title: Functional Generalized Additive Models.
abstract: We introduce the functional generalized additive model (FGAM), a novel regression model for association studies between a scalar response and a functional predictor. We model the link-transformed mean response as the integral with respect totofF{X(t),t} whereF(·,·) is an unknown regression function andX(t) is a functional covariate. Rather than having an additive model in a finite number of principal components as in Müller and Yao (2008), our model incorporates the functional predictor directly and thus our model can be viewed as the natural functional extension of generalized additive models. We estimateF(·,·) using tensor-product B-splines with roughness penalties. A pointwise quantile transformation of the functional predictor is also considered to ensure each tensor-product B-spline has observed data on its support. The methods are evaluated using simulated data and their predictive performance is compared with other competing scalar-on-function regression alternatives. We illustrate the usefulness of our approach through an application to brain tractography, whereX(t) is a signal from diffusion tensor imaging at position,t, along a tract in the brain. In one example, the response is disease-status (case or control) and in a second example, it is the score on a cognitive test. R code for performing the simulations and fitting the FGAM can be found in supplemental materials available online.
authors.affiliation: PhD Student, School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853.
authors.forename: Mathew W
authors.initials: MW
authors.lastname: McLean
authors.affiliation: Assistant Professor, Department of Biological Statistics and Computational Biology, Cornell University, Ithaca, NY, 14853.
authors.forename: Giles
authors.initials: G
authors.lastname: Hooker
authors.affiliation: Assistant Professor, Department of Statistics, North Carolina State University, Raleigh, North Carolina 27695.
authors.forename: Ana-Maria
authors.initials: AM
authors.lastname: Staicu
authors.affiliation: Postdoc, Department of Statistics, Ludwig Maximilian University of Munich, Munich, Germany, 80333.
authors.forename: Fabian
authors.initials: F
authors.lastname: Scheipl
authors.affiliation: Andrew Schultz Jr. Professor of Engineering and Professor of Statistical Science, School of Operations Research and Information Engineering and Department of Statistical Science, Cornell University, 1170 Comstock Hall, Ithaca, NY 14853, USA.
authors.forename: David
authors.initials: D
authors.lastname: Ruppert
keywords: Diffusion tensor imaging; Functional data analysis; Functional regression; Generalized additive model; P-spline
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