Confounder Adjustment in Multiple Hypothesis Testing

TL;DR

This paper unifies and generalizes confounder adjustment methods in large-scale hypothesis testing, providing theoretical guarantees and demonstrating their effectiveness in controlling false discoveries and type I error.

Contribution

It introduces a unified framework for confounder adjustment, extends existing methods to multiple variables, and offers theoretical analysis of their statistical properties.

Findings

Estimators based on RUV-4 and LEAPP are asymptotically powerful under strong confounding.

Asymptotic z-tests control type I error in the presence of confounders.

Benjamini-Hochberg procedure effectively controls false discovery rate with large samples.

Abstract

We consider large-scale studies in which thousands of significance tests are performed simultaneously. In some of these studies, the multiple testing procedure can be severely biased by latent confounding factors such as batch effects and unmeasured covariates that correlate with both primary variable(s) of interest (e.g. treatment variable, phenotype) and the outcome. Over the past decade, many statistical methods have been proposed to adjust for the confounders in hypothesis testing. We unify these methods in the same framework, generalize them to include multiple primary variables and multiple nuisance variables, and analyze their statistical properties. In particular, we provide theoretical guarantees for RUV-4 and LEAPP, which correspond to two different identification conditions in the framework: the first requires a set of "negative controls" that are known a priori to follow the null distribution; the second requires the true non-nulls to be sparse. Two different estimators which are based on RUV-4 and LEAPP are then applied to these two scenarios. We show that if the confounding factors are strong, the resulting estimators can be asymptotically as powerful as the oracle estimator which observes the latent confounding factors. For hypothesis testing, we show the asymptotic z-tests based on the estimators can control the type I error. Numerical experiments show that the false discovery rate is also controlled by the Benjamini-Hochberg procedure when the sample size is reasonably large.