Hypothesis testing is an essential statistical method in experimental 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 toward 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, various Bayesian posterior indices which have been proposed in the literature are compared and their benefits and limitations are discussed. The 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, the comparison also reveals that there exist at least two candidates among the available Bayesian posterior indices which have appealing theoretical properties and are widely underused in the cognitive sciences.
Keywords: Bayes factor; Bayesian hypothesis testing; Bayesian posterior indices; MAP-based p-value; ROPE; e-value; equivalence testing; probability of direction (PD).