Objective Bayesian Comparison of Constrained Analysis of Variance Models

In the social sciences we are often interested in comparing models specified by parametric equality or inequality constraints. For instance, when examining three group means {μ1, μ2, μ3} through an analysis of variance (ANOVA), a model may specify that μ1< μ2< μ3, while another one may state that {μ1= μ3} < μ2, and finally a third model may instead suggest that all means are unrestricted. This is a challenging problem, because it involves a combination of non-nested models, as well as nested models having the same dimension. We adopt an objective Bayesian approach, and derive the posterior probability of each model under consideration, focussing on ANOVA models, although our procedure is general. More specifically, our method combines in a unified way both the intrinsic prior methodology -which is especially appropriate for comparing nested models having different dimensions- and the encompassing prior approach -which was devised to compare models specified by inequality constraints. Our strategy presents several advantages with respect to current methods: i) it requires no prior inputs, and is thus fully objective; ii) it does not call for training the prior with empirical data; iii) it deals naturally with equality constraints; iv) is computationally efficient. A comparative evaluation of the performance of our method with respect to currently available techniques is performed through some simulation studies. We also present an application to real data collected through a psychological experiment aimed at measuring the perception of competence of a leader among five groups of students