Supplemental Material for Bayes Factors for Logistic (Mixed-Effect) Models
Psychological Methods · 2024
Supplemental Materials for 'Bayes factors for mixed-effects models'1 Alternative approach to Case 2: using the main effectAs noted in the paper, when deciding on an H 1 distribution for an interaction effect, there is more than one approach we could take.The approach in the paper uses the intercept from the mixed-effects model as a basis for generating a motivated maximum.An alternative approach is to base our estimate on the size of one of the main effects.This is related to the approach suggested by Gallistel (2009), described in Dienes (2019) as a special case of the 'room-to-move' heuristic.However, where that approach uses the simple effect (e.g., the difference between pre-test and post-test in the LV condition), the current approach uses the main effect t (the difference between pre-test and post-test, averaged across conditions).The logic here is the same as that outlined for the intercept-based approach in the paper: for the maximum interaction d, we assume that all improvement from pre-test to post-test happens in the HV condition.If this is the case, then improvement from pre-test to post-test in the LV condition is 0, and the difference that represents the interaction effect d is equal to the improvement from pre-test to post-test in the HV condition.In a centered design, the main effect of test-session t is the average of these two values, or d/2.The main effect t is therefore half the maximum effect we might observe.We set the standard deviation of the half-normal distribution that is our model of H 1 to equal t, the main effect of test-session from our mixed effects model.It is an open question which of these two versions of the motivated-maximum approach (using twice the intercept or using the main effect) performs better on average for returning appropriate Bayes factors.In the plot below, we contrast these two approaches.In the situation where performance in the pre-test in both conditions is at chance, the estimate and hence the results are the same for the two approaches.Figure 1 shows the results in the situation where performance in the pre-test in both conditions is above chance (.73 proportion correct, similar to average pre-test performance in Logan et al. (1991)).Here, the estimates from the two approaches diverge, and we can observe which estimate enables us to disentangle H 0 and H 1 most effectively.From Figure 1, two things are apparent: 1) the estimate based on the intercept can be used in a wider range of situations, because the grand mean remains positive even where the main effect of session is not; 2) the