On the proportional hazards model with last observation carried forward covariates

Standard partial likelihood methodology for the proportional hazards model with time-dependent covariates requires knowledge of the covariates at the observed failure times, which is not realistic in practice. A simple and commonly used estimator imputes the most recently observed covariate prior to each failure time, which is known to be biased. In this paper, we show that a weighted last observation carried forward approach may yield valid estimation. We establish the consistency and asymptotic normality of the weighted partial likelihood estimators and provide a closed form variance estimator for inference. The estimator may be conveniently implemented using standard software. Interestingly, the convergence rate of the estimator is slower than the parametric rate achieved with fully observed covariates but the same as that obtained with all lagged covariate values. Simulation studies provide numerical support for the theoretical findings. Data from an Alzheimer's study illustrate the practical utility of the methodology.

Automated summary

This paper introduces a weighted partial likelihood estimation method for handling time-dependent covariates in the proportional hazards model, demonstrating consistency and asymptotic normality in theory and providing a closed form variance estimator for inference conveniently implemented using standard software. The convergence rate is shown to be slower than fully observed covariates but equivalent to that of all lagged covariate values, with simulation studies supporting the theoretical findings. Data from an Alzheimer's study demonstrates the practical utility of the methodology.