Estimating the Distribution of True Rates of Visual Field Progression in Glaucoma.

Abstract

The purpose of this study was to estimate the distribution of the true rates of progression (RoP) of visual field (VF) loss. We analyzed the progression of mean deviation over time in series of ≥ 10 tests from 3352 eyes (one per patient) from 5 glaucoma clinics, using a novel Bayesian hierarchical Linear Mixed Model (LMM); this modeled the random-effect distribution of RoPs as the sum of 2 independent processes following, respectively, a negative exponential distribution (the "true" distribution of RoPs) and a Gaussian distribution (the "noise"), resulting in a skewed exGaussian distribution. The exGaussian-LMM was compared to a standard Gaussian-LMM using the Watanabe-Akaike Information Criterion (WAIC). The random-effect distributions were compared to the empirical cumulative distribution function (eCDF) of linear regression RoPs using a Kolmogorov-Smirnov test. The WAIC indicated a better fit with the exGaussian-LMM (estimate [standard error]: 192174.4 [721.2]) than with the Gaussian-LMM (192595 [697.4], with a difference of 157.2 [22.6]). There was a significant difference between the eCDF and the Gaussian-LMM distribution (P < 0.0001), but not with the exGaussian-LMM distribution (P = 0.108). The estimated mean (95% credible intervals, CIs) "true" RoP (-0.377, 95% CI = -0.396 to -0.359 dB/year) was more negative than the observed mean RoP (-0.283, 95% CI = -0.299 to -0.268 dB/year), indicating a bias likely due to learning in standard LMMs. The distribution of "true" RoPs can be estimated with an exGaussian-LMM, improving model accuracy. We used these results to develop a fast and accurate analytical approximation for sample-size calculations in clinical trials using standard LMMs, which was integrated in a freely available web application.