Bayesian semiparametric methods for functional data

Motivated by studies of reproductive hormone profiles in the menstrual cycle, we develop methods for hierarchical functional data analysis. The data come from the North Carolina Early Pregnancy Study, in which measurements of urinary progesterone metabolites are available from a cohort of women who were trying to become pregnant. Methods for menstrual hormone data are needed that avoid standardizing menstrual cycle lengths while also allowing for flexible relationships between the hormones and covariates. In addition, it is necessary to account for within-woman dependency in the hormone trajectories from multiple cycles. All of the methods are developed for and applied to menstrual hormone data, but they are general enough to be applied in many other settings.; The statistical approach is based on a hierarchical generalization of Bayesian multivariate adaptive regression splines. The generalization allows for an unknown set of basis functions characterizing both the overall trajectory means and woman-specific covariate effects and allows for the complex dependency structure of the data. To relax distributional assumptions, we use a Dirichlet process prior on the unknown distribution of the random basis coefficients in the spline model. This requires the development of methodology for the use of the Dirichlet process on the distribution of a parameter of varying dimension. While modeling the curves nonparametrically, the Dirichlet process also identifies clusters of similar curves. Finally, we combine our approach with Bayesian methods for generalized linear models, developing a procedure that clusters trajectories while jointly estimating the response distribution of each cluster.; In all of the models, a reversible jump Markov chain Monte Carlo algorithm is developed for posterior computation. Applying the methods to the progesterone data, we investigate differences in progesterone profiles between conception and non-conception cycles, identify clusters of pre-ovulatory progesterone, and demonstrate the ability of the joint model to distinguish early pregnancy losses from clinical pregnancies.