How to choose between different Bayesian posterior indices for hypothesis testing in practice

TL;DR

This paper compares various Bayesian posterior indices for hypothesis testing, highlighting their benefits, limitations, and suitability depending on study design, and identifies two promising yet underused indices in psychology.

Contribution

It provides a systematic comparison of Bayesian posterior indices for hypothesis testing, clarifying their advantages and guiding their selection in practice.

Findings

Not all Bayesian indices are equally beneficial for hypothesis testing.

Some indices have strong theoretical properties but are underused.

The usefulness of indices depends on study design and research goals.

Abstract

Hypothesis testing is an essential statistical method in psychology and the cognitive sciences. The problems of traditional null hypothesis significance testing (NHST) have been discussed widely, and among the proposed solutions to the replication problems caused by the inappropriate use of significance tests and $p$-values is a shift towards Bayesian data analysis. However, Bayesian hypothesis testing is concerned with various posterior indices for significance and the size of an effect. This complicates Bayesian hypothesis testing in practice, as the availability of multiple Bayesian alternatives to the traditional $p$-value causes confusion which one to select and why. In this paper, we compare various Bayesian posterior indices which have been proposed in the literature and discuss their benefits and limitations. Our comparison shows that conceptually not all proposed Bayesian alternatives to NHST and $p$-values are beneficial, and the usefulness of some indices strongly depends on the study design and research goal. However, our comparison also reveals that there exist at least two candidates among the available Bayesian posterior indices which have appealing theoretical properties and are, to our best knowledge, widely underused among psychologists.