A0218
Title: Proper correlation coefficients for discrete random variables
Authors: Jan-Lukas Wermuth - Goethe University Frankfurt (Germany) [presenting]
Marc-Oliver Pohle - Heidelberg Institute for Theoretical Studies (Germany)
Abstract: Contrary to Pearson correlation, Kendalls Tau and Spearmans Rho fulfill all important desirable properties for directed dependence measures, at least for continuous random variables. In the discrete case, however, they lose the property of attainability, meaning that they do not attain the values -1 and 1 under perfect negative and positive dependence and often severely understate the strength of dependence, impeding their usefulness as dependence measures. We show that their widely-used generalizations for the discrete case, Tau-B and Grade Correlation, can, to a certain extent, mitigate the severity of the attainability problem but are still non-attainable. We discuss attainable versions of rank correlations. For Spearmans Rho, we look into its several population definitions and demonstrate that an attainable version of it comes at the price of other shortcomings. For Kendalls Tau, on the contrary, we show that Goodman-Kruskals Gamma is a theoretically appealing and simple attainable generalization, which we consequently recommend as a suitable general-purpose dependence measure. We discuss further attainable versions of Tau and Rho and introduce an attainable version of Blomqvists Beta. In theoretical and empirical examples, we analyze and illustrate the attainability problem of classical correlation measures and how it is solved by attainable generalizations.
