Notes on the interpretation of dependence measures

Besides the classical distinction of correlation and dependence, many dependence measures bear further pitfalls in their application and interpretation. The aim of this paper is to raise and recall awareness of some of these limitations by explicitly discussing Pearson's correlation and the multivariate dependence measures: distance correlation, distance multicorrelations and their copula versions. The discussed aspects include types of dependence, bias of empirical measures, influence of marginal distributions and dimensions. In general it is recommended to use a proper dependence measure instead of Pearson's correlation. Moreover, a measure which is distribution-free (at least in some sense) can help to avoid certain systematic errors. Nevertheless, in a truly multivariate setting only the p-values of the corresponding independence tests provide always values with indubitable interpretation.