Generalized Local Kendall's τ: a novel framework for uncovering nonlinear local dependence

Identifying patient subgroups is increasingly vital for advancing personalized and precision medicine, where patient heterogeneity plays a central role. Subgroups can be revealed by examining how predictors depend on each other. However, traditional global dependence measures, such as Kendall's τ, often overlook localized and nonlinear relationships, potentially missing critical patterns that define subpopulations. Existing local dependence measures are typically constrained to linear structures or lack flexibility in capturing complex local behaviors. We propose a novel, intuitive, and highly flexible framework, the Generalized Local Kendall's τ, for assessing nonlinear local dependence. This method allows local regions to take diverse shapes (e.g., rectangles, squares, extreme-value neighborhoods), thereby unifying global, local, and tail dependence within a flexible and interpretable structure. The framework includes U-statistic-based and copula model-based estimators, introducing four new quantile dependence measures and novel tail dependence coefficients. We develop two visualization tools-Type I and Type II local Kendall's τ surfaces-applied to Archimedean copulas to aid interpretation. Comparisons show that global Kendall's τ often differs markedly from its local versions, emphasizing the added value of local analysis. Simulation results indicate that when the model is correctly specified, the copula-based estimator exhibits significantly superior performance. We illustrate the framework's practical utility using Parkinson's disease and COVID-19 datasets, showing its ability to detect subgroup-specific dependence patterns vital for precision medicine.