Bayes factors for testing order-constrained hypotheses on correlations

Correlation coefficients play a key role in the social and behavioral sciences for quantifying the degree of linear association between variables. A Bayes factor is proposed that allows researchers to test hypotheses with order constraints on correlation coefficients in a direct manner. This Bayes factor balances between fit and complexity of order-constrained hypotheses in a natural way. A diffuse prior on the correlation matrix is used that minimizes prior shrinkage and results in most evidence for an order-constrained hypothesis that is supported by the data. An efficient method is proposed for the computation of the Bayes factor. A key aspect in the computation is a Fisher Z transformation on the posterior distribution of the correlations such that an approximately normal distribution is obtained. The methodology is implemented in a freely downloadable software program called “BOCOR”. The methods are applied to a multitrait-multimethod analysis, a repeated measures study, and a study on directed moderator effects.