Mathematical Sciences: Semi-parametric Methods for Longitudinal Data Analysis

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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.