Application of Minimum Bayes Factor to a Balanced Two-Way Anova with Random Effects

It is a common practice in statistical analysis to draw conclusions based on significance. P-values often reflect the probability of incorrectly concluding that a null hypothesized model is true; they do not provide information about other types of error that are also important for interpreting statistical results. Standard model selection criteria and test procedures are often inappropriate for comparing models with different numbers of random effects, due to constraints on the parameter space of the variance components. In this paper, we focused on a minimum Bayes factor proposed by Held and Ott (2018) and applied it to a balanced two way analysis of variance (ANOVA) with random effects under three cases namely: Case 1: both factors are fixed; Case 2: both factors are random; Case 3: factor A is fixed and factor B is random. We realized that in all the three cases, considered the Bayes factor indicates weak evidence against the null hypothesis of zero variability in the effects of the levels of the factors as well as the interactions. This result is due to the conservative nature of the minimum Bayes factor.