FarmTest: Factor-Adjusted Robust Multiple Testing With Approximate False Discovery Control

ABSTRACT

Large-scale multiple testing with correlated and heavy-tailed data arises in a wide range of research areas from genomics, medical imaging to finance. Conventional methods for estimating the false discovery proportion (FDP) often ignore the effect of heavy-tailedness and the dependence structure among test statistics, and thus may lead to inefficient or even inconsistent estimation. Also, the commonly imposed joint normality assumption is arguably too stringent for many applications. To address these challenges, in this article we propose a factor-adjusted robust multiple testing (FarmTest) procedure for large-scale simultaneous inference with control of the FDP. We demonstrate that robust factor adjustments are extremely important in both controlling the FDP and improving the power. We identify general conditions under which the proposed method produces consistent estimate of the FDP. As a byproduct that is of independent interest, we establish an exponential-type deviation inequality for a robust U-type covariance estimator under the spectral norm. Extensive numerical experiments demonstrate the advantage of the proposed method over several state-of-the-art methods especially when the data are generated from heavy-tailed distributions. The proposed procedures are implemented in the R-package FarmTest. Supplementary materials for this article are available online.

KEYWORDS:

• Factor adjustment
• False discovery proportion
• Huber loss
• Large-scale multiple testing
• Robustness

Acknowledgments

The authors would like to thank the Editor, Associate Editor, and two anonymous referees for their valuable comments. The bulk of the research were conducted while Yuan Ke, Qiang Sun and Wen-Xin Zhou were postdoctoral fellows at Department of Operations Research and Financial Engineering, Princeton University.

Funding

Additional information

Funding

This work is supported by NSERC Grant RGPIN-2018-06484, a Connaught Award, NSF Grants DMS-1662139, DMS-1712591, DMS-1811376, NIH funding: R01-GM072611, and NSFC Grant 11690014.