Unification of Rare/Weak Detection Models using Moderate Deviations Analysis and Log-Chisquared P-values

Rare/Weak models for multiple hypothesis testing assume that only a small proportion of the tested hypotheses concern non-null effects and the individual effects are only moderately large, so that they generally do not stand out individually, for example in a Bonferroni analysis. Such rare/weak models have been studied in quite a few settings, for example in some cases studies focused on underlying Gaussian means model for the hypotheses being tested; in some others, Poisson. It seems not to have been noticed before that such seemingly different models have asymptotically the following common structure: Summarizing the evidence each test provides by the negative logarithm of its P-value, previous rare/weak model settings are asymptotically equivalent to detection where most negative log P-values have a standard exponential distribution but a small fraction of the P-values might have an alternative distribution which is moderately larger; we do not know which individual tests those might be, or even if there are any such. Moreover, the alternative distribution is noncentral chisquared on one degree of freedom. We characterize the asymptotic performance of global tests combining these P-values in terms of the chisquared mixture parameters: the scaling parameters controlling heteroscedasticity, the non-centrality parameter describing the effect size whenever it exists, and the parameter controlling the rarity of the non-null effects. Specifically, in a phase space involving the last two parameters, we derive a region where all tests are asymptotically powerless. Outside of this region, the Berk-Jones and the Higher Criticism tests have maximal power. Inference techniques based on the minimal P-value, false-discovery rate controlling, and Fisher's test have sub-optimal asymptotic phase diagrams. We provide various examples for multiple testing problems of the said common structure.