Information Borrowing for Cox Regression with an Auxiliary Outcome

2026-09-27

AI Summary

This study addresses how to effectively leverage auxiliary outcome information to enhance the estimation efficiency of Cox regression in survival analysis. To this end, it proposes a joint-likelihood-free semiparametric framework built upon the Cox proportional hazards and partial linear models. This approach facilitates cross-source information borrowing by linking baseline covariate effects through a quadratic penalty, while an adaptive penalization mechanism is designed to optimize the bias–variance trade-off. The asymptotic properties of the proposed estimator are rigorously derived. Both simulation studies and real-data applications demonstrate that the method substantially improves estimation efficiency while preserving unbiasedness.

Abstract

Time-to-event outcomes are often collected together with auxiliary outcomes that may provide additional information about baseline risk. We develop a semiparametric approach for incorporating such information into Cox regression without specifying a joint likelihood. The survival outcome follows a Cox proportional hazards model and a continuous auxiliary outcome follows a partially linear model, with separate nonparametric baseline covariate effects linked through a quadratic penalty. When these effects coincide and the outcome scores satisfy the information identities and a first-order orthogonality condition, we derive the sandwich covariance of the penalized estimator and show that, for any fixed penalty level, the asymptotic variance of the Cox regression estimator is no greater than that under separate estimation. We further characterize departures from the shared-effect setting. Local differences of order \(n^{-1/2}\) induce an explicit mean shift in the limiting distribution, yielding a direct bias--variance trade-off, whereas fixed differences generally alter the population target under nonvanishing penalization. These results motivate an adaptive penalty that borrows information when the fitted covariate effects are close and approaches separate estimation when a persistent difference is detected. Simulations and a real world data analysis demonstrate the validity and effectiveness of the proposed method.

Problem

• Cox regression
• auxiliary outcome
• information borrowing
• semiparametric approach
• penalized estimation

Innovation

• Cox regression
• auxiliary outcome
• semiparametric approach
• quadratic penalty
• adaptive penalty