Title: Bayesian Evaluation of Equality and Inequality Constrained
Hypotheses on Variances

Authors: Florian Boeing-Messing, Marcel A. L. M. van Assen, Joris Mulder,
Herbert Hoijtink

Affiliation: Utrecht University

Abstract: Researchers are frequently interested in comparing the
heterogeneity of multiple independent populations.  For example, suppose we
would like to compare the heterogeneity of educational performances of
students of age 8, 10, and 14.  We might expect the 8-year-olds to be less
heterogeneous than the 10-year-olds, who we expect to be less heterogeneous
than the 14-year-olds.  Alternatively, we might expect students of age 8
and 10 to be equally heterogeneous, but less heterogeneous than students of
age 14.  Such expectations can be translated into equality and inequality
constrained hypotheses on variances.  In this article we consider a
Bayesian approach to testing such (in)equality constrained hypotheses using
Bayes factors.  The latter have two distinct strengths: First, using Bayes
factors, it is straightforward to simultaneously test multiple (non)nested
hypotheses.  Second, Bayes factors function as Occam’s razor by
automatically balancing fit and complexity of (in)equality constrained
hypotheses.  Our method is highly flexible in that it can handle hypotheses
with arbitrary equality and inequality constraints on J > 2 variances.  We
provide an R program for computing the Bayes factors.

Keywords: Bayes factor, homogeneity of variance, inequality constrained
hypothesis, posterior model probability, prior distribution