Distributions of glaucomatous visual field progression rates: a comparison of three parametric models

Abstract

Purpose: Parametric models of the distribution of glaucomatous visual field progression rates (in dB/year) can be used in Bayesian methods for improving progression rate estimation, for summarising how distributions differ between different populations and as a function of glaucoma risk factors, and for overcoming the problems that arise when zero-frequency histogram bins occur at intermediate progression rates in empirical data. Here we compare three parametric models to test if one is clearly preferred for fitting published distributions of visual field progression rates for glaucoma.

Methods: We used a modified Gaussian model, a modified Cauchy model and a modified hyperbolic secant model, each of which had three free parameters. The modification allowed the shape of the model’s distribution either side of the mode to be independently varied to allow for the asymmetric tails seen in visual field progression rate distributions. Models were fit to published distributions of the overall (i.e. non-pointwise) rates of visual field loss in glaucoma cohorts from Canada, Sweden and the USA, using a maximum-likelihood procedure. Goodness-of-fit was quantified with a coefficient of determination, R².

Results: For the Canadian data, log₁₀ likelihoods for the modified Gaussian, Cauchy and hyperbolic secant models were -3036.8, -2891.1 & -2893.9 (R²: 0.77, 0.99 & 0.94), respectively, indicating that the modified Cauchy model fitted best and was 583 times (-2891.1 minus -2893.9 = 2.8 log units) more likely than the next best fitting model, the modified hyperbolic secant. For the Swedish data, likelihoods were -695.2, -701.3 and -684.1 (R²: 0.92, 0.96 & 0.98), indicating the modified hyperbolic secant gave the best fit (by 1.0x10⁶ times). For the USA data, likelihoods were -412.3, -412.1 and -402.8 (R²: 1.00, 0.94 & 0.92), indicating the modified hyperbolic secant gave the best fit (by 2.8x10⁹ times). Summing likelihoods across datasets, the hyperbolic secant was strongly favoured (by 26.7 log units) compared to the next best fitting model, the modified Cauchy.

Conclusions: Parametric models can describe well the distribution of visual field progression rates in treated glaucoma. Although the optimum model differs depending upon the particular dataset fitted, a modified hyperbolic secant performed well for all distributions investigated and was strongly favoured when evidence was summed across datasets.