A simple consistent Bayes factor for testing the Kendall rank correlation coefficient

Abstract

In this paper, we propose a simple and easy-to-implement Bayesian hypothesis test for the presence of an association, described by Kendall's τ coefficient, between two variables measured on at least an ordinal scale. Owing to the absence of the likelihood functions for the data, we employ the asymptotic sampling distributions of the test statistic as the working likelihoods and then specify a truncated normal prior distribution on the noncentrality parameter of the alternative hypothesis, which results in the Bayes factor available in closed form in terms of the cumulative distribution function of the standard normal distribution. Investigating the asymptotic behaviour of the Bayes factor we find the conditions of the priors so that it is consistent to whichever the hypothesis is true. Simulation studies and a real-data application are used to illustrate the effectiveness of the proposed Bayes factor. It deserves mentioning that the proposed method can be easily covered in undergraduate and graduate courses in nonparametric statistics with an emphasis on students' Bayesian thinking for data analysis.

Keywords:

• Bayes factor
• consistency
• Kendall's τ
• nonparametric hypothesis testing
• prior elicitation

Acknowledgments

The authors greatly appreciate the reviewers' thoughtful comments and suggestions, which have significantly improved the quality of the manuscript. This work was based on the first author's dissertation research which was supervised by the corresponding author.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of Dr. Min Wang was partially supported by the Internal Research Awards (INTRA) program from the UTSA Vice President for Research, Economic Development, and Knowledge Enterprise.