We propose a new regression model and inferential tools for the case when both the outcome and the functional exposures are observed at multiple visits. This data structure is new but increasingly present in applications where functions or images are recorded at multiple times. This raises new inferential challenges that cannot be addressed with current methods and software. Our proposed model generalizes the Generalized Linear Mixed Effects Model (GLMM) by adding functional predictors. Smoothness of the functional coefficients is ensured using roughness penalties estimated by Restricted Maximum Likelihood (REML) in a corresponding mixed effects model. This method is computationally feasible and is applicable when the functional predictors are measured densely, sparsely or with error; code implementing the proposed procedure is freely available. Methods are applied to a longitudinal diffusion tensor imaging (DTI) study relating changes in the microstructure of intracranial white matter tracts to cognitive disability in multiple sclerosis patients, but we note that the discussed data structure is increasingly common and our methods apply generally. An online appendix compares two implementations, one likelihood-based and the other Bayesian, and provides the software used in simulations.


Statistical Methodology | Statistical Theory