Higher Criticism for Detecting Sparse Heterogeneous Mixtures

Higher Criticism, or second-level significance testing, is a multiple comparisons concept mentioned in passing by Tukey (1976). It concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis. Tukey suggested to compare the fraction of observed significances at a given α-level to the expected fraction under the joint null, in fact he suggested to standardize the difference of the two quantities and form a z-score; the resulting z-score tests the significance of the body of significance tests. We consider a generalization, where we maximize this z-score over a range of significance levels 0 < α ≤ α0. We are able to show that the resulting Higher Criticism statistic is effective at resolving a very subtle testing problem: testing whether n normal means are all zero versus the alternative that a small fraction is nonzero. The subtlety of this ‘sparse normal means’ testing problem can be seen from work of Ingster (1999) and Jin (2002), who studied such problems in great detail. In their studies, they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution on the p-values either for the bulk of the tests or for the few most highly significant tests. In this range, when the amplitude of nonzero means is calibrated with the fraction of nonzero means, the likelihood ratio test for a precisely-specified alternative would still succeed in separating the two hypotheses. We show that the Higher Criticism is successful throughout the same region of amplitude vs. sparsity where the likelihood ratio test would succeed. Since it does not require a specification of the alternative, this shows that Higher Criticism is in a sense optimally adaptive to unknown sparsity and size of the non-null effects. While our theoretical work is largely asymptotic, we provide simulations in finite samples, and suggest some possible applications. We also show that Higher Critcism works well over a range of nonGaussian cases.