Investigating the relationship between the Bayes factor and the separation of credible intervals.

We examined the relationship between the Bayes factor and the separation of credible intervals in between- and within-subject designs under a range of effect and sample sizes. For the within-subject case, we considered five intervals: (1) the within-subject confidence interval of Loftus and Masson (1994); (2) the within-subject Bayesian interval developed by Nathoo et al. (2018), whose derivation conditions on estimated random effects; (3) and (4) two modifications of (2) based on a proposal by Heck (2019) to allow for shrinkage and account for uncertainty in the estimation of random effects; and (5) the standard Bayesian highest-density interval. We derived and observed through simulations a clear and consistent relationship between the Bayes factor and the separation of credible intervals. Remarkably, for a given sample size, this relationship is described well by a simple quadratic exponential curve and is most precise in case (4). In contrast, interval (5) is relatively wide due to between-subjects variability and is likely to obscure effects when used in within-subject designs, rendering its relationship with the Bayes factor unclear in that case. We discuss how the separation percentage of (4), combined with knowledge of the sample size, could provide evidence in support of either a null or an alternative hypothesis. We also present a case study with example data and provide an R package 'rmBayes' to enable computation of each of the within-subject credible intervals investigated here using a number of possible prior distributions.