Measures of dependence between random vectors and tests of independence. Literature review

Simple correlation coefficients between two variables have been generalized to measure association between two matrices in many ways. Coefficients such as the RV coefficient, the distance covariance (dCov) coefficient and kernel based coefficients have been adopted by different research communities. Scientists use these coefficients to test whether two random vectors are linked. If they are, it is important to uncover what patterns exist in these associations. We discuss the topic of measures of dependence between random vectors and tests of independence and show links between different approaches. We document some of the interesting rediscoveries and lack of interconnection between bodies of literature. After providing definitions of the coefficients and associated tests, we present the recent improvements that enhance their statistical properties and ease of interpretation. We summarize multi-table approaches and provide scenarii where the indices can provide useful summaries of heterogeneous multi-block data. We illustrate these different strategies on several examples of real data and suggest directions for future research.