Functional Additive Mixed Models

Abstract

We propose an extensive framework for additive regression models for correlated functional responses, allowing for multiple partially nested or crossed functional random effects with flexible correlation structures for, for example, spatial, temporal, or longitudinal functional data. Additionally, our framework includes linear and nonlinear effects of functional and scalar covariates that may vary smoothly over the index of the functional response. It accommodates densely or sparsely observed functional responses and predictors which may be observed with additional error and includes both spline-based and functional principal component-based terms. Estimation and inference in this framework is based on standard additive mixed models, allowing us to take advantage of established methods and robust, flexible algorithms. We provide easy-to-use open source software in the pffr() function for the R package refund. Simulations show that the proposed method recovers relevant effects reliably, handles small sample sizes well, and also scales to larger datasets. Applications with spatially and longitudinally observed functional data demonstrate the flexibility in modeling and interpretability of results of our approach.

Key Words:

• Functional data analysis
• Functional principal component analysis
• P-splines
• Smoothing
• Varying coefficient models

ACKNOWLEDGMENTS

The authors thank Danny Reich and Peter Calabresi for supplying and explaining the DTI tractography data. Richard Herrick and Jeffrey S. Morris provided a Linux-version of WFMM. Sonja Greven and Fabian Scheipl were funded by Emmy Noether grant GR 3793/1-1 from the German Research Foundation. Ana-Maria Staicu’s research was supported by U.S. National Science Foundation grant number DMS 1007466 and by U.S. National Institute of Health grant number 1R01 NS085211-01.