Sufficient Dimension Reduction for Missing, Censored, and Correlated Data

这一建议旨在为充分降维(SDR)发展新的理论和方法。特别是，研究集中在生物统计学问题上，通常包括缺失数据、生存数据和纵向数据分析。该方法可以有效地将高维回归问题转化为低维投影，保留完整的回归信息，并能施加很少的或不施加概率模型。这项研究包括三个部分。首先，当预报器有缺失观测值时，研究者提出了一类增广逆概率加权SDR估值器。这种新方法允许比现有解决方案更通用的缺失数据机制，并允许超越同方差线性模型的更灵活的回归形式。研究的第二部分针对生存数据的SDR，其中对死亡或疾病复发的时间的反应受到审查。将截尾响应视为一种特定类型的缺失数据，调查者将逆概率加权估计策略与各种SDR方法相结合。第三，研究人员研究了一种纵向数据，其中所有研究对象的测量都是在相同的预定时间点收集的。建立了总体基础和相关的估计程序。这三个组成部分都围绕着常见的生物统计学问题展开，三个组成部分的发展是相互关联的。现代技术以产生和收集海量、高维数据的能力推动了科学的前沿。大量高维数据集的例子出现在许多研究领域，如环境研究、人类健康和医学研究以及国土安全。充分降维(SDR)方法有效地将高维数据问题转化为低维数据问题。因此，SDR允许许多现有的分析方法，这些方法过去受到维度诅咒的阻碍，现在可以处理高维问题。此外，数据的信息性可视化通常在降维后成为可能，从而促进对数据的理解和分析。通过为缺失、审查和相关数据开发新的理论和方法，研究人员的研究将SDR的边界扩展到生物统计学以及其他学科，如计量经济学、金融学和生物信息学。这项研究的影响预计将是广泛的，因为高维数据普遍存在，而且迫切需要有效的分析工具来解决这些问题。

This proposal aims to develop new theory and methodology for sufficient dimension reduction (SDR). In particular, the research focuses on biostatistical problems which commonly include missing data, survival data, and longitudinal data analysis. The proposed methodology can effectively transform a high dimensional regression problem to a low dimensional projection, retain full regression information, and impose few or no probabilistic models. There are three components to this research. First, the investigator proposes a family of augmented inverse probability weighted SDR estimators when predictors have missing observations. This new approach allows a more general missing data mechanism than the existing solution and permits more flexible regression forms beyond the homoscedastic linear model. The second component of the research targets SDR for survival data, where the response of time to death or disease recurrence is subject to censoring. Viewing the censored response as a specific type of missing data, the investigator integrates an inverse probability weighted estimation strategy with a variety of SDR methods. Thirdly, the investigator studies a type of longitudinal data where measurements for all the study subjects are collected at the same scheduled time points. Both a population foundation and the associated estimation procedure are developed. All three components center around commonly encountered biostatistical problems and the development of the three components are interrelated.Modern technologies have pushed the frontier of science with the capability of generating and collecting data in large quantity and high dimensionality. Examples of large high dimensional data sets arise in a great number of research areas, such as environmental studies, human health and medical research, and homeland security. Sufficient dimension reduction (SDR) methodology effectively transforms a high dimensional data problem to a low dimensional one. Consequently, SDR allows many existing analytical methods, which used to be hindered by the curse of dimensionality, to now work for the high dimensional problems. In addition, informative visualization of the data often becomes possible after dimension reduction, facilitating both the understanding and the analysis of the data. By developing new theory and methodology for missing, censored, and correlated data, the investigator's research extends the boundary of SDR to biostatistics as well as other disciplines such as econometrics, finance and bioinformatics. The impact of this research is anticipated to be widespread, due to the prevalence of the high dimensional data and the urgent demand for effective analytical tools to tackle those problems.