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.


Statistical Methodology | Statistical Theory