An association measure for mixed-type variables

Quantifying the association between a real-valued variable and a categorical variable is a fundamental task in data analysis.

Existing methods often rely on parametric assumptions or arbitrary integer encoding, which may lead to unstable results.

We propose a label-invariant population measure of association, $\xi'$, specifically designed for the mixed real-valued-categorical setting.

The proposed measure is normalized between 0 and 1; it equals 0 if and only if the variables are independent and 1 if and only if the categorical variable is a measurable function of the real-valued one.

We also introduce a corresponding sample estimator, $\xi_n'$, computable in $O(n \log n)$ time.

These measures are invariant to permutations of category labels and strictly monotone transformations of the real-valued variable.

We establish the strong consistency and asymptotic normality of the estimator $\xi_n'$, enabling a computationally efficient, permutation-free Wald test for independence, and an asymptotic confidence interval for the population measure $\xi'$.

Extensive simulations and an application to The Cancer Genome Atlas (TCGA) data demonstrate that the proposed method provides coding stability, competitive power, and substantial computational advantages in nominal mixed-type settings.