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We describe and analyze a longitudinal diffusion tensor imaging (DTI) study relating changes in the microstructure of intracranial white matter tracts to cognitive disability in multiple sclerosis patients. In this application the scalar outcome and the functional exposure are measured longitudinally. This data structure is new and raises challenges that cannot be addressed with current methods and software. To analyze the data, we introduce a penalized functional regression model and inferential tools designed specifically for these emerging types of data. Our proposed model extends the Generalized Linear Mixed Model by adding functional predictors; this method is computationally feasible and is applicable when the functional predictors are measured densely, sparsely or with error. An online appendix compares two implementations, one likelihood-based and the other Bayesian, and provides the software used in simulations; the likelihood-based implementation is included as the lpfr() function in the R package refund available on CRAN.
Goldsmith et al. (Thu,) studied this question.