Testing a Large Number of Composite Null Hypotheses Using Conditionally Symmetric Multidimensional Gaussian Mixtures in Genome-Wide Studies

TL;DR

This paper introduces the csmGmm framework for large-scale testing of composite null hypotheses in genetic studies, unifying Bayesian and frequentist methods with improved robustness and interpretability.

Contribution

The work develops the conditionally symmetric multidimensional Gaussian mixture model (csmGmm), unifying lfdr and z-statistic testing rules across multiple genetic analysis settings.

Findings

csmGmm outperforms recent alternatives in robustness

It ensures lfdr and z-statistic testing rule agreement

Applied successfully to lung cancer genetic studies

Abstract

Causal mediation analysis, pleiotropy analysis, and replication analysis are three highly popular genetic study designs. Although these analyses address different scientific questions, the underlying inference problems all involve large-scale testing of composite null hypotheses. The goal is to determine whether all null hypotheses - as opposed to at least one - in a set of individual tests should simultaneously be rejected. Various recent methodology has been proposed for the aforementioned situations, and an appealing empirical Bayes strategy is to apply the popular two-group model, calculating local false discovery rates (lfdr) for each set of hypotheses. However, such a strategy is difficult due to the need for multivariate density estimation. Furthermore, the multiple testing rules for the empirical Bayes lfdr approach and conventional frequentist z-statistics can disagree, which is troubling for a field that ubiquitously utilizes summary statistics. This work proposes a framework to unify two-group testing in genetic association composite null settings, the conditionally symmetric multidimensional Gaussian mixture model (csmGmm). The csmGmm is shown to demonstrate more robust operating characteristics than recently-proposed alternatives. Crucially, the csmGmm also offers strong interpretability guarantees by harmonizing lfdr and z-statistic testing rules. We extend the base csmGmm to cover each of the mediation, pleiotropy, and replication settings, and we prove that the lfdr z-statistic agreement holds in each situation. We apply the model to a collection of translational lung cancer genetic association studies that motivated this work.