2026/07/29 by Yongjae Kim, Haeun Moon, Sungkyu Jung
Mathematics · #math.ST #msc:62G10 #msc:62G30 #msc:62H20 #stat.ME #stat.TH
82 pages, 17 figures. Submitted to the Electronic Journal of Statistics
arxiv created 2026/07/29 · arxiv updated 2026/07/30
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, ξ', 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, ξn', 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 ξn', enabling a computationally efficient, permutation-free Wald test for independence, and an asymptotic confidence interval for the population measure ξ'. 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.