Objective Bayesian Comparison of Constrained Analysis of Variance Models

TL;DR

This paper develops an objective Bayesian method to compare constrained ANOVA models involving equality and inequality constraints, enabling accurate model selection even with low frequentist power.

Contribution

It introduces a novel intrinsic prior approach tailored for constrained models, facilitating Bayesian comparison of non-nested and nested ANOVA models with constraints.

Findings

Bayesian model comparison effectively identifies true models in simulations.

Method performs well even with low frequentist power.

Application demonstrates practical utility in psychological data analysis.

Abstract

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 $\{ \mu_1, \mu_2, \mu_3\}$ through an analysis of variance (ANOVA), a model may specify that $\mu_1<\mu_2<\mu_3$, while another one may state that $\{ \mu_1=\mu_3\} <\mu_2$, 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. Our method is based on the intrinsic prior methodology, with suitably modifications to accommodate equality and inequality constraints. Focussing on normal ANOVA models, a comparative assessment is carried out through simulation studies, showing that correct model identification is possible even in situations where frequentist power is low. We also present an application to real data collected in a psychological experiment.