Proper Correlation Coefficients for Nominal Random Variables

Abstract: This paper develops an intuitive concept of perfect dependence between two variables of which at least one has a nominal scale that is attainable for all marginal distributions and proposes a set of dependence measures that are 1 if and only if this perfect dependence is satisfied. The advantages of these dependence measures relative to classical dependence measures like contingency coefficients, Goodman-Kruskal's lambda and tau and the so-called uncertainty coefficient are twofold. Firstly, they are defined if one of the variables is real-valued and exhibits continuities. Secondly, they satisfy the property of attainability. That is, they can take all values in the interval [0,1] irrespective of the marginals involved. Both properties are not shared by the classical dependence measures which need two discrete marginal distributions and can in some situations yield values close to 0 even though the dependence is strong or even perfect. Additionally, I provide a consistent estimator for one of the new dependence measures together with its asymptotic distribution under independence as well as in the general case. This allows to construct confidence intervals and an independence test, whose finite sample performance I subsequently examine in a simulation study. Finally, I illustrate the use of the new dependence measure in two applications on the dependence between the variables country and income or country and religion, respectively.