We propose a class of estimation techniques for scalar-on-function regression
in longitudinal studies where both outcomes and functional predictors may be
observed at multiple visits. Our methods are motivated by a longitudinal brain
diffusion tensor imaging (DTI) tractography study. One of the primary goals of
the study is to evaluate the contemporaneous association between human function
and brain imaging over time. The complexity of the study requires development of
methods that can simultaneously incorporate: (1) multiple functional (and scalar)
regressors; (2) longitudinal outcome and functional predictors measurements per
patient; (3) Gaussian or non-Gaussian outcomes; and, (4) missing values within
functional predictors. We propose two versions of a new method, longitudinal
functional principal components regression. These methods extend functional
principal component regression and allow for different effects of subject-specific
trends in curves and of visit-specific deviations from that trend. The presented
methods are compared in simulation studies, and the most promising approaches
are used for analyzing the tractography data. Joint work with Jeff Goldsmith,
Ciprian Crainiceanu and Sonja Greven.