Analysis of the Proportional Hazards Model With Sparse Longitudinal Covariates

Regression analysis of censored failure observations via the proportional hazards model permits time-varying covariates that are observed at death times. In practice, such longitudinal covariates are typically sparse and only measured at infrequent and irregularly spaced follow-up times. Full likelihood analyses of joint models for longitudinal and survival data impose stringent modeling assumptions that are difficult to verify in practice and that are complicated both inferentially and computationally. In this article, a simple kernel weighted score function is proposed with minimal assumptions. Two scenarios are considered: half kernel estimation in which observation ceases at the time of the event and full kernel estimation for data where observation may continue after the event, as with recurrent events data. It is established that these estimators are consistent and asymptotically normal. However, they converge at rates that are slower than the parametric rates that may be achieved with fully observed covariates, with the full kernel method achieving an optimal convergence rate that is superior to that of the half kernel method. Simulation results demonstrate that the large sample approximations are adequate for practical use and may yield improved performance relative to last value carried forward approach and joint modeling method. The analysis of the data from a cardiac arrest study demonstrates the utility of the proposed methods. Supplementary materials for this article are available online.

基于比例风险模型（Proportional Hazards Model）开展删失失效观测的回归分析，可纳入死亡时刻观测得到的时变协变量（time-varying covariates）。实际应用中，这类纵向协变量（longitudinal covariates）通常较为稀疏，仅在频次稀少且间隔不规则的随访时点进行测量。针对纵向与生存数据联合模型（joint models for longitudinal and survival data）的全似然分析，往往需要施加严苛的建模假设——此类假设不仅在实际中难以验证，且在推断与计算层面均较为复杂。本文提出一种仅需极弱假设的简洁核加权得分函数（kernel weighted score function）。本文考虑两种分析场景：其一为半核估计（half kernel estimation），即观测在事件发生时刻终止；其二为全核估计（full kernel estimation），适用于事件发生后仍可继续观测的场景，例如复发事件数据（recurrent events data）。研究证明，此类估计量具有相合性与渐近正态性。但与完全观测协变量下可获得的参数化收敛速率相比，其收敛速率更慢；其中全核估计法可达到优于半核估计法的最优收敛速率。模拟结果表明，大样本近似方法足以满足实际应用需求，且相较于末次观测值结转法（last value carried forward）与联合建模法（joint modeling method），所提方法可实现更优的性能表现。对一项心脏骤停研究（cardiac arrest study）的数据分析验证了所提方法的实用性。本文补充材料可在线获取。