Dupuy Jean-François, Mesbah Mounir
Laboratoire de Statistiques Appliquées, l'Université de Bretagne-Sud (Sabres), 56000 Vannes, France.
Lifetime Data Anal. 2002 Jun;8(2):99-115. doi: 10.1023/a:1014871806118.
Survival studies usually collect on each participant, both duration until some terminal event and repeated measures of a time-dependent covariate. Such a covariate is referred to as an internal time-dependent covariate. Usually, some subjects drop out of the study before occurrence of the terminal event of interest. One may then wish to evaluate the relationship between time to dropout and the internal covariate. The Cox model is a standard framework for that purpose. Here, we address this problem in situations where the value of the covariate at dropout is unobserved. We suggest a joint model which combines a first-order Markov model for the longitudinally measured covariate with a time-dependent Cox model for the dropout process. We consider maximum likelihood estimation in this model and show how estimation can be carried out via the EM-algorithm. We state that the suggested joint model may have applications in the context of longitudinal data with nonignorable dropout. Indeed, it can be viewed as generalizing Diggle and Kenward's model (1994) to situations where dropout may occur at any point in time and may be censored. Hence we apply both models and compare their results on a data set concerning longitudinal measurements among patients in a cancer clinical trial.
生存研究通常会收集每个参与者直到某个终点事件的持续时间以及一个随时间变化的协变量的重复测量值。这样的协变量被称为内部随时间变化的协变量。通常，一些受试者在感兴趣的终点事件发生之前就退出了研究。于是人们可能希望评估退出时间与内部协变量之间的关系。Cox模型就是用于此目的的一个标准框架。在此，我们针对协变量在退出时的值未被观测到的情况来解决这个问题。我们提出一个联合模型，该模型将用于纵向测量协变量的一阶马尔可夫模型与用于退出过程的随时间变化的Cox模型相结合。我们考虑此模型中的最大似然估计，并展示如何通过期望最大化（EM）算法进行估计。我们指出所建议的联合模型可能在具有不可忽略的退出情况的纵向数据背景中有应用。实际上，它可以被视为将迪格勒（Diggle）和肯沃德（Kenward）的模型（1994年）推广到退出可能在任何时间点发生且可能被截尾的情况。因此，我们应用这两个模型，并在一个关于癌症临床试验中患者纵向测量的数据集上比较它们的结果。
