In nonparametric regression, understanding how specific covariates Z influence a response variable Y is crucial, particularly when this effect varies with certain characteristic variables S, such as age in physiological studies. We refer to this varying effect on the response distribution as “distributional heterogeneous variable importance”. To quantify it, we introduce a measure of the dependence between Y and Z given additional covariates that need to be controlled, evaluated locally at a specific value S = s. To our knowledge, such a localized notion of dependence has not been explored in the literature. Conceptually, our measure generalizes several widely used measures of variable association, offering a unified framework for studying variable relationships across diverse settings. Based on the K-nearest neighbors algorithm, we propose a nonparametric estimator for our measure, featuring a simple and intuitive form. We establish its consistency and derive its convergence rate. Building on this estimator, we develop a forward selection procedure to identify covariates that are locally relevant to Y at S = s, and prove its selection consistency. Extensive simulation studies demonstrate strong finite-sample performance, while two real data applications yield scientifically meaningful findings that highlight the practical utility of our framework.
