Heteroscedasticity-Adjusted Ranking and Thresholding for Large-Scale Multiple Testing

Abstract

Standardization has been a widely adopted practice in multiple testing, for it takes into account the variability in sampling and makes the test statistics comparable across different study units. However, despite conventional wisdom to the contrary, we show that there can be a significant loss in information from basing hypothesis tests on standardized statistics rather than the full data. We develop a new class of heteroscedasticity-adjusted ranking and thresholding (HART) rules that aim to improve existing methods by simultaneously exploiting commonalities and adjusting heterogeneities among the study units. The main idea of HART is to bypass standardization by directly incorporating both the summary statistic and its variance into the testing procedure. A key message is that the variance structure of the alternative distribution, which is subsumed under standardized statistics, is highly informative and can be exploited to achieve higher power. The proposed HART procedure is shown to be asymptotically valid and optimal for false discovery rate (FDR) control. Our simulation results demonstrate that HART achieves substantial power gain over existing methods at the same FDR level. We illustrate the implementation through a microarray analysis of myeloma.

KEYWORDS:

• Covariate-assisted inference
• Data processing and information loss
• False discovery rate
• Heteroscedasticity
• Multiple testing with side information
• Structured multiple testing

Supplementary Materials

The supplementary material contains the formulas for the illustrative example; proofs of main theorems, propositions, and lemma;s and additional numerical results.

Funding

The research of Wenguang Sun was supported in part by NSF grant DMS-1712983.