Default Bayes Factors for Testing the (In)equality of Several Population Variances

TL;DR

This paper introduces default Bayes factor tests for evaluating the equality or inequality of multiple population variances, providing a flexible and practical Bayesian approach for variance testing in various statistical contexts.

Contribution

The authors develop a novel default Bayes factor methodology for testing multiple variances, accommodating directed hypotheses, null regions, and complex hypotheses, with implementation in R.

Findings

The Bayes factor test performs well in practical examples.

The method allows for flexible hypotheses including inequalities and null regions.

Implementation is available in the R package bfvartest.

Abstract

Testing the (in)equality of variances is an important problem in many statistical applications. We develop default Bayes factor tests to assess the (in)equality of two or more population variances, as well as a test for whether the population variances equal a specific value. The resulting test can be used to check assumptions for commonly used procedures such as the $t$-test or ANOVA, or test substantive hypotheses concerning variances directly. We show that our Bayes factor fulfills a number of desiderata. Researchers may have directed hypotheses such as $\sigma_{1}^{2} > \sigma_{2}^{2}$, they may want to extend $\mathcal{H}_{0}$ to have a null-region, or wish to combine hypotheses about equality with hypotheses about inequality, for example $\sigma_{1}^{2} = \sigma_{2}^{2} > (\sigma_{3}^{2}, \sigma_{4}^{2})$. We extend our Bayes factor test to allow for these deviations from our proposed default and illustrate it on a number of practical examples. Our procedure is implemented in the R package $bfvartest$.