Generalized Semiparametric Varying-Coefficient Models for Longitudinal Data

由于艾滋病临床试验和HIV疫苗疗效试验中存在的问题，本研究致力于开发新的理论、方法和计算算法来分析纵向数据。这些方法将增强我们对艾滋病临床试验中转换治疗的好处，以及艾滋病毒疫苗接种如何随着时间的推移改变艾滋病毒感染者的疾病进展的理解。这项研究的更广泛影响包括发展纵向数据的统计理论，将其应用于医学、公共卫生和社会科学，促进本科生和研究生的统计学学习和培训，以及促进妇女参与科学研究。这些统计模型和方法将为研究复杂的协变量效应，包括线性和非线性效应，时变效应，以及协变量之间的非线性相互作用提供一个广阔的平台。这些模型可以通过链接函数分析分类和连续的纵向反应，并灵活地模拟三种类型的协变量效应：恒定效应、时变效应和协变量变化效应。第一部分研究了广义半参数变系数模型(GSVCM)，其中的协变量变化效应是暴露变量的参数函数，直到有限个未知参数；第二部分研究了GSVCM模型，其中协变量变化效应和时变效应都是未指定的函数。第三部分研究了广义半参数单指数变系数模型(Single-Index GSVCM)。该模型对于评估多个协变量之间的交互作用更为有效。提出了基于多元局部线性平滑和广义加权最小二乘的模型估计方法。将为拟议的估计和假设检验程序开发大样本理论。将研究的其他重要问题包括方差估计、协变量效应的假设检验、权重函数和带宽选择、拟合度诊断以及链路函数的估计和假设检验。将开发计算算法，以促进将所建议的方法应用于艾滋病临床试验和疫苗效力试验的真实数据。通过遵循项目概述的方向，可能会在建立生物学上可解释的模型方面取得重大进展，并在开发统计上有效的方法来处理纵向数据的复杂性方面取得重大进展。

This research focuses on the development of new theory, methods, and computational algorithms for analysis of longitudinal data, motivated by problems in AIDS clinical trials and HIV vaccine efficacy trials. The methods will enhance our understanding of the benefits of treatment switching in an AIDS clinical trial, and how HIV vaccination modifies the disease progression in HIV infected individuals over time. The broader impacts of the research include the development of statistical theory for longitudinal data, applications to medicine, public health and the social sciences, advancing statistical learning and training for undergraduate and graduate students, and promoting the participation of women in scientific research. The statistical models and methods will provide a broad platform for investigating complex covariate effects including linear and nonlinear effects, time-varying effects, and nonlinear interactions among the covariates.The project investigates a class of generalized semiparametric varying-coefficient models for longitudinal data. These models can be used to analyze both categorical and continuous longitudinal responses through a link function, and flexibly model three types of covariate effects: constant effects, time-varying effects, and covariate-varying effects. Part I studies the generalized semiparametric varying-coefficients model (GSVCM), where the covariate-varying effects are parametric functions of an exposure variable specified up to a finite number of unknown parameters, and Part II investigates the GSVCM model, where both the covariate-varying effects and the time-varying effects are unspecified functions. In Part III, the generalized semiparametric single-index varying-coefficients model (single-index GSVCM) is investigated. The model is more powerful for assessing interactions among multiple covariates. Estimation procedures for the model based on multivariate local linear smoothing and generalized weighted least squares are proposed. Large sample theory will be developed for the proposed estimation and hypothesis testing procedures. Other important problems that will be investigated include variance estimation, hypothesis testing of covariate effects, weight function and bandwidth selection, goodness-of-fit diagnostics, and estimation and hypothesis testing of link functions. Computational algorithms will be developed to facilitate the applications of the proposed methods to real data from AIDS clinical trials and vaccine efficacy trials. By pursuing the directions outlined in the project, significant progress may be made in building biologically interpretable models, and in developing statistically efficient methods to handle the complexity of longitudinal data.