Selective inference is easier with p-values

TL;DR

This paper introduces selectively dominant p-values, simplifying post-selection inference across various statistical tests by ensuring a unified dominance property, and extends these ideas to p-value combination methods.

Contribution

It defines the concept of selectively dominant p-values, demonstrating their applicability to many common tests and providing a unified framework for selective inference.

Findings

All common p-values are shown to be selectively dominant.

Simpler derivations and deeper understanding of selective inference problems.

New variants of p-value combination methods are introduced.

Abstract

Selective inference is a subfield of statistics that enables valid inference after selection of a data-dependent question. In this paper, we introduce selectively dominant p-values, a class of p-values that allow practitioners to easily perform inference after arbitrary selection procedures. Unlike a traditional p-value, whose distribution must stochastically dominate the uniform distribution under the null, a selectively dominant p-value must have a post-selection distribution that stochastically dominates that of a uniform having undergone the same selection process; moreover, this property must hold simultaneously for all possible selection processes. Despite the strength of this condition, we show that all commonly used p-values (e.g., p-values from two-sided testing in parametric families, one-sided testing in monotone likelihood ratio and exponential families, $F$-tests for linear regression, and permutation tests) are selectively dominant. By recasting two canonical selective inference problems-inference on winners and rank verification-in our selective dominance framework, we provide simpler derivations, a deeper conceptual understanding, and new generalizations and variations of these methods. Additionally, we use our insights to introduce selective variants of methods that combine p-values, such as Fisher's combination test.