Testing for Conditional Independence in Binary Single-Index Models

TL;DR

This paper develops a new statistical test to determine if a variable $Z$ adds explanatory power for a binary outcome $Y$ beyond a multivariate predictor $X$, using a single-index model to avoid high-dimensional issues.

Contribution

It introduces a novel distribution-free testing procedure for conditional independence in single-index models, addressing the challenge of non-standard empirical process convergence.

Findings

The proposed test effectively detects the additional explanatory power of $Z$.

The test is distribution-free and does not rely on parametric assumptions.

Simulation studies demonstrate the test's good finite-sample performance.

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 $X^\top\beta$ 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.