Proper Correlation Coefficients for Nominal Random Variables

TL;DR

This paper introduces a new notion of perfect dependence for pairs of variables where at least one is nominal, ensuring attainability irrespective of marginals. It defines proper dependence measures built from permutations of nominal categories, with a focus on a gamma-based coefficient $\gamma^{*}$ that attains 1 if and only if perfect dependence holds (in the nominal sense). The authors develop a consistent estimator for the measure and derive its non-standard, permutation-based asymptotic distribution, enabling confidence intervals and an independence test, complemented by a simulation study and two real-data applications (countries with income and with religion). The approach addresses limitations of classical measures (e.g., Cramér's V, GK-lambda/tau, and the uncertainty coefficient) by handling nominal–continuous cases and guaranteeing attainability, offering a practical and theoretically grounded toolkit for dependence assessment in nominal settings.

Abstract

This paper develops an intuitive concept of perfect dependence between two variables of which at least one has a nominal scale. Perfect dependence is attainable for all marginal distributions. It furthermore 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 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 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, the paper provides 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 with good finite sample properties, as a subsequent simulation study shows. Finally, two applications on the dependence between the variables country and income, and country and religion, respectively, illustrate the use of the new measure.

Figures (5)

• Figure 1: This figure shows empirical confidence interval coverage rates (nominal level: $0.9$) and mean bias plots for $\gamma^*$ with $MC = 1,000$ simulation replications. The sample sizes $n\in \{50, 200, 800\}$ are given in the column and the DGPs are given in the row descriptions, respectively. The x-axis displays the true values of $\gamma^*$ that arise for a specific choice of $\alpha$.
• Figure 2: This figure shows p-value histograms that arise either from the newly introduced testing scheme (left subplot) or from traditional testing approaches (right subplot). It considers three different sample sizes and six different DGPs which are explained in Subsection \ref{['subsec:DGPs']}. For all DGPs, sampling happens under independence $(\alpha = 0)$ with 1,000 Monte Carlo repetitions.
• Figure 3: This figure shows $\widehat{\gamma}^*$ for the comparison between the variables country and income chosen from the Luxembourg Income Study (LIS) Database (2011 PPP).
• Figure 4: This figure shows Cramér's $V$ (left plot) and $\gamma^*$ (right plot) for the dependence between the variables country and religion. Each colored dot represents the value of the respective coefficient if the variable country is restricted to the three countries featuring in the border triangle at which the dot is located. The variable religion is always restricted to the three monotheistic world religions Christianity, Islam and Judaism.
• Figure E.1: This figure shows $PC$ (upper left plot), $\tau$ (upper right plot), $\lambda$ (lower left plot) and $U$ (lower right plot) for the dependence between the variables country and religion. Each colored dot represents the value of the respective coefficient if the variable country is restricted to the three countries featuring in the border triangle at which the dot is located. The variable religion is always restricted to the three monotheistic world religions Christianity, Islam and Judaism.

Theorems & Definitions (42)

• Definition 2.1: Perfect Dependence
• Lemma 2.2
• Definition 3.1: Dependence Measure
• Definition 3.2: Proper Dependence Measure
• Lemma 4.1: $MSC$ attainability
• Lemma 4.2: attainability
• Lemma 4.3: dependence concepts
• Definition 5.1: Proper Measure
• Proposition 5.2
• Proposition 5.3