A Spatially Varying Hierarchical Random Effects Model for Longitudinal Macular Structural Data in Glaucoma Patients

Abstract: We model longitudinal macular thickness measurements to monitor the course of glaucoma and prevent vision loss due to disease progression. The macular thickness varies over a 6$\times$6 grid of locations on the retina with additional variability arising from the imaging process at each visit. Currently, ophthalmologists estimate slopes using repeated simple linear regression for each subject and location. To estimate slopes more precisely, we develop a novel Bayesian hierarchical model for multiple subjects with spatially varying population-level and subject-level coefficients, borrowing information over subjects and measurement locations. We augment the model with visit effects to account for observed spatially correlated visit-specific errors. We model spatially varying (a) intercepts, (b) slopes, and (c) log residual standard deviations (SD) with multivariate Gaussian process priors with Mat\'ern cross-covariance functions. Each marginal process assumes an exponential kernel with its own SD and spatial correlation matrix. We develop our models for and apply them to data from the Advanced Glaucoma Progression Study. We show that including visit effects in the model reduces error in predicting future thickness measurements and greatly improves model fit.