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.

中文翻译：

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上次观测结转协变量的比例风险模型

具有时间相关协变量的比例风险模型的标准部分似然方法需要了解观察到的故障时间的协变量，这在实践中是不现实的。一个简单且常用的估计器在每个故障时间之前估算最近观察到的协变量，这已知是有偏差的。在本文中，我们表明加权最后一次观察结转方法可能会产生有效的估计。我们建立了加权偏似然估计量的一致性和渐近正态性，并为推理提供了一个封闭形式的方差估计量。估计器可以使用标准软件方便地实现。有趣的是，估计器的收敛速度比使用完全观察到的协变量实现的参数速度慢，但与使用所有滞后协变量值获得的相同。模拟研究为理论发现提供了数值支持。来自阿尔茨海默氏症研究的数据说明了该方法的实际效用。