Measures of dependence between random vectors and tests of independence.

Summary The simple correlation coefficient between two variables has been generalized to measures of association between two matrices several times. Coefficients such as the RV coefficient, the distance covariance (dCov) coefficient or the Hilbert Schmidt Information Criterion (HSIC) have all been adopted by different communities. Scientists also use tests to measure whether two random variables are linked and then interpret the coefficients in context. many branches of science currently need multiway measures of association. The aim of this paper is to provide a small state of the art on the topic of measures of dependence between random vectors and tests of independence and to show links between different approaches. We document some of the interesting rediscoveries and lack of interconnection between bodies of literature. This review starts with a short history of randomization tests using distance matrices and some motivating examples. We then provide definition of the coefficients and associated tests. Finally we review a few of the recent modifications that have been proposed that provide improved properties and enhance ease of interpretation, as well as some prospective directions for future research.