Estimating the distribution of true rates of visual field progression in glaucoma

PURPOSE: The purpose of this study was to estimate the distribution of the true rates of progression (RoP) of visual field (VF) loss.

METHODS: 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.

RESULTS: 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.

CONCLUSIONS: The distribution of "true" RoPs can be estimated with an exGaussian-LMM, improving model accuracy.

TRANSLATIONAL RELEVANCE: 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.