A measure of evidence based on the likelihood-ratio statistics

TL;DR

This paper introduces a likelihood-ratio based measure of evidence that maintains invariance, aligns with frequentist error control, and can serve as an upper bound for posterior probabilities, offering advantages over traditional p-values.

Contribution

It demonstrates the properties of the likelihood-ratio measure, including invariance, logical consistency, and its potential as an upper bound for posterior probabilities, with applications in genetic equilibrium testing.

Findings

Likelihood-ratio measure is invariant under sigma-finite measures.

It satisfies logical properties not held by p-values.

It can control Type I error and bound posterior probabilities.

Abstract

In this paper, we show that the likelihood-ratio measure (a) is invariant with respect to dominating sigma-finite measures, (b) satisfies logical consequences which are not satisfied by standard $p$-values, (c) respects frequentist properties, i.e., the type I error can be properly controlled, and, under mild regularity conditions, (d) can be used as an upper bound for posterior probabilities. We also discuss a generic application to test whether the genotype frequencies of a given population are under the Hardy-Weinberg equilibrium, under inbreeding restrictions or under outbreeding restrictions.