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Susan Bryan, Koenraad Vermeer, Paul Eilers, Hans Lemij, Emmanuel Lesaffre; Bayesian Hierarchical Modelling of Longitudinal Visual Fields to Quantify Glaucoma Progression. Invest. Ophthalmol. Vis. Sci. 2013;54(15):1900.
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© ARVO (1962-2015); The Authors (2016-present)
Evaluation of a longitudinal series of visual fields (VF) provides a method to quantify functional deterioration. Previous research focused on regression models for each location in the eye independently, ignoring that measurements belong to the same eye. To improve predictive ability, a method which explicitly takes into account the hierarchical structure of the data is needed. We propose such a model and evaluate its assumptions.
We propose the following Bayesian mixed effects model which takes into account the hierarchical structure of the data: yijk(t) = β0 + β1*t + b0i + b1i*t + α0k(i) + α1k(i)*t + εijk, where i corresponds to individual, j to visit, k to location and t represents time. Glaucoma is a progressive disease, and the model takes this latent monotonic nature of the data into account by enforcing a negative slope: β1*t + b1i*t + α1k(i)*t ≤ 0. We used the slopes of 52 locations per eye (excluding blind spot) estimated by linear regression to evaluate possible assumptions for the priors in our model. 134 glaucoma patients from the Rotterdam Eye Hospital were included, with a minimum of 8 years follow-up and at least 15 HFA SITA standard white on white 24-2 fields per person. Both eyes were used in the analysis.
We investigated the slopes estimated from linear regression models considering each location independently. An example of the distribution in 3 different eyes is shown in Figure 1. This shows that the magnitude of the slope is highly variable and the distribution of the slopes sometimes includes positive values (red) and may either be uni-modal (green) or a mixture of distributions (blue).
We proposed a method for modelling the progression of VFs which takes into account the hierarchical structure of the data and uses assumptions based on clinical properties of the data. The distributions of the estimated slopes vary. Hence the need for a mixture versus a particular parametric distribution for the location specific slopes should be considered. Even though some of the estimated slopes are positive, glaucoma is a progressive disease and hence we attribute these positive estimates to measurement error. Including factors which have been shown in the literature to influence measurements, such as test reliability, time of day and season, will take this error into account using our model.
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