Phase Transitions in Genome-wide Association Studies and Categorical Variable Screenings

TL;DR

This paper explores phase transitions in high-dimensional chi-square models relevant to GWAS, identifying new boundaries for detection and control of false discoveries, and analyzing the impact of design balance on power.

Contribution

It characterizes four new phase transitions in high-dimensional chi-square models and relates signal sizes to marginal frequencies, odds ratios, and sample sizes in GWAS.

Findings

Four new phase transitions identified in chi-square models.

Well-known procedures attain these phase transition boundaries.

Balanced designs are often suboptimal for rare variant detection.

Abstract

Motivated by genome-wide association screening studies (GWAS), we study high-dimensional marginal screenings of categorical variables where test statistics have approximate chi-square distributions. We characterize four new phase transitions in high-dimensional chi-square models, and derive the signal sizes necessary and sufficient for statistical procedures to simultaneously control false discovery (in terms of family-wise error rate or false discovery rate) and missed detection (in terms of family-wise non-discovery rate or false non-discovery rate) in large dimensions. Remarkably, degrees of freedom in the chi-square distributions do not affect the boundaries in all four phase transitions. Several well-known procedures are shown to attain these boundaries. Two new phase transitions are also identified in the Gaussian location model under one-sided alternatives. We then elucidate on the nature of signal sizes in association tests by characterizing its relationship with marginal frequencies, odds ratio, and sample sizes in $2\times2$ contingency tables. This allows us to illustrate an interesting manifestation of the phase transition phenomena in genome-wide association studies (GWAS). We also show, perhaps surprisingly, that given total sample sizes, balanced designs in such association studies rarely deliver optimal power for detecting the effects of rare genetic variants.