  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.
