TFisher Tests: Optimal and Adaptive Thresholding for Combining $p$-Values

Abstract

For testing a group of hypotheses, tremendous p-value combination methods have been developed and widely applied since 1930's. Some methods (e.g., the minimal p-value) are optimal for sparse signals, and some others (e.g., Fisher's combination) are optimal for dense signals. To address a wide spectrum of signal patterns, this paper proposes a unifying family of statistics, called TFisher, with general p-value truncation and weighting schemes. Analytical calculations for the p-value and the statistical power of TFisher under general hypotheses are given. Optimal truncation and weighting parameters are studied based on Bahadur Efficiency (BE) and the proposed Asymptotic Power Efficiency (APE), which is superior to BE for studying the signal detection problem. A soft-thresholding scheme is shown to be optimal for signal detection in a large space of signal patterns. When prior information of signal pattern is unavailable, an omnibus test, oTFisher, can adapt to the given data. Simulations evidenced the accuracy of calculations and validated the theoretical properties. The TFisher tests were applied to analyzing a whole exome sequencing data of amyotrophic lateral sclerosis. Relevant tests and calculations have been implemented into an R package TFisher and published on the CRAN.

Gamma Approximation of Stratified Truncated Exact test (GASTE-test) & Application

“…Additionally, inspired by Zaykin et al (2002) and Zhang et al (2018), to increase the statistical power of the test a truncated version of the Fisher's method can be used to combine only the p-values that contribute a significant amount of information to the overall signal, taking into account that some p-values are discarded. To preserve a certain softness and re-spread the retained p-values of the combination between 0 and 1, only the amount of signal below the truncation threshold τ is taken into account.…”