonferroni, Holm, Hochberg, Hommel) provide valid control but sacrifice power, increasingly so as $K$ grows, because they base decisions on marginal $p$-value ranks rather than the joint likelihood. Rosset et al. (2022) formulated the most powerful family-wise-error-rate-controlling test as a dual program and proved the existence of an optimal dual vector $μ^*$, but left its computation as an open problem. We solve this problem for $K$ exchangeable hypotheses. The key insight is that the family-wise error rate constraint coefficients $b_{l,k}(\vec{u})$ admit closed-form expressions through elementary symmetric polynomials of the likelihood-ratio values $g(u_1), \ldots, g(u_K)$. This algebraic structure implies a global monotonicity theorem: the target functions $F_γ(μ) = {\rm FWER}_γ(\vec{D}^μ)$ are simultaneously non-increasing in every component of $μ$, for arbitrary $K$, which guarantees unique coordinate-wise roots and enables a bisection-based coordinate-descent algorithm with $O(\log \varepsilon^{-1})$ convergence rate. The relative power gain over Hommel's method grows from 15\% at $K{=}3$ to 84\% at $K{=}12$. Applications to replication studies, a clinical trial, and a replicability assessment illustrate both the power gains and the role of the exchangeability assumption.

Problem

• multiple testing
• family-wise error rate
• optimal test
• exchangeable hypotheses
• statistical power

Innovation

• elementary symmetric polynomials
• family-wise error rate
• optimal multiple testing
• coordinate-descent algorithm
• exchangeable hypotheses