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An exciting recent development is the uptake of deep neural networks in many scientific fields, where the main objective is outcome prediction with a black-box nature. Significance testing is promising to address the black-box issue and explore novel scientific insights and interpretations of the decision-making process based on a deep learning model. However, testing for a neural network poses a challenge because of its black-box nature and unknown limiting distributions of parameter estimates while existing methods require strong assumptions or excessive computation. In this article, we derive one-split and two-split tests relaxing the assumptions and computational complexity of existing black-box tests and extending to examine the significance of a collection of features of interest in a dataset of possibly a complex type, such as an image. The one-split test estimates and evaluates a black-box model based on estimation and inference subsets through sample splitting and data perturbation. The two-split test further splits the inference subset into two but requires no perturbation. Also, we develop their combined versions by aggregating the p-values based on repeated sample splitting. By deflating the bias-sd-ratio, we establish asymptotic null distributions of the test statistics and the consistency in terms of Type II error. Numerically, we demonstrate the utility of the proposed tests on seven simulated examples and six real datasets. Accompanying this article is our python library dnn-inference (https://dnninference.readthedocs.io/en/latest/) that implements the proposed tests.
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Aggregating Dependent Signals with Heavy-Tailed Combination Tests
Combining dependent p-values poses a long-standing challenge in statistical inference, particularly when aggregating findings from multiple methods to enhance signal detection. Recently, p-value combination tests based on regularly varying-tailed distributions, such as the Cauchy combination test and harmonic mean p-value, have attracted attention for their robustness to unknown dependence. This paper provides a theoretical and empirical evaluation of these methods under an asymptotic regime where the number of p-values is fixed and the global test significance level approaches zero. We examine two types of dependence among the p-values. First, when p-values are pairwise asymptotically independent, such as with bivariate normal test statistics with no perfect correlation, we prove that these combination tests are asymptotically valid. However, they become equivalent to the Bonferroni test as the significance level tends to zero for both one-sided and two-sided p-values. Empirical investigations suggest that this equivalence can emerge at moderately small significance levels. Second, under pairwise quasi-asymptotic dependence, such as with bivariate t-distributed test statistics, our simulations suggest that these combination tests can remain valid and exhibit notable power gains over Bonferroni, even as the significance level diminishes. These findings highlight the potential advantages of these combination tests in scenarios where p-values exhibit substantial dependence. Our simulations also examine how test performance depends on the support and tail heaviness of the underlying distributions.
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- Award ID(s):
- 2238656
- PAR ID:
- 10597135
- Publisher / Repository:
- Biometrika
- Date Published:
- Journal Name:
- Biometrika
- ISSN:
- 0006-3444
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1093/biomet/asaf038
