Combining P-Values Under Dependence and Optimal Transport

RUODU WANG – UNIVERSITY OF WATERLOO 

ABSTRACT

The problem of combining p-values is an old and fundamental one, and the classic assumption of independence is often violated or unverifiable in many modern applications. There are deep connections between p-merging under dependence, multi-marginal optimal transport, robust risk management, and e-values (e for “expectation”). We discuss these connections from different areas of mathematics, statistics, and economics. We obtain several results on the representation and admissibility of p-merging functions under arbitrary dependence. Moreover, we improve many p-merging methods when the p-values are exchangeable, or when external randomization is allowed (or both). The main technical advance is to show that all existing combination rules can be obtained by calibrating the p-values to e-values, averaging those e-values, converting to a test using Markov’s inequality, and finally obtaining p-values by combining this family of tests. Improvements are then available via randomized and exchangeable variants of Markov’s inequality.

Related Papers: 

• https://doi.org/10.1093/biomet/asaa027
• http://dx.doi.org/10.1214/21-AOS2109
• https://arxiv.org/abs/2404.03484