The goal of our article is to provide a transparent, robust, and computationally feasible statistical approach for testing in the context of functional linear models. In particular, we are interested in testing for the necessity of functional effects against standard linear models. Our approach is to express the coefficient function so that the null model includes the average of functional predictors as a scalar covariate. Two specific methods are utilized: the first is a modified version of the now-standard functional principal components regression; and the second is a flexible spline approach that induces smoothness in a linear mixed model framework. In the first method testing is accomplished using a standard likelihood ratio test, while in the second testing is performed using (restricted) likelihood ratio tests for zero variance components. We extend the methodology to be of use when multiple functional predictors are observed and when observations are made longitudinally. 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 which support recent findings regarding the relationship between cognitive impairment and white matter demyelination. Accompanying R code from an upcoming release of the R-package refund is provided.
Biostatistics | Neurosciences
Swihart, Bruce J.; Goldsmith, Jeff; and Crainiceanu, Ciprian M., "Testing For Functional Effects" (July 2012). Johns Hopkins University, Dept. of Biostatistics Working Papers. Working Paper 247.