Some Results on Generalized Familywise Error Rate Controlling Procedures under Dependence

TL;DR

This paper studys $k$-FWER control in a Gaussian sequence model $X_i\sim N(\mu_i,1)$ with correlation $\rho_{ij}$, addressing how different dependence structures affect generalized error rates. It develops improved upper bounds and a new general probability inequality for the event of at least $k$ rejections, leveraging independence, negative dependence, arbitrary dependence, equicorrelation, and nearly independent regimes. Key contributions include explicit $\alpha^*$ scaling rules under various dependence settings, asymptotic results for nearly independent data, and empirical validation showing sharper bounds and increased power relative to existing methods. The results have practical impact for multiple testing in genomics and related fields by enabling more powerful procedures with robust control of $k$-FWER.

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

Theorems & Definitions (20)

• Theorem 2.1
• Theorem 3.1
• proof
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Remark 1
• Theorem 4.1
• Lemma 4.1
• proof