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, e.g., 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 data sets. Applications with spatially and longitudinally observed functional data demonstrate the flexibility in modeling and interpretability of results of our approach.
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 to t of F{X(t), t} where F(·,·) is an unknown regression function and X(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 estimate F(·,·) 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, where X(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.
We introduce Generalized Multilevel Functional Linear Models (GMFLM), a novel statistical framework motivated by and applied to the Sleep Heart Health Study (SHHS), the largest community cohort study of sleep. The primary goal of SHHS is to study the association between sleep disrupted breathing (SDB) and adverse health effects. An exposure of primary interest is the sleep electroencephalogram (EEG), which was observed for thousands of individuals at two visits, roughly 5 years apart. This unique study design led to the development of models where the outcome, e.g. hypertension, is in an exponential family and the exposure, e.g. sleep EEG, is multilevel functional data. We show that GMFLMs are, in fact, generalized multilevel mixed effect models. Two consequences of this result are that: 1) the mixed effects inferential machinery can be used for GMFLM and 2) functional regression models can be extended naturally to include, for example, additional covariates, random effects and nonparametric components. We propose and compare two inferential methods based on the parsimonious decomposition of the functional space.
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