Restricted Likelihood Ratio Tests for Functional Effects in the Functional Linear Model

Abstract

The goal of our article is to provide a transparent, robust, and computationally feasible statistical approach for testing in the context of scalar-on-function linear regression models. Assuming linearity between response and predictors, we are interested in testing for the necessity of functional effects. Our methods are motivated by and applied to a large longitudinal study involving diffusion tensor imaging of intracranial white matter tracts in a susceptible cohort. In the context of this study, we conduct hypothesis tests that are motivated by anatomical knowledge and support recent findings regarding the relationship between cognitive impairment and white matter demyelination. R code and data are in the examples of refund::rlrt.pfr(). Supplementary materials for this article are available online.

KEY WORDS:

• Functional data analysis
• Nonparametric smoothing
• Nonregular problem
• Penalized splines
• Variance components

ACKNOWLEDGMENTS

The diffusion tensor imaging data were collected at Johns Hopkins University and the Kennedy-Krieger Institute under the direction of Peter A. Calabresi, MD. Crainiceanu and Swihart were supported by Grant Number R01NS060910 from the National Institute of Neurological Disorders and Stroke. Crainiceanu was also supported by Grant Number R01EB012547 from the National Institute of Biomedical Imaging And Bioengineering.