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来监测青光眼病程并预防因疾病进展导致的视力丧失。黄斑厚度在视网膜上6$\times$6网格位置间存在差异，且在每次就诊时成像过程会产生额外变异性。目前，眼科医师通过为每个受试者和位置重复进行简单线性回归来估计斜率。为更精确地估计斜率，我们提出了一种新颖的贝叶斯分层模型，该模型适用于包含空间变化群体水平和受试者水平系数的多个受试者，通过借用受试者和测量位置间的信息。我们加入了就诊效应以增强模型，从而解释观测到的空间相关就诊特定误差。我们利用具有Mat\'ern交叉协方差函数的多变量高斯过程先验对空间变化的(a)截距、(b)斜率以及(c)对数残差标准差进行建模。每个边际过程假设具有自身标准差和空间相关矩阵的指数核。我们为高级青光眼进展研究数据开发并应用模型。研究表明，在模型中包含就诊效应可降低预测未来厚度测量值的误差，并显著改善模型拟合效果。