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. Supplementary materials for this article are available online.

Keywords:

• Functional data analysis
• Functional mixed models
• Functional regression
• Longitudinal functional data
• Nonparametric effects
• Smoothing splines

Supplementary Materials

• Supplement.pdf: This document is organized as follows. Section 1 describes details of MCMC update steps. Section 2 provides derivation of DF(t) for Model (23(23) $$\begin{array}{ccc}\hfill Y_{ijp}\left(\mathbf{t}\right)& =& B_0\left(\mathbf{t}\right)+B_1\left(\mathbf{t}\right)G_1\left(p\right)+B_2\left(\mathbf{t}\right)G_2\left(p\right)+f(X_{{\mathrm{age}}_i},t)\hfill \\ & & +U_{ij}\left(\mathbf{t}\right)+U_{ij1}\left(\mathbf{t}\right)G_1\left(p\right)+U_{ij2}\left(\mathbf{t}\right)G_2\left(p\right)+E_{ijp}\left(\mathbf{t}\right),\hfill \end{array}$$(23) ). Sections 3 and 4 include details of model selection results. In Section 5, we assess whether the smoothing parameter for the nonparametric age effect should be constant or vary around the scleral surface. In Section 6, we provide description of overall procedure to fit our model and obtain inferential results with a simulation dataset. Section 7 contains sensitivity analyses to various modeling assumptions, including results for other basis (wavelet-regularized PC), other model (model that also includes left vs. right eye effect), and other regularization hyperparameters (choice of values with no additional shrinkage provided by prior). Section 8 describes some supplementary files presenting additional results discussed in the article, and Section 9 describes the simulated pseudo-data and demonstrates that it appears to capture the features of the real MPS scleral strain data reasonably well.

• RawMPScurves.zip: It includes plots of raw MPS curves, results after tensor wavelet compression, and results after robust filtering to remove spiky artifacts.

• movies.zip: Includes various .mp4 movie files illustrating various detailed results from the article, including the following:

• MPSvsAge-wave.mp4: Movie of MPS versus age for each IOP based on the tensor wavelet basis function.

• Combo_plots.mp4: Movie of key summary results based on the tensor wavelet basis functions.

• Intrafunctional_correlations.mp4 Movie showing intrafunctional correlations induced by tensor wavelet basis functions.

• Intra_IOP_corr.mp4 Movie showing interfunctional variance and serial correlation across IOP from same eye based on the tensor wavelet basis functions.

• MPSvsAge-pc.mp4: Movie of MPS versus age for each IOP based on the principal component basis functions.

• AUCvsAge-pc.mp4: Movie of AUC versus age based on the principal component basis functions.

• Intrafunctional_correlations-pc.mp4: Movie showing intrafunctional correlations induced by the principal component basis functions.

• MPSvsAge-eye.mp4: Movie of MPS versus age for each IOP based on tensor wavelet basis functions and model including left versus right eye effect.

• AUCvsAge-eye.mp4: Movie of AUC versus age based on tensor wavelet basis functions and model including left versus right eye effect.

• Intrafunctional_correlations-eye.mp4: Movie showing intrafunctional correlations induced by the tensor wavelet basis functions and model including left versus right eye effect.

• MPSvsAge-nosmooth.mp4: Movie of MPS versus age for each IOP based on tensor wavelet basis functions and model with no smoothing (π_· = 1, τ_· = 10⁶).

• AUCvsAge-nosmooth.mp4: Movie of AUC versus age based on tensor wavelet basis functions and model with no smoothing (π_· = 1, τ_· = 10⁶).

• Intrafunctional_correlations-nosmooth.mp4: Movie showing intrafunctional correlations induced by the tensor wavelet basis functions and model with no smoothing (π_· = 1, τ_· = 10⁶).

• EYE_toolbox.zip: Contains all of the files necessary to run the methods presented in the article, including the raw glaucoma data, pseudo data, and full scripts to run the analyses and produce the plots contained in the article. This includes wfmm_install.pdf that contains step-by-step instructions on how to install the R package wfmm and associated executable, and Analysis_of_Pseudo_Data.pdf that contains detailed step-by-step instructions for running a complete analysis on pseudo data generated to mimic the real data in the application in Matlab, including the basis transform, model selection heuristic, MCMC in basis space, projection of posterior samples back to data space, MCMC convergence diagnostics, and producing all inferential summaries and plots contained in this article that present results and illustrate properties of the model, with run time estimates for each step. We also include Producing Plots for Main Data Analysis in Paper.pdf that gives instructions for running scripts to reproduce the figures for the real data analysis for the main model used for the MPS scleral strain data analysis contained in the article. This toolbox and data are available on our github (https://github.com/MorrisStatLab/SemiparametricFMM).

Acknowledgment

The authors thank Richard Herrick and Nan Chen for computational assistance, and two very thorough reviewers and associate editor whose insightful queries led to a greatly improved article.

Additional information

Funding

This work is supported by grants from the National Cancer Institute (R01-CA178744, P30-CA016672, R01-CA160736), the National Science Foundation (1550088), and the National Eye Institute (R01-EY18926).