A note on p-values interpreted as plausibilities

TL;DR

This paper reinterprets p-values as plausibilities within the inferential model framework, aligning their interpretation with practical usage and highlighting limitations in constrained parameter problems.

Contribution

It introduces a formal plausibility interpretation of p-values, showing their equivalence to plausibility functions in the inferential model framework.

Findings

P-values can be viewed as plausibility measures.

This interpretation aligns with practitioners' understanding of p-values.

Standard p-values have limitations with constrained parameters.

Abstract

P-values are a mainstay in statistics but are often misinterpreted. We propose a new interpretation of p-value as a meaningful plausibility, where this is to be interpreted formally within the inferential model framework. We show that, for most practical hypothesis testing problems, there exists an inferential model such that the corresponding plausibility function, evaluated at the null hypothesis, is exactly the p-value. The advantages of this representation are that the notion of plausibility is consistent with the way practitioners use and interpret p-values, and the plausibility calculation avoids the troublesome conditioning on the truthfulness of the null. This connection with plausibilities also reveals a shortcoming of standard p-values in problems with non-trivial parameter constraints.