Samuel Berchuck is a Postdoctoral Associate in Duke University’s Department of Statistical Science and Forge-Duke’s Center for Actionable Health Data Science.
Joshua L. Warren is an Assistant Professor of Biostatistics at Yale University.
Looking Forward in Glaucoma Progression Research
The contribution of the womblR package and corresponding statistical methodology is a technique for correctly accounting for the complex spatial structure of the visual field. The purpose of this method is to properly model visual field data, so that an effective diagnostic is derived that discriminates progression status. This is one of many important clinical questions that needs to be addressed in glaucoma research. Others include: quantifying visual field variability to create simulations of healthy and progression patients, combining multiple data modalities to obtain a composite diagnostic, and predicting the timing and spatial location of future vision loss. There is opportunity within the glaucoma literature for the development of quantitative methods that answer important clinical questions, are easy to understand, and are simple to use. To this end, in closing this three-part series, we present a final example of a new method that uses change points to assess future vision loss.
Modeling Changes on the Visual Field Using Spatial Change Points
To motivate the use of change points, we note that patients diagnosed with glaucoma are often monitored for years with slow changes in visual functionality. It is not until disease progression that notable vision loss occurs, and the deterioration is often swift. This disease course inspires a modeling framework that can identify a point of functional change in the course of follow-up, thus change points are employed. This is an appealing modeling framework, because the concept of disease progression becomes intrinsically parameterized into the model, with the change point representing the point of functional change. In this model, the time of the change point triggers a simultaneous change in both the mean and variance process. Furthermore, to account for the typical trend of a long period of monitoring with little change followed by abrupt vision loss, we force the mean and variance to be constant before the change point. For the mean process, and assuming data from a patient with nine visual field tests, this results in \[\mu_t\left(\mathbf{s}_i\right)=\left\{ \begin{array}{ll}
{\beta}_0\left(\mathbf{s}_i\right) &amp; x_t \leq \theta\left(\mathbf{s}_i\right),\\
{\beta}_0\left(\mathbf{s}_i\right) + {\beta}_1\left(\mathbf{s}_i\right)\left\{x_t-\theta\left(\mathbf{s}_i\right)\right\} &amp; \theta\left(\mathbf{s}_i\right) \leq x_t.\end{array} \right. \quad t = 1,\ldots,9 \quad i = 1,\ldots,52 \] Here, the change point at location \(\mathbf{s}_i\) is given by \(\theta(\mathbf{s}_i)\), and \(x_t\) is the days from baseline visit for follow-up visit \(t\). A final important detail is that the change points \(\theta(\mathbf{s}_i)\) are truncated in the observed follow-up range, \((x_1, x_9)\). In practice, the true change point can occur outside of this region, so we define a latent process, \(\eta(\mathbf{s}_i)\), that defines the true change point, \(\theta(\mathbf{s}_i) = \max\{\min\{\eta(\mathbf{s}_i), x_9\}, x_1\}\). Finally, all of the location-specific effects are modeled using a novel multivariate conditional autoregressive (MCAR) prior that incorporates the anatomy detailed in the womblR method. More details can be found in Berchuck et al. 2018.
We once again rely on MCMC methods for inference, and a package similar to womblR was developed that implements the spatially varying change point model, spCP. This package has much of the same functionality as womblR, and we demonstrate its functionality below.
We begin by loading spCP. All of the visual field data (VFSeries), adjacencies (HFAII_Queen), and anatomical angles (GarwayHeath) are included in the spCP package.
###Load package
library(spCP)
###Format data
blind_spot