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Gertheiss, Jan and Goldsmith, Jeff and Crainiceanu, Ciprian and Greven, Sonja (2012): Longitudinal Scalar-on-Function Regression with Application to Tractography Data. Department of Statistics: Technical Reports, No.118
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Abstract

We propose a class of estimation techniques for scalar-on-function regression in longitudinal studies where both outcomes, such as test results on motor functions, and functional predictors, such as brain images, 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 review existing approaches designed to handle such types of data and discuss their limitations. We propose two versions of a new method, longitudinal functional principal components regression. These methods extend the well-known functional principal component regression and allow for different effects of subject-specific trends in curves and of visit-specific deviations from that trend. The different methods are compared in simulation studies, and the most promising approaches are used for analyzing the tractography data.