Fused Multinomial Logistic Regression Utilizing Summary-Level External Machine-learning Information

TL;DR

This paper introduces a novel empirical-likelihood framework that leverages external machine-learning predictions to enhance multinomial logistic regression inference, addressing data-quality issues and demonstrating efficiency gains.

Contribution

It develops a robust method to incorporate summary-level external predictions into primary analysis, improving inference without requiring explicit density-ratio modeling.

Findings

The fused estimator is consistent and asymptotically normal.

Incorporating external predictions can yield efficiency gains.

The method handles coarsened data, covariate shift, and concept shift.

Abstract

In many modern applications, a carefully designed primary study provides individual-level data for interpretable modeling, while summary-level external information is available through black-box, efficient, and nonparametric machine-learning predictions. Although summary-level external information has been studied in the data integration literature, there is limited methodology for leveraging external nonparametric machine-learning predictions to improve statistical inference in the primary study. We propose a general empirical-likelihood framework that incorporates external predictions through moment constraints. An advantage of nonparametric machine-learning prediction is that it induces a rich class of valid moment restrictions that remain robust to covariate shift under a mild overlap condition without requiring explicit density-ratio modeling. We focus on multinomial logistic regression as the primary model and address common data-quality issues in external sources, including coarsened outcomes, partially observed covariates, covariate shift, and heterogeneity in generating mechanisms known as concept shift. We establish large-sample properties of the resulting fused estimator, including consistency and asymptotic normality under regularity conditions. Moreover, we provide mild sufficient conditions under which incorporating external predictions delivers a strict efficiency gain relative to the primary-only estimator. Simulation studies and an application to the National Health and Nutrition Examination Survey on multiclass blood-pressure classification.