Five Shades of Grey: Phase Transitions in High-dimensional Multiple Testing
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We are motivated by marginal screenings of categorical variables, and study high-dimensional multiple testing problems 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 2×2 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.
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