A Minimax Optimal Ridge-Type Set Test for Global Hypothesis With Applications in Whole Genome Sequencing Association Studies

Abstract

Testing a global hypothesis for a set of variables is a fundamental problem in statistics with a wide range of applications. A few well-known classical tests include the Hotelling’s T ² test, the F-test, and the empirical Bayes based score test. These classical tests, however, are not robust to the signal strength and could have a substantial loss of power when signals are weak or moderate, a situation we commonly encounter in contemporary applications. In this article, we propose a minimax optimal ridge-type set test (MORST), a simple and generic method for testing a global hypothesis. The power of MORST is robust and considerably higher than that of the classical tests when the strength of signals is weak or moderate. In the meantime, MORST only requires a slight increase in computation compared to these existing tests, making it applicable to the analysis of massive genome-wide data. We also provide the generalizations of MORST that are parallel to the traditional Wald test and Rao’s score test in asymptotic settings. Extensive simulations demonstrated the robust power of MORST and that the Type I error of MORST was well controlled. We applied MORST to the analysis of the whole-genome sequencing data from the Atherosclerosis Risk in Communities study, where MORST detected 20%–250% more signal regions than the classical tests. Supplementary materials for this article are available online.

Keywords:

• F-test
• Global hypothesis testing
• Robust power
• Score test
• Signal strength
• Whole genome sequencing studies

Supplementary Materials

Supplementary materials include the derivation of $Q(\tau )$, the proofs of Theorems 1 and 2, Corollaries 1 and 2, and other technical lemmas, the algorithm to compute ${\tau }_c^{*}$, additional simulation results, as well as the genomic landscapes of significant sliding windows in ARIC WGS data analysis.

Acknowledgments

The authors thank the referees for their constructive comments that have helped greatly improve the article. The authors also thank Dr. Eric Boerwinkle for providing the Atherosclerosis Risk in Communities whole genome sequencing data.

Additional information

Funding

This work was supported by grants R35-CA197449, U19CA203654, and P01-CA134294 from the National Cancer Institute, U01-HG009088 from the National Human Genome Research Institute, and R01-HL113338 from the National Heart, Lung, and Blood Institute. The Atherosclerosis Risk in Communities (ARIC) study has been funded in whole or in part with Federal funds from the National Heart, Lung, and Blood Institute, National Institutes of Health, Department of Health and Human Services (contract numbers HHSN268201700001I, HHSN268201700002I, HHSN268201700003I, HHSN268201700004I and Accepted Manuscript HHSN268201700005I). The authors thank the staff and participants of the ARIC study for their contributions. Funding support for Building on GWAS for NHLBI diseases: the U.S. CHARGE consortium was provided by the NIH through the American Recovery and Reinvestment Act of 2009 (ARRA) (5RC2HL102419). Sequencing was carried out at the Baylor College of Medicine Human Genome Sequencing Center and supported by the National Human Genome Research Institute grants U54 HG003273 and UM1 HG008898.