Balancing central and marginal rejection when combining independent significance tests

• URL: http://arxiv.org/abs/2310.16600v2
• Date: Mon, 13 Nov 2023 20:22:03 GMT
• Title: Balancing central and marginal rejection when combining independent significance tests
• Authors: Chris Salahub and Wayne Oldford
• Abstract summary: A common approach to evaluating the significance of a collection of $p$-values combines them with a pooling function. A series of alternative hypotheses are introduced that communicate the strength and prevalence of non-null evidence in the $p$-values. It is proven that central rejection is always greater than or equal to marginal rejection, motivating a quotient to measure the balance between the two.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: A common approach to evaluating the significance of a collection of $p$-values combines them with a pooling function, in particular when the original data are not available. These pooled $p$-values convert a sample of $p$-values into a single number which behaves like a univariate $p$-value. To clarify discussion of these functions, a telescoping series of alternative hypotheses are introduced that communicate the strength and prevalence of non-null evidence in the $p$-values before general pooling formulae are discussed. A pattern noticed in the UMP pooled $p$-value for a particular alternative motivates the definition and discussion of central and marginal rejection levels at $\alpha$. It is proven that central rejection is always greater than or equal to marginal rejection, motivating a quotient to measure the balance between the two for pooled $p$-values. A combining function based on the $\chi^2_{\kappa}$ quantile transformation is proposed to control this quotient and shown to be robust to mis-specified parameters relative to the UMP. Different powers for different parameter settings motivate a map of plausible alternatives based on where this pooled $p$-value is minimized.

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