Comparing crossing hazard rate functions by joint modelling of survival and longitudinal data

It is one of the important issues in survival analysis to compare two hazard rate functions to evaluate treatment effect. It is quite common that the two hazard rate functions cross each other at one or more unknown time points, representing temporal changes of the treatment effect. In certain applications, besides survival data, we also have related longitudinal data available regarding some time-dependent covariates. In such cases, a joint model that accommodates both types of data can allow us to infer the association between the survival and longitudinal data and to assess the treatment effect better. In this paper, we propose a modelling approach for comparing two crossing hazard rate functions by joint modelling survival and longitudinal data. Maximum likelihood estimation is used in estimating the parameters of the proposed joint model using the EM algorithm. Asymptotic properties of the maximum likelihood estimators are studied. To illustrate the virtues of the proposed method, we compare the performance of the proposed method with several existing methods in a simulation study. Our proposed method is also demonstrated using a real dataset obtained from an HIV clinical trial.比较两个风险率函数（hazard rate function）以评估治疗效果，是生存分析（survival analysis）中的重要议题之一。当两个风险率函数在一个或多个未知时间点相交时，这种情况十分常见，其反映了治疗效果的时序变化。在某些应用场景中，除生存数据外，我们还可获取与某些时变协变量（time-dependent covariates）相关的纵向数据（longitudinal data）。在此类情形下，同时兼容两类数据的联合模型（joint model），能够帮助我们推断生存数据与纵向数据间的关联，并更精准地评估治疗效果。本文提出一种建模方法，通过联合建模生存数据与纵向数据，以实现两个相交风险率函数的比较。本文所提联合模型的参数估计采用极大似然估计（maximum likelihood estimation）法，并借助EM算法完成参数求解。本文还研究了极大似然估计量的渐近性质（asymptotic properties）。为验证所提方法的优势，我们通过仿真实验（simulation study）将本文方法与多种现有方法的性能进行了对比。此外，我们利用一项HIV临床试验（HIV clinical trial）的真实数据集，对本文所提方法进行了实例演示。