| Longitudinal studies are common because they provide more information about the evolutions of the underlying system. With the advance of technology, high frequency data can be collected from the studied subjects and how to balance the flexibility and interpretability in extracting the information from the individual profiles is the key challenge. In literature, linear mixed effects models have clear interpretations but only limited flexibility. Functional mixed effects models, on the other hand, are extremely flexible but the results are hard to interpret. To simultaneously obtain flexibility and interpretability, this dissertation utilizes state space method as the basic unit of data analysis. We first develop a data driven spline smoothing method by extending the classical smoothing spline to allow the roughness to adapt to the underlying signal. We also propose an equivalent state space model to ease to computational demand. We then develop a new class of mixed effects model where state space method is used to specify both the population effects and individual effects. The resultant models can handle a wide range of individual profiles and have clear interpretation. We further extend the mixed effects state space models to study various types of relationships across multivariate outcomes. The proposed methods are motivated by and applied to two data sets: (1) adrenocorticotropic hormone and cortisol from a study of chronic fatigue syndrome and fibromyalgia syndrome: and (2) the electroencephalogram data from epilepsy patients. |
