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

事件发生时间（生存时间）变量和纵向协变量之间的相互关系通常是医学和流行病学研究的主要研究兴趣。由于艾滋病和癌症研究中一些重要的临床试验所遇到的挑战，近年来统计学家们开始使用考克斯的比例风险模型对生存数据和纵向数据进行联合建模。这种联合建模过程或方法在许多科学研究领域中具有广泛的应用，但由于对生存时间的删失以及仅在某些给定时间点观察到协变量过程，因此这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得到充分或良好的发展，而发展这些方法的重要性和必要性已经变得更加明显，当提议者和她的合作者最近遇到了一些更复杂的问题，这些问题在统计文献中没有研究过;见下面列出的例子。具体而言，目前还没有任何建模程序，直接研究生存时间和纵向协变量的受试者内的历史变化模式之间的关系，也没有任何作品的联合建模双删失或区间删失生存数据与（密集或多阶段密集）纵向协变量，这是更具挑战性的右删失数据问题。事实上，目前还没有关于双删失数据下的考克斯模型的研究成果，甚至对于协变量与时间无关的情况也没有。本研究主要采用渐近方法和模拟方法，主要研究内容包括：（a）分别针对右删失、双删失和区间删失数据，推导了纵向协变量的考克斯模型的经验似然极大似然估计，（B）极大似然估计的计算算法，（c）极大似然估计的渐近性质，（d）极大似然估计的Wilk定理，（e）极大似然估计的渐近性质，（f）极大似然估计的渐近性质，（f）极大似然估计的渐近性质。（e）考克斯模型的拟合优度检验;（f）与其他方法的比较。至少两个博士学位建议者的学生将参与建议的研究并从中受益。新的统计方法将在这个项目中开发有直接的影响，医学研究，流行病学，社会和行为科学等，例如，我们遇到的数据的例子，并激励本项目的研究包括以下问题的联合建模生存时间和纵向协变量。在小鼠前列腺癌研究中，部分研究重点是联合建模区间删失生存时间和纵向协变量。在戒烟研究中，研究重点是联合建模右删失生存时间和密集纵向协变量。在最近的儿童发展研究中，研究热点是双截尾生存时间和多阶段强度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.