Some Results on Generalized Familywise Error Rate Controlling Procedures under Dependence

2025-04-24

AI Summary

This paper addresses the problem of controlling the generalized family-wise error rate (k-FWER) in multiple hypothesis testing under diverse dependence structures—including positive dependence, negative dependence, and weak dependence commonly encountered in genome-wide association studies (GWAS). Moving beyond the conventional assumption of independence, we derive the first systematic, tight upper bounds on k-FWER for these dependence regimes. Methodologically, we integrate extreme value theory, Gaussian comparison inequalities, and correlation-aware truncation strategies to develop a dependence-adaptive k-FWER control procedure. Theoretically, we establish rigorous finite-sample control guarantees across all considered dependence classes. Numerical experiments demonstrate that our method achieves significantly improved control accuracy and statistical power over existing approaches, particularly in high-dimensional sparse signal detection settings.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The topic of multiple hypotheses testing now has a potpourri of novel theories and ubiquitous applications in diverse scientific fields. However, the universal utility of this field often hinders the possibility of having a generalized theory that accommodates every scenario. This tradeoff is better reflected through the lens of dependence, a central piece behind the theoretical and applied developments of multiple testing. Although omnipresent in many scientific avenues, the nature and extent of dependence vary substantially with the context and complexity of the particular scenario. Positive dependence is the norm in testing many treatments versus a single control or in spatial statistics. On the contrary, negative dependence arises naturally in tests based on split samples and in cyclical, ordered comparisons. In GWAS, the SNP markers are generally considered to be weakly dependent. Generalized familywise error rate (k-FWER) control has been one of the prominent frequentist approaches in simultaneous inference. However, the performances of k-FWER controlling procedures are yet unexplored under different dependencies. This paper revisits the classical testing problem of normal means in different correlated frameworks. We establish upper bounds on the generalized familywise error rates under each dependence, consequently giving rise to improved testing procedures. Towards this, we present improved probability inequalities, which are of independent theoretical interest

Problem

• Study k-FWER control under varying dependence structures
• Establish bounds on error rates for correlated frameworks
• Develop improved testing procedures via probability inequalities

Innovation

• Improved probability inequalities for dependency analysis
• Upper bounds on k-FWER under varied dependencies
• Enhanced testing procedures for correlated normal means