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.

翻译：我们建立纵向黄斑厚度测量模型，用于监测青光眼病程进展并预防因疾病恶化导致的视力丧失。黄斑厚度在视网膜6×6网格位置间存在变异，且每次随访的成像过程会引入额外变异性。目前，眼科医生通过在每个受试者和位置重复进行简单线性回归来估算斜率。为更精确地估算斜率，我们开发了一种新颖的贝叶斯分层模型，该模型针对多个受试者设置空间变化的群体水平和个体水平系数，通过跨受试者和测量位置的信息借用来提高精度。我们在模型中纳入随访效应，以解释观测到的空间相关随访特异性误差。我们采用具有马特恩互协方差函数的多元高斯过程先验，对空间变化的(a)截距、(b)斜率及(c)对数残差标准差进行建模。每个边缘过程假设使用具有自身标准差和空间相关矩阵的指数核。我们针对高级青光眼进展研究的数据开发并应用本模型。研究表明，在模型中纳入随访效应可降低未来厚度测量值的预测误差，并显著改善模型拟合度。