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

TL;DR

The paper introduces TFisher, a flexible family of p-value combination tests with adaptive thresholding, providing optimal detection across various signal patterns, supported by theoretical analysis, simulations, and real data application.

Contribution

It proposes the TFisher family with adaptive truncation and weighting, along with optimal parameter selection methods, unifying existing tests and enhancing detection power.

Findings

Soft-thresholding is optimal for diverse signal patterns.

The oTFisher test adapts to unknown signal structures.

The methods outperform traditional tests in simulations and real data.

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.