Optimal Adjustment and Combination of Independent Discrete $p$-Values

TL;DR

This paper introduces an optimal transport-based method for combining independent discrete p-values, improving Type I error control and power in statistical tests, with applications demonstrated in genetic association studies.

Contribution

It extends a recent framework to unify and improve classical p-value combination methods for discrete data using optimal transport techniques.

Findings

Accurate Type I error control when variance matches continuous case

Power comparable to UMP test for monotonic likelihood ratio tests

Method successfully applied to genetic association data

Abstract

Combining p-values from multiple independent tests is a fundamental task in statistical inference, but presents unique challenges when the p-values are discrete. We extend a recent optimal transport-based framework for combining discrete p-values, which constructs a continuous surrogate distribution by minimizing the Wasserstein distance between the transformed discrete null and its continuous analogue. We provide a unified approach for several classical combination methods, including Fisher's, Pearson's, George's, Stouffer's, and Edgington's statistics. Our theoretical analysis and extensive simulations show that accurate Type I error control is achieved when the variance of the adjusted discrete statistic closely matches that of the continuous case. We further demonstrate that, when the likelihood ratio test is a monotonic function of a combination statistic, the proposed approximation achieves power comparable to the uniformly most powerful (UMP) test. The methodology is illustrated with a genetic association study of rare variants using case-control data, and is implemented in the R package DPComb.