Semiparametric Regression Modeling for Longitudinal Data

在这个方案中，研究者提出了一种半参数方法，用于纵向响应和测量时间的联合建模。半参数变系数回归模型假定部分协变量的影响随时间非参数变化，而其余协变量的影响服从一定的时间参数函数。反应的测量时间由条件平均率的Cox比例模型模拟。对Aalen的加性模型的建模也进行了研究。研究者采用了Lin和Ying(2001)的方法来估计非参数回归系数。参数分量估计采用加权最小二乘法。响应过程使用了更高效的平滑技术。研究者提出了一种基于残差平方和的数据驱动的带宽选择准则。这种选择方法将通过仿真进行进一步的研究和实证检验。给出了参数分量、非参数回归系数函数及其累积量的置信带的构造。通过比较非参数估计和零假设下累积回归系数函数的相应估计，提出了检验某些回归系数函数是否符合某些参数形式的拟合度检验方法。初步的仿真研究表明，所提出的估计和假设检验方法具有良好的应用前景。研究者将研究所有所提出的估计量的渐近性质，如相合性、弱收敛和相合性。我们将通过广泛的数值模拟研究来评估所提出的估计方法和拟合度检验的有限样本特性。研究人员还建议在权重函数的最优选择、测量时间的最优设计和信息审查等方面进行相关问题的研究。研究人员建议寻找具有物理或生物学基础和生物可解释参数的统计模型，并开发统计有效的方法来更好地理解反应过程的线性或非线性行为。提出的半参数变系数回归模型允许更多的灵活性来探索协变量效应是如何随时间变化的，并为检验具有科学相关性的简单模型是否成立提供了基础。拟议的研究一旦进行，将有助于丰富对纵向数据分析有重要影响的统计工具。将制定实际应用的指导方针。这些方法将用于分析艾滋病相关医学研究中的纵向数据，以开发更有效的治疗方法，并用于医学研究中的其他真实例子。对这些问题的研究还将产生许多不同层次的研究课题，适合研究生和本科生学习，从而促进学生参与当前的科学研究。

In this proposal, the investigator proposes a semiparametric method for joint modeling of longitudinal responses and measurement times. The semiparametric varying-coefficient regression model postulates that the influences of some covariates vary nonparametrically with time while the effects of the remaining covariates follow certain parametric functions of time. The measurement times of responses are modeled by Cox's proportional model for conditional mean rate. The modeling by Aalen's additive model will also be studied. The investigator employs the approach of Lin and Ying (2001) to estimate the nonparametric regression coefficients. The parametric components are estimated by a weighted least squares method. More efficient smoothing techniques are used for response processes. The investigator proposes a data-driven bandwidth selection criterion based on the sum of squares of residuals. This selection method will be further studied and tested empirically through simulations. Constructions of confidence bands for the parametric components, nonparametric regression coefficient functions and their cumulatives are proposed. Goodness-of-fit test procedures are proposed to check whether some regression coefficient functions follow certain parametric forms by comparing the nonparametric estimators and the corresponding estimators of the cumulative regression coefficient functions under the null hypothesis. A preliminary simulation study demonstrates that the proposed estimation and hypothesis testing methods are promising. The investigator will study the asymptotic properties, such as consistency, weak convergence and consistency of the estimators of the asymptotic variances, for all the proposed estimators. Finite sample properties of the proposed estimation methods and goodness-of-fit tests will be evaluated through extensive numerical simulation studies. The investigator also proposes to study related problems in optimal choice of the weight function, optimal design of measurement times and informative censoring.The investigator proposes to seek statistical models with a physical or biological basis and biologically interpretable parameters and to develop statistically efficient methods to better understand linear or nonlinear behavior of response process. The proposed semiparametric varying-coefficient regression model allows for additional flexibility to explore how covariate effects change over time and provides a base to test whether simpler models with scientific relevance hold. The proposed research when carried out would help to enrich a collection of statistical tools which have important impact on the analysis of longitudinal data. Giudelines for practical applications will be developed. The methods will be used to analyze longitudinal data in AIDS related medical research to develop more effective treatments and to other real examples in medical studies. The research of the problems proposed here will also generate many research topics at different levels suitable for graduate and undergraduate studies, therefore promotes involvement of students in current scientific research.