Proportional Hazards Model for Various Types of Censored Survival Data with Longitudinal Covariates

时间到事件（生存时间）变量和纵向协变量之间的相互关系往往是医学和流行病学研究的主要研究兴趣。由于在艾滋病和癌症研究的一些重要临床试验中遇到的挑战，最近统计学家开始通过Cox比例风险模型对生存数据和纵向数据进行联合建模。这种联合建模程序或方法在许多科学研究领域有广泛的应用，但由于对生存时间的审查和协变量过程仅在某些给定的时间点观察到，因此是一个相当困难的问题。到目前为止，关于这一主题的统计方法还没有完全或很好地发展起来，而当提出者和她的合作者最近遇到一些统计文献中尚未研究的更复杂的问题时，发展这些方法的重要性和必要性变得更加明显；请参阅下面列出的示例。具体而言，目前还没有任何直接研究生存时间与纵向协变量的受试者内历史变化模式之间关系的建模程序，也没有任何与（密集或多相密集）纵向协变量联合建模双删减或间隔删减生存数据的工作，这比正确删减数据问题更具挑战性。事实上，目前还没有关于双重审查数据的Cox模型的出版作品，甚至没有关于具有时间独立协变量的情况。本研究将主要采用渐近方法和模拟方法进行研究，考虑的问题包括：(a)分别为右截尾、双截尾和间隔截尾生存数据导出纵向协变量Cox模型的基于经验似然的最大似然；(b)最大似然值的计算算法；(c)最大似然函数的渐近性质；(d)最大似然值的Wilk定理；(e) Cox模型的拟合优度检验；(f)与备选方法的比较。至少有两名博士生参与并受益于拟进行的研究。在这个项目中发展的新的统计方法对医学研究、流行病学、社会和行为科学等有直接影响。例如，我们遇到并激励本项目研究的数据示例包括以下关于联合建模生存时间和纵向协变量的问题。在一项小鼠前列腺癌研究中，部分研究重点是联合建模间隔剔除生存时间和纵向协变量。在一项戒烟研究中，研究的重点是对删节生存时间和密集纵向协变量的联合建模。在最近的儿童发展研究中，研究重点是双截尾生存时间和多相密集纵向协变量的联合建模。

The interrelationships between time-to-event (survival time) variable and longitudinal covariates is often the primary research interest in medical and epidemiological studies. Due to the challenges encountered in some important clinical trials on AIDS and cancer research, recently the statisticians started modeling survival data and longitudinal data jointly via Cox's proportional hazards model. Such a joint modeling procedure or methodology has broad applications in many scientific research fields, but it is a considerably difficult problem due to censoring on the survival time and that the covariate process is only observed at some given time points. Up to now, statistical methods on this topic have not been fully or well developed, while the importance and needs for developing these methods have become more evident when the proposer and her collaborators recently encountered some more complicated problems which have not been studied in statistical literature; see examples listed below. Specifically, there have not been any modeling procedures that directly study the relationship between survival time and within-subject historic patterns of change in longitudinal covariates, nor have there been any works on joint modeling doubly censored or interval censored survival data together with (intensive or multi-phase intensive) longitudinal covariates, which is far more challenging than right censored data problem. In fact, there have been no published works on the Cox model with doubly censored data, not even for the case with time-independent covariates. In this research, asymptotic methods and simulations will be mainly used in the studies, and the issues under consideration include: (a) derivation of the empirical likelihood based MLE for the Cox model with longitudinal covariates for right censored, doubly censored and interval censored survival data, respectively; (b) computation algorithms for the MLE; (c) asymptotic properties of the MLE; (d) Wilk's theorem for the MLE; (e) goodness-of-fit tests for the Cox model; (f) comparison with alternative methods. At least two Ph.D. students of the proposer will be involved in and benefit from the proposed research. The new statistical methodology to be developed in this project has direct impact to medical research, epidemiology, social and behavioral sciences, etc. For instance, the data examples which we have encountered and motivate the research of this project include the following problems on joint modeling survival time and longitudinal covariates. In a prostate cancer study on mice, part of the research focus is joint modeling interval censored survival time and longitudinal covariates. In a smoking cessation study, the research focus is joint modeling right censored survival time and intensive longitudinal covariates. In a recent study of child development, the research focus is joint modeling doubly censored survival time and multi-phase intensive longitudinal covariates.