Bayesian t-tests for correlations and partial correlations

Abstract

In this paper, we develop Bayes factor based testing procedures for the presence of a correlation or a partial correlation. The proposed Bayesian tests are obtained by restricting the class of the alternative hypotheses to maximize the probability of rejecting the null hypothesis when the Bayes factor is larger than a specified threshold. It turns out that they depend simply on the frequentist t-statistics with the associated critical values and can thus be easily calculated by using a spreadsheet in Excel and in fact by just adding one more step after one has performed the frequentist correlation tests. In addition, they are able to yield an identical decision with the frequentist paradigm, provided that the evidence threshold of the Bayesian tests is determined by the significance level of the frequentist paradigm. We illustrate the performance of the proposed procedures through simulated and real-data examples.

Keywords:

• Bayes factor
• restricted most powerful Bayesian tests
• statistical evidence
• t-test
• Zellner's g-prior

Acknowledgments

The authors thank the Editor, an associate editor, and the two reviewers for their detailed and constructive comments, which have led to a significant improvement of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.