On the Decision-Theoretic Foundations and the Asymptotic Bayes Risk of the Region of Practical Equivalence for Testing Interval Hypotheses

Testing interval hypotheses is of huge relevance in the biomedical and cognitive sciences; for example, in clinical trials. Frequentist approaches include the proposal of equivalence tests, which have been used to study if there is a predetermined meaningful treatment effect. In the Bayesian paradigm, two popular approaches exist: The first is the region of practical equivalence (ROPE), which has become increasingly popular in the cognitive sciences. The second is the Bayes factor for interval null hypotheses, which was proposed by Morey et al. One advantage of the ROPE procedure is that, in contrast to the Bayes factor, it is quite robust to the prior specification. However, while the ROPE is conceptually appealing, it lacks a clear decision-theoretic foundation like the Bayes factor. In this paper, a decision-theoretic justification for the ROPE procedure is derived for the first time, which shows that the Bayes risk of a decision rule based on the highest-posterior density interval (HPD) and the ROPE is asymptotically minimized for increasing sample size. To show this, a specific loss function is introduced. This result provides an important decision-theoretic justification for testing the interval hypothesis in the Bayesian approach based on the ROPE and HPD, in particular, when sample size is large.