Mathematical Sciences: Semi-parametric Methods for Longitudinal Data Analysis

线性和广义线性混合模型是纵向响应曲线分析的有力工具。本研究的重点是这些模型的半参数扩展：通过自建模方法恢复底层曲线，对总结协变量效应的参数进行估计和推理，并将曲线作为数据用于统计程序，如设计实验分析，判别分析和聚类。该表示的一个新特征是时间轴和响应轴的单独参数化，这使得协变量可以通过响应水平的变化和时间膨胀或收缩来作用于响应。估计和推理技术正在开发中。该方法在使用混合模型的许多领域都有应用：医学和流行病学研究、环境研究、公共政策评估和经济学，仅举几例。在许多研究中，每个人的反应都可以看作是一条随时间变化的曲线。例子包括不同治疗方案下患者的HIV感染进展，以及不同环境条件下不同土壤中农药的降解。类似类型的数据被用于公共政策评估、经济学、心理学、药物动力学和许多其他领域。了解反应随时间的演变对于解释治疗和其他影响的效果至关重要。统计建模的最新进展大大提高了估计治疗效果的效率，但要求研究者在分析数据之前指定响应曲线的形状和预期的治疗效果类型。为本项目开发的方法允许根据观测数据确定响应曲线的形状，同时保留处理效果的简单措施。一个新颖的特点是，延长反应时间的治疗（例如，通过减缓疾病进展）以一种自然的方式进行。本项目涉及的相关工作包括使用反应曲线来寻找具有相似反应演变的子群体，并将个体分为健康和患病等群体。

DMS 9625350 Altman Linear and generalized linear mixed models are powerful tools for analysis of longitudinal response curves. This research focuses on semi-parametric extensions of these models: recovery of the underlying curve via self-modeling methods, estimation and inference for parameters which summarize covariate effects and the use of the curves as data in statistical routines such as analysis of designed experiments, discriminant analysis and clustering. A novel feature of the representation is separate parametrization of the time and response axes which allows the covariates to act on the response both by changes in the level of response and by time dilation or contraction. Estimation and inferential techniques are under development. The methodology has applications in the many areas in which mixed models are used: medical and epidemiological research, environmental studies, public policy assessment, and economics, to name just a few. In many studies the response of each individual can be thought of as a curve over time. Examples include the the progression of HIV infection in patients under different treatment programs and the degradation of pesticides in different soils under different environmental conditions. Similar types of data are used for public policy assessment, economics, psychology, pharmokinetics and numerous other fields. Understanding the evolution of response over time can be critical to interpreting the effects of treatments and other influences. Recent advances in statistical modeling have greatly improved the efficiency of estimating treatment effects but require that the investigator specify the shape of the response curve and the types of treatment effects expected prior to analyzing the data. The methods developed for this project allows the shape of the response curve to be determined from the observed data, while retaining simple measures of treatment effects. A novel feature is that treatments which stretch the time scale of the response (for example, by slowing disease progression) are handled in a natural way. Related work covered by this project involves the use of response curves to find subgroups with similar response evolution and for classification of individuals into groups, such as healthy and diseased.