Assessment of τ * in Partially Linear Logistic Regression with Missing and Complete Data

Abstract

Generalized partially linear models (GPLMs) provide a versatile regression framework that blends parametric and nonparametric components, allowing flexible modeling of complex data structures. In binary response settings, particularly within logistic frameworks, verifying the independence between covariates and error terms is essential for ensuring model adequacy and validity. This paper develops a nonparametric diagnostic based on the Bergsma–Dassios measure of association, ${\tau }^{*}$ , to assess the independence between the regressors $(\mathbf{X},\mathbf{W})$ and the random error component in logistic GPLMs. Unlike traditional correlation measures, ${\tau }^{*}$ captures broad classes of dependencies, including nonlinear and nonmonotonic associations, thus offering a powerful and robust diagnostic tool. Both complete data and missing-response scenarios are considered, where responses are missing completely at random (MCAR) or missing at random (MAR). Consistent and asymptotically efficient estimators for the parametric vector $\mathit{\beta }$ and the nonparametric function $m(\mathbf{W})$ are constructed under these settings. Theoretical properties of the proposed ${\tau }^{*}$ -based test are established, including its asymptotic distribution and power against local alternatives. Simulation studies and real-data analyses further confirm the practical effectiveness and robustness of the proposed method, demonstrating its utility in semiparametric logistic regression with incomplete or potentially misspecified data.