Bayesian Semiparametric Functional Mixed Models for Serially Correlated Functional Data, With Application to Glaucoma Data

Abstract

Glaucoma, a leading cause of blindness, is characterized by optic nerve damage related to intraocular pressure (IOP), but its full etiology is unknown. Researchers at UAB have devised a custom device to measure scleral strain continuously around the eye under fixed levels of IOP, which here is used to assess how strain varies around the posterior pole, with IOP, and across glaucoma risk factors such as age. The hypothesis is that scleral strain decreases with age, which could alter biomechanics of the optic nerve head and cause damage that could eventually lead to glaucoma. To evaluate this hypothesis, we adapted Bayesian Functional Mixed Models to model these complex data consisting of correlated functions on spherical scleral surface, with nonparametric age effects allowed to vary in magnitude and smoothness across the scleral surface, multi-level random effect functions to capture within-subject correlation, and functional growth curve terms to capture serial correlation across IOPs that can vary around the scleral surface. Our method yields fully Bayesian inference on the scleral surface or any aggregation or transformation thereof, and reveals interesting insights into the biomechanical etiology of glaucoma. The general modeling framework described is very flexible and applicable to many complex, high-dimensional functional data.

Comparison and contrast of two general functional regression modelling frameworks

“…Lee, et al (2016) showed how to fit semiparametric FMMs that include smooth functional terms f ( X ia , t ) that are nonparametric in both x a and t . These terms can be directly accommodated in model (5) by specifying a fixed effect predictor that is linear in x a and a single random effect level whose design matrix Z h involves Demmler-Reinsch orthonormal spline basis functions (Demmler and Reinsch 1975) evaluated on X ia .…”

“…The form of the penalty for the FLAM involves just a scalar smoothing parameter and fixed penalty matrix, while the parameterization of the FMM can allow a broader class of regularization parameters, which can potentially yield more flexibility in modeling within-function correlations and also enables nonstationary structures, e.g. allowing the smoothing parameter of nonparametric term f ( x, t ) regulating smoothness of x to vary over k , which subsequently allows it to vary over t (Lee, et al 2016). …”

“…This involves splitting the data into training and validation subsets, fitting an MCMC for each proposed model to the training data, and then calculating the predictive likelihood for the validation data (Zhu et al, 2016c). To select fixed and random covariates and their functional forms, Lee et al (2016) propose a faster alternative that involves a weighted Bayesian information criterion (BIC) comparison across models based on preliminary basis-space fits of the proposed models using italiclme in R.…”

“…Its accommodation of a broader class of basis functions enables the FMM to immediately be applied to functional data defined on higher dimensional (e.g., 2D or 3D) and potentially non-Euclidean domains, making it well equipped to model more complex functional objects. The form of the penalty for the FLAM involves just a scalar smoothing parameter and fixed penalty matrix, while the parameterization of the FMM can allow a broader class of regularization parameters, which can potentially yield more flexibility in modelling within-function correlations and also enables nonstationary structures, for example, allowing the smoothing parameter of nonparametric term ffalse(x,false) regulating smoothness of x to vary over k, which subsequently allows it to vary over t (Lee et al, 2016).…”

“…Zhang et al (2016a) extended this framework to model functions observed on a spatial lattice, introducing functional CAR processes, and Zhu et al (2016c) presented a general functional regression model for spatially or temporally correlated functions and introduced predictive likelihood-based model selection techniques. Lee et al (2016) demonstrated how to perform semiparametric functional regression with terms nonparametric in both x and t using this framework, and how it can accommodate longitudinally correlated functional data. Zhu et al (2016a) developed unified methods for functional regression and discrimination for sonar-terrain data, and Zhu et al (2016b) developed methods for multivariate functional response regression for fluorescence spectroscopy data.…”