Discussion of ‘Multi-scale Fisher’s independence test for multivariate dependence’

This is the authors' abstract. We don't add key points for this paper.

We would like to congratulate the authors on an interesting paper that provides many avenues for discussion. The reduction of a nonparametric problem to a sequence of |2 × 2| contingency tables is very elegant, and Fisher’s exact test is used to great effect. There are now a large number of methods for multivariate independence testing, but the proposed procedure stands out by providing uniform Type I error controls without a large computational burden, combining the speed of an asymptotic test with the rigorous validity of a permutation test. Moreover, the ability of the test to provide information on the form of the possible dependence is a feature that should be appreciated in practice, and one that is worthy of further consideration. The problem of independence testing has received a great deal of attention throughout the history of statistics, and is a canonical example of a nonparametric testing problem where the null hypothesis is composite. Further, it is a problem where departures from the null hypothesis can take many forms. Beyond the obvious practical importance of independence testing, these two features of the problem are key reasons for the development of such a vast literature on the topic. In this discussion we describe how the new work fits in with this literature and outline criteria to consider when choosing an independence test.