In the current study, a methodology was proposed for the integration of structural and risk factor information to improve the assessment of progression and rates of visual field loss in glaucoma. Our approach resulted in more precise estimates of slopes of functional change, which were also able to more accurately predict future visual field observations. To our knowledge, this is the first study to report a successful method for integrating risk factor information into the assessment of glaucomatous progression with Bayesian statistics.
An initial evaluation was performed to determine which risk factors were associated with rates of glaucomatous visual field loss in order to subsequently use the information on these factors to improve estimates of rates of change. The results confirmed previous published findings showing that high IOP, thinner CCT, and presence of optic disc hemorrhages are significantly associated with increased risk of progression.
5–13 In addition, presence of progressive optic disc damage was also found to be associated with faster rates of visual field loss. This is in agreement with a report by Medeiros et al.
14 showing that progressive optic disc damage is highly associated with development of future visual field loss with a hazard ratio of 25.8. This is an expected result as glaucoma is a progressive optic neuropathy and the associated visual field damage is essentially the result of retinal ganglion cell loss. In that sense, progressive optic disc damage should not be strictly considered a risk factor for the disease, as it is actually part of its definition. However, the use of information on the presence of progressive optic disc damage may help improve inferences derived from functional measurements, as shown in the current study. In contrast to some other previous studies, older age was not found to significantly influence rates of visual field loss. The relationship between age and glaucoma progression is likely complex. Although previous reports have shown that age is associated with increased risk of progression, as measured by event-based assessments, the association between age and rates of change has not been clarified.
After obtaining the coefficients associated with the effect of the different risk/predictive factors on the rates of change, this information was incorporated into the Bayesian model in order to improve inferences on the slopes of functional change. It is important to note that a leave-one-out approach was used in order to prevent model overfitting, that is, the model coefficients were obtained from n−1 eyes and applied to the estimation of the slope of the nth eye. In addition, the information on risk factors was collected only during the first half of the follow-up period and used to influence calculation of slopes obtained from MD measurements available at the same interval of time. The estimated slopes were then used to predict future observations collected during the second half of the follow-up period. Our results showed that Bayesian slopes performed significantly better than slopes calculated from the OLS method to predict future observations. Predictions obtained by extrapolating slopes of change calculated from the Bayesian model were closer to the actual MD values than those obtained by extrapolating OLS regression lines, as shown by differences in the mean square error of the predictions.
The use of Bayesian statistics helps improve the estimates of slopes of change by incorporating “prior” information that contributes to their final estimation. In our methodology, the prior is composed of a multivariate distribution relating slopes and intercepts of functional change and influenced by the values of risk factors and presence of progressive optic disc damage. For example, an eye that had mean IOP of 30 mm Hg during follow-up and evidence of progressive optic disc damage will have a prior distribution that corresponds to the expected distribution of slopes of visual field loss of eyes with the same characteristics. Such distribution will obviously be concentrated towards negative values, indicating that these eyes are in general expected to have relatively fast rates of progression. In contrast, an eye that had mean IOP of 15 mm Hg and no evidence of progressive optic disc damage during follow-up will have a prior distribution that will be concentrated towards values close to zero, indicating that on average these eyes are not expected to have much visual field change over time. The prior information will exert influence into the calculation of the final slope, but the amount of influence will depend on the degree of information that can be derived solely from the visual field measurements of the eye under consideration. If the visual fields show large variability, or only a few measurements are available over time, then the slope of change would be poorly estimated, using only the visual field data for that particular eye. In this situation, the prior will exert great influence on the estimation of the final slope of change. This is a very desirable property as it will allow better evaluation of progression in those who have large visual field variability over time, a common situation in clinical practice. A patient with high IOP and evidence of progressive disc damage may still be declared as progressing and have a fast estimated rate of change despite having relatively little information that can be derived from visual field data only. On the other hand, if a large amount of reliable visual field information is available in the eye under consideration, then its slope of change can be calculated with great precision and the prior will exert little influence on the final calculation. Again, this is also a desirable property. An eye that has sufficient visual field information to be declared as nonprogressor should not be deemed as progressing just because of the presence of risk factors. Similarly, an eye with significant evidence of progression based only on visual field data should still be declared as progressing despite the absence of apparent risk factors for progression.
Besides better accuracy, slopes of change obtained by Bayesian regression had also greater precision. The precision is a measure of how closely an estimator is expected to be to the true value of the parameter and can be measured by the standard error or confidence interval of the slopes. Large imprecision, that is, large standard errors, will confound the interpretation of the clinical relevance of an estimated slope, as there will be great uncertainty about where the true slope of change is likely to be. In the presence of imprecise slopes, more tests will increase the precision of the estimate. Although an improvement in the predictions and standard error of the slopes was seen for both the Bayesian and OLS methods with an increase in the number of visual fields used for the calculation of the slopes, Bayesian predictions still outperformed OLS predictions for large number of tests, as indicated by
Figure 5. It is known that as the number of measurements increases, the OLS estimate approaches the true underlying latent variable. However, in clinical practice, there is a cost associated with obtaining more measurements over time, including the expense of the test itself, the cost in patient time, and the cost related to delaying detection of change. By providing more precise estimates of rates of change with fewer tests, the Bayesian approach potentially allows more confident clinical decision-making to be made earlier with regard to the clinical and statistical significance of the calculated slopes. It is important to emphasize, however, that when assessing the clinical relevance of an estimated slope, clinicians also need to consider other factors, such as life expectancy and the patient's expectations with regard to treatment.
In previous studies, the authors used Bayesian statistics to incorporate longitudinal information on optic disc topography and retinal nerve fiber layer change obtained from imaging devices into the assessment of rates of visual field loss.
21,22 A better agreement was shown between rates of structural and functional change estimated with Bayesian regression than with the OLS method. In the current investigation, it was also shown that Bayesian slopes of change detected a significantly higher proportion of eyes, which showed progressive optic disc damage based on stereophotographs. The evaluation of the accuracy of any new method for detection of glaucoma progression is hampered by the inexistence of a perfect reference standard for progression. However, our results suggest a higher sensitivity of the Bayesian method than OLS because it detected more cases of progressive optic disc damage. Additionally, the smaller number of eyes with significant positive slopes, as compared to the OLS method, also suggests that the Bayesian method had a higher specificity for detection of progression. This conclusion is supported by the demonstration that Bayesian slopes were also better predictors of future observations.
Our proposed model provides individualized estimates of rates of change, which could potentially be used in clinical practice for individual management. The estimation of parameters of the model depends on previously acquired data in order to evaluate the relationship between rates of structural and functional change and the impact of risk factors in glaucoma. Accurate assessment of these relationships is essential in order to define how risk factors and structural information will influence rates of functional change. However, once this information is available, the model can easily be used to estimate individualized slopes by taking into account risk factors and the results of structural and functional tests performed in the eye under evaluation. Construction of built-in databases would be required to implement this methodology in currently available instruments, in a similar approach to the built-in progression packages available for some instruments. Another potential advantage of our methodology is that data from patients tested over time could be continuously incorporated into the Bayes model, leading to improved estimates that would more likely reflect the progression rates in a particular clinical setting, generating a “customized” database for comparison.
This study had limitations. A linear rate of functional change over time was assumed. Previous studies using cross-sectional data, however, suggest that functional changes over the whole course of the disease would probably not be linear.
31–35 In fact, the use of decibel scale may ultimately underestimate the rate of retinal ganglion cell loss in early stages of the disease.
36 However, extensions of our methodology to incorporate nonlinear change or change assessed by other metrics are also possible. Changes related only to the parameter MD were also evaluated. It is known that change in MD values can be influenced by media opacities, such as development of cataract over time. However, such effect would affect calculation of the slopes with both the Bayesian and OLS methods and would likely not interfere with the comparison between the two methods. The MD is also a global parameter that may not fully capture localized changes in the visual field. However, extension of our method for evaluation of changes in sectoral parameters should be straightforward and will be the subject of future investigations.
In conclusion, a Bayesian regression model incorporating structural and risk factor information into the estimation of glaucomatous visual field progression resulted in more accurate and precise estimates of slopes of functional change than the conventional method of OLS regression. The proposed strategy may result in better assessment of the global risk for development of functional impairment in individual patients.