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June 2004 Higher criticism for detecting sparse heterogeneous mixtures
David Donoho, Jiashun Jin
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Ann. Statist. 32(3): 962-994 (June 2004). DOI: 10.1214/009053604000000265


Higher criticism, or second-level significance testing, is a multiple-comparisons concept mentioned in passing by Tukey. It concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis. Tukey suggested comparing the fraction of observed significances at a given α-level to the expected fraction under the joint null. In fact, he suggested standardizing the difference of the two quantities and forming 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 and Jin, 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 of 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 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 nonnull 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 non-Gaussian cases.


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David Donoho. Jiashun Jin. "Higher criticism for detecting sparse heterogeneous mixtures." Ann. Statist. 32 (3) 962 - 994, June 2004.


Published: June 2004
First available in Project Euclid: 24 May 2004

zbMATH: 1092.62051
MathSciNet: MR2065195
Digital Object Identifier: 10.1214/009053604000000265

Primary: 62G10
Secondary: 62G20 , 62G32

Keywords: combining many p-values , Multiple comparsions , normalized empirical process , sparse normal means , thresholding

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 3 • June 2004
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