An eigenfunction approach to multivariate and high-dimensional survival analysis

生存分析是对所谓的事件发生时间数据的研究，典型的多变量生存数据可以是并行数据，其中一个人试图观察几个相关个人(如家庭成员、人体器官、系统组件)发生感兴趣事件(如疾病发作、死亡、衰竭)的时间，或序列数据，其中跟踪一系列连续事件(如疾病的阶段)，或可能作为这两种方案的组合。事件间隔时间数据通常与高维协变量相结合，例如基因表达数据中可用遗传标记的信息。事件发生时间数据复杂性的两个主要来源是随机审查和随机截断，其中某些事件由于跟踪不足而不能被观察到，而随机截断是只有在阈值事件(例如研究开始时年龄之后发生的疾病)之后才能观察到的事件。 对于服从随机截尾和截尾的单变量生存数据(即，当只有一个个体或成分被跟踪时)，存在两个著名的、至今仍是经典的估计过程，分别称为Kaplan-Meier估计和Lynden-Bell估计。这些估计是广泛适用的，因为它们是在没有任何参数模型假设的情况下得出的。此外，基于这些估计值，可以开发一系列此类数据的统计程序，例如研究协变量对事件发生时间的影响。不幸的是，到目前为止还没有令人满意的后者的多变量版本。因此，统计学家被迫依赖限制性模型假设来分析删减或截断的多变量生存数据。例如，在协变量不受删减或截断的假设下，研究了协变量对生存时间的影响。 最近，我和我的合著者构造了随机截尾下的多元Kaplan-Meier估计，它具有单变量Kaplan-Meier估计的所有实用和理论性质。事实上，我们的方法是无量纲的，因为人们需要在任何维度上求解相同的方程，并且当应用于单变量数据时，它产生经典的单变量估计。本课题的主要目的正是要将这一过程推广到多元截尾和更复杂的半截尾模型，这两个模型在生存研究中都是非常重要的。就像单变量Kaplan-Meier和Lynden-Bell估计量一样，我们的新方法也有可能为多变量生存数据产生丰富的推断过程，我们建议对它们进行系统的探索。特别是，我们将在比目前所尝试的更一般的设置中研究高维协变量。除了开辟了一条新的研究路线，我们的项目还为培养研究生提供了巨大的可能性。

Survival analysis is the study of so-called time-to-event data, and typically multivariate survival data arise either as parallel data, where one tries to observe times to events of interest (such as disease onset, death, failure) for several related individuals (such as family members, organs of human body, components of a system), or serial data, where a series of successive events are tracked (such as stages of a disease), or possibly as a combination of these two schemes. Often time to event data is combined with high-dimensional covariates, such as information on genetic markers available from gene expression data. Two major sources of complication in time to event data are random censoring, whereby certain events cannot be observed due to insufficient followup, and random truncation, whereby an event can be observed only if it occurs after a threshold event (such as a disease occuring after age at beginning of study). For univariate survival data (i.e., when only one individual or component is being tracked) subject to random censoring and truncation, there exist two famous, and by now classical, estimation procedures called Kaplan-Meier and Lynden-Bell estimators, respectively. These estimates are widely applicable since they were derived without any parametric model assumption. Moreover, a host of statistical procedures for such data, such as studying effects of covariates on the time to event, can be developed based on these estimators. Unfortunately, no satisfactory multivariate versions of the latter were available until now. Consequently, statisticians were forced to rely on restrictive model assumptions in order to analyze censored or truncated multivariate survival data. For instance, effects of covariates on survival times were studied under the assumption that the former were not subject to censoring or truncation. Recently, my co-author and I have constructed a multivariate Kaplan-Meier estimator under random censoring, which shares all the nice practical and theoretical properties of its univariate counterpart. In fact, our method is dimension-free, in that one needs to solve the same equation in any dimension, and it produces the classical univariate estimator when applied to univariate data. The main goal of the present project is precisely to extend the procedure to multivariate censoring-cum-truncation and a more complicated semi-censoring model, both of which are of great importance in survival studies. Just like the univariate Kaplan-Meier and Lynden-Bell estimators our new method too has the potential to generate a wealth of inference procedures for multivariate survival data, and we propose to explore them systematically. In particular, we shall study high-dimensional covariates in a much more general set-up than has been attempted so far. In addition to opening up a new line of research, our project has immense possibilities for training graduate students.