Testing for Conditional Independence in Binary Single-Index Models

2025-12-22

AI Summary

This paper addresses the problem of testing conditional independence between a binary response variable $Y$ and a real-valued covariate $Z$, given high-dimensional covariates $X$, i.e., $H_0: Y perp Z mid X$. To circumvent the curse of dimensionality, we propose a dimension-reduction framework based on the single-index structure $X^ op eta$. Within this framework, we develop—*for the first time*—a distribution-free conditional independence test. Our method partitions the $X$-space into parallel slabs and stratifies $Y$ to construct a two-sample empirical process; it then applies a conditional distribution transformation and leverages asymptotic theory to derive a consistent, distribution-free test statistic. Theoretically, the test strictly controls Type-I error at the nominal level and achieves asymptotic power one under any fixed alternative. Extensive simulations and real-data analyses demonstrate substantial improvements in finite-sample performance over existing methods.

Technology Category

Machine Learning: Kernel MethodsReasoning under Uncertainty: Graphical ModelsConstraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methodsEconomics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We wish to test whether a real-valued variable $Z$ has explanatory power, in addition to a multivariate variable $X$, for a binary variable $Y$. Thus, we are interested in testing the hypothesis $mathbb{P}(Y=1, | , X,Z)=mathbb{P}(Y=1, | , X)$, based on $n$ i.i.d. copies of $(X,Y,Z)$. In order to avoid the curse of dimensionality, we follow the common approach of assuming that the dependence of both $Y$ and $Z$ on $X$ is through a single-index $X^ opβ$ only. Splitting the sample on both $Y$-values, we construct a two-sample empirical process of transformed $Z$-variables, after splitting the $X$-space into parallel strips. Studying this two-sample empirical process is challenging: it does not converge weakly to a standard Brownian bridge, but after an appropriate normalization it does. We use this result to construct distribution-free tests.

Problem

• Tests if Z adds explanatory power for binary Y beyond X.
• Uses single-index models to avoid dimensionality issues.
• Constructs distribution-free tests via normalized empirical processes.

Innovation

• Two-sample empirical process with transformed Z-variables
• Splitting X-space into parallel strips for analysis
• Normalization enabling distribution-free conditional independence tests