Decisions about equivalence: A comparison of TOST, HDI-ROPE, and the Bayes factor

Some important research questions require the ability to find evidence for two conditions being practically equivalent. This is impossible to accomplish within the traditional frequentist null hypothesis significance testing framework; hence, other methodologies must be utilized. We explain and illustrate three approaches for finding evidence for equivalence: The frequentist two one-sided tests procedure, the Bayesian highest density interval region of practical equivalence procedure, and the Bayes factor interval null procedure. We compare the classification performances of these three approaches for various plausible scenarios. The results indicate that the Bayes factor interval null approach compares favorably to the other two approaches in terms of statistical power. Critically, compared to the Bayes factor interval null procedure, the two one-sided tests and the highest density interval region of practical equivalence procedures have limited discrimination capabilities when the sample size is relatively small: specifically, in order to be practically useful, these two methods generally require over 250 cases within each condition when rather large equivalence margins of approximately 0.2 or 0.3 are used; for smaller equivalence margins even more cases are required. Because of these results, we recommend that researchers rely more on the Bayes factor interval null approach for quantifying evidence for equivalence, especially for studies that are constrained on sample size.