A consistent test of independence based on a sign covariance related to Kendall’s tau

Abstract

The most popular ways to test for independence of two ordinal random variables are by means of Kendall’s tau and Spearman’s rho. However, such tests are not consistent, only having power for alternatives with “monotonic” association. In this paper, we introduce a natural extension of Kendall’s tau, called ${\tau }^{\ast }$, which is non-negative and zero if and only if independence holds, thus leading to a consistent independence test. Furthermore, normalization gives a rank correlation which can be used as a measure of dependence, taking values between zero and one. A comparison with alternative measures of dependence for ordinal random variables is given, and it is shown that, in a well-defined sense, ${\tau }^{\ast }$ is the simplest, similarly to Kendall’s tau being the simplest of ordinal measures of monotone association. Simulation studies show our test compares well with the alternatives in terms of average $p$-values.