Unification of Rare and Weak Detection Models using Moderate Deviations Analysis and Log-Chisquared P-values
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arXiv
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| Format: | Preprint |
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2021
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| author | Kipnis, Alon |
| author_facet | Kipnis, Alon |
| contents | Rare and 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 they generally do not stand out individually, for example in a Bonferroni analysis. Such models have been studied in quite a few settings, for example in some cases studies focused on an underlying Gaussian means model for the hypotheses being tested; in others, Poisson and Binomial. Such seemingly different models have the following common structure. Summarizing the evidence of individual tests by the negative logarithm of its P-value, the model is asymptotically equivalent to a situation in which most negative log P-values have a standard exponential distribution but a small fraction might have an alternative distribution which is approximately noncentral chisquared on one degree of freedom. We characterize the asymptotic performance of global tests combining asymptotic log-chisquared P-values in terms of the chisquared mixture parameters: the scaling parameter controlling heteroscedasticity, the non-centrality parameter, and the parameter controlling the rarity of individual non-null effects. 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 combination test have sub-optimal asymptotic phase diagrams. Our analysis yields the asymptotic power of global testing in various new rare and weak models, including two-sample heteroscedastic normal mixtures and binomial experiments with perturbed probabilities of success. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2103_03999 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Unification of Rare and Weak Detection Models using Moderate Deviations Analysis and Log-Chisquared P-values Kipnis, Alon Statistics Theory 62H17 (Primary) 62H15, 62G10 (Secondary) Rare and 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 they generally do not stand out individually, for example in a Bonferroni analysis. Such models have been studied in quite a few settings, for example in some cases studies focused on an underlying Gaussian means model for the hypotheses being tested; in others, Poisson and Binomial. Such seemingly different models have the following common structure. Summarizing the evidence of individual tests by the negative logarithm of its P-value, the model is asymptotically equivalent to a situation in which most negative log P-values have a standard exponential distribution but a small fraction might have an alternative distribution which is approximately noncentral chisquared on one degree of freedom. We characterize the asymptotic performance of global tests combining asymptotic log-chisquared P-values in terms of the chisquared mixture parameters: the scaling parameter controlling heteroscedasticity, the non-centrality parameter, and the parameter controlling the rarity of individual non-null effects. 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 combination test have sub-optimal asymptotic phase diagrams. Our analysis yields the asymptotic power of global testing in various new rare and weak models, including two-sample heteroscedastic normal mixtures and binomial experiments with perturbed probabilities of success. |
| title | Unification of Rare and Weak Detection Models using Moderate Deviations Analysis and Log-Chisquared P-values |
| topic | Statistics Theory 62H17 (Primary) 62H15, 62G10 (Secondary) |
| url | https://arxiv.org/abs/2103.03999 |