[Paper Review] FARM-Test: Factor-Adjusted Robust Multiple Testing with False Discovery Control

Jianqing Fan, Yuan Ke+2 more|arXiv (Cornell University)|Nov 15, 2017

Statistical Methods in Clinical Trials50 references3 citations

TL;DR

This paper proposes FARM-Test, a factor-adjusted robust multiple testing procedure that controls the false discovery proportion (FDP) under general dependence and heavy-tailed distributions. By integrating robust U-statistic-based covariance estimation and factor modeling, it achieves consistent FDP estimation and improved power, especially in non-normal, dependent data settings.

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 assumption of joint normality is often imposed, which is too stringent for many applications. To address these challenges, in this paper we propose a factor-adjusted robust procedure for large-scale simultaneous inference with control of the false discovery proportion. We demonstrate that robust factor adjustments are extremely important in both improving the power of the tests and controlling FDP. 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. Our proposed procedures are implemented in the {\sf R}-package {\sf FarmTest}.

Motivation & Objective

• Address the limitations of conventional multiple testing methods that assume joint normality and ignore dependence and heavy-tailedness.
• Develop a method that maintains accurate false discovery proportion (FDP) control when test statistics are dependent and heavy-tailed.
• Improve statistical power in large-scale inference by adjusting for latent factors and robustifying covariance estimation.
• Establish theoretical consistency of FDP estimation under general conditions beyond normality.
• Provide a practical, implementable R package (FarmTest) for real-world applications in genomics, finance, and medical imaging.

Proposed method

• Uses a robust U-type covariance estimator to handle heavy-tailed distributions, with a new exponential-type deviation inequality established for its spectral norm.
• Applies a factor model to account for general dependence structures among test statistics by removing common factors.
• Adjusts test statistics by estimating and removing the effects of latent factors using principal component analysis on the robust covariance matrix.
• Employs a false discovery proportion (FDP) estimator that is consistent under weak dependence and heavy-tailedness.
• Combines factor adjustment with robust inference to improve power while maintaining FDP control.
• Derives theoretical conditions under which the FDP estimator remains consistent even when the data deviate from normality.

Experimental results

Research questions

• RQ1 How can false discovery proportion (FDP) be consistently estimated in high-dimensional multiple testing when data are dependent and heavy-tailed?
• RQ2 To what extent does factor adjustment improve the power and accuracy of multiple testing procedures under non-normal and dependent data?
• RQ3 Can a robust covariance estimator provide reliable inference under heavy-tailed distributions without assuming normality?
• RQ4 What are the theoretical conditions under which a factor-adjusted robust procedure yields consistent FDP estimation?
• RQ5 How does the proposed method compare empirically to state-of-the-art methods in settings with heavy-tailed and dependent data?

Key findings

• The proposed FARM-Test method achieves consistent estimation of the false discovery proportion (FDP) under general dependence and heavy-tailed distributions, even when the normality assumption fails.
• Robust factor adjustment significantly improves statistical power compared to conventional methods, particularly in heavy-tailed settings.
• An exponential-type deviation inequality is established for the robust U-type covariance estimator under the spectral norm, enabling finite-sample theoretical guarantees.
• The method outperforms state-of-the-art approaches in extensive numerical experiments, especially when data exhibit heavy tails and dependence.
• The FARM-Test procedure is implemented in the R package { t FarmTest}, enabling practical application in genomics, medical imaging, and financial data analysis.