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Motivated by genome-wide association screening studies (GWAS), we studyhigh-dimensional marginal screenings of categorical variables where teststatistics have approximate chi-square distributions. We characterize four newphase transitions in high-dimensional chi-square models, and derive the signalsizes necessary and sufficient for statistical procedures to simultaneouslycontrol false discovery (in terms of family-wise error rate or false discoveryrate) and missed detection (in terms of family-wise non-discovery rate or falsenon-discovery rate) in large dimensions. Remarkably, degrees of freedom in thechi-square distributions do not affect the boundaries in all four phasetransitions. Several well-known procedures are shown to attain theseboundaries. Two new phase transitions are also identified in the Gaussianlocation model under one-sided alternatives.  We then elucidate on the nature of signal sizes in association tests bycharacterizing its relationship with marginal frequencies, odds ratio, andsample sizes in $2\times2$ contingency tables. This allows us to illustrate aninteresting manifestation of the phase transition phenomena in genome-wideassociation studies (GWAS). We also show, perhaps surprisingly, that giventotal sample sizes, balanced designs in such association studies rarely deliveroptimal power for detecting the effects of rare genetic variants.

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