**Purpose**:
To present and evaluate a new method of integrating risk factors into the analysis of rates of visual field progression in glaucoma.

**Methods**:
The study included 352 eyes of 250 glaucoma patients followed up for an average of 8.1 ± 3.5 years. Slopes of change over time were evaluated by the mean deviation (MD) from standard automated perimetry. For each eye, the follow-up time was divided into two equal periods: the first half was used to obtain the slopes of change and the second period was used to test the predictions. Slopes of change were calculated with two methods: the conventional approach of ordinary least squares (OLS) linear regression and a Bayesian regression model incorporating information on risk factors and presence of progressive optic disc damage on stereophotographs. The mean square error (MSE) of the predictions was used to compare the predictive performance of the different methods.

**Results**:
Higher mean IOP, thinner central corneal thickness (CCT), and presence of progressive optic disc damage were associated with faster rates of MD change. Incorporation of risk factor information into the calculation of individual slopes of MD change with the Bayesian method resulted in better prediction of future MD values than with the OLS method (MSE: 4.31 vs. 8.03, respectively; *P* < 0.001).

**Conclusions**:
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. (ClinicalTrials.gov number, NCT00221897.)

^{ 1–6 }

^{ 5–14 }In addition, progressive optic disc damage has been shown to be highly predictive of future development of standard visual field loss.

^{ 14 }Therefore, the presence of high IOP or progressive optic disc damage, for example, would increase the probability that a change detected in SAP is indeed real. On the other hand, a change in SAP in the presence of low IOP and absence of optic disc changes may have a higher probability of being just fluctuation over time, and not related to progression. Such clinical assessment is often done subjectively and the relationship between SAP changes, progressive optic disc damage, and risk factors, such as IOP, is not formally taken into account in the decision-making process.

^{ 15–19 }The true rate of change, however, is actually an unobservable variable, and the slope of change obtained from OLS is just an imprecise estimate that is confounded by noise and influenced by the number and intervals of measurements during follow-up.

^{ 20 }OLS estimates are obtained by taking into account only the measurements of an individual patient, without considering the influence of other existing information. Although other assembled data are rarely used to improve the accuracy of a patient's estimated rate of visual field loss, it could be argued that improved accuracy would be possible by incorporating information on risk factors and presence of concomitant structural damage. For example, it is reasonable to assume that the best estimator of the rate of visual field change in a patient for whom there are no SAP measurements collected over time is the average rate of change in the overall population where he or she comes from, that is, a population with a similar risk profile. As measurements are acquired for this patient, however, his or her rate of SAP change will most likely deviate from the population average. For patients with fewer measurements, the accuracy and precision of the estimates can be increased by “borrowing strength” from the population with similar risk factors, whereas for patients with large number of measurements, accurate and precise estimates can be obtained by relying almost only on the individual data, and the need to borrow strength from the population decreases.

^{ 20 }

^{ 21,22 }to improve estimation of rates of visual field loss,

^{ 20,22 }as well as for merging event- and trend-based methods for assessment of change.

^{ 23 }In the current study, Bayesian models were used to incorporate risk factors, in addition to structural information, into the assessment of rates of SAP change over time. It was demonstrated that these rates can more accurately predict future observations and show increased precision than do traditional OLS estimates.

^{ 24 }Visual fields with more than 33% fixation losses or false-negative errors, or more than 15% false-positive errors were excluded. The only exception was the inclusion of visual fields with false-negative errors of more than 33% when the field showed advanced disease. Visual fields exhibiting a learning effect were also excluded. Visual fields were further reviewed for the following artifacts: lid and rim artifacts, fatigue effects, inappropriate fixation, evidence that the visual field results were due to a disease other than glaucoma (such as homonymous hemianopia), and inattention. The VisFACT requested repeats of unreliable visual field test results, and these were obtained whenever possible.

**Step 1: Identification of Risk Factors Associated with Visual Field Progression.**To identify and quantify the factors associated with visual field progression, a regression model was initially built that evaluated the relationship between rates of visual field loss and putative predictive factors. The potential risk factors evaluated were age at baseline, mean IOP, CCT, and presence of optic disc hemorrhages. The relationship between progressive optic disc damage detected on stereophotographs and rates of visual field progression was also evaluated.

**Step 2: Bayesian Regression Model.**For the Bayesian analysis of trend in the visual fields over time, a random-intercept random-slope Bayesian hierarchical model was fitted for the MD data. In these models, the average evolution of MD values was described using a linear function of time, and eye-specific deviations from this average evolution were introduced by random intercepts and random slopes, allowing for different baseline values and different rates of change for each eye.

^{ 25–27 }In our application, the Bayesian models incorporated results from risk factors and presence of optic disc progression in the prior distribution for the slopes, which allowed them to influence the MD slopes.

*y*represents the MD data and the α

_{i}*s and β*

_{j}*s represent the intercepts and slopes for each one of the*

_{j}*j*eyes, respectively. Each eye had n

*visual field tests over time. Equation (1) is called the data model or the likelihood. The α*

_{j}*s and β*

_{j}*s were modeled by using a bivariate normal distribution, that is, in Bayesian inference, the bivariate distribution was used as the “prior” distribution for the pairs of slopes and intercepts. To allow predictive factors to influence inferences made on MD slopes, a group-level regression on the slopes was added. This can be written as:*

_{j}*u*represents the value of the

_{kj}*k*th predictive factor on the

*j*th eye. By doing that, different prior distributions are permitted for the slope of MD change for each eye. For any particular eye, its slope β

*has a prior distribution with mean*

_{j}*= γ*Display Formula

_{j}_{0}+ γ

*. The γ*

_{k}u_{kj}*s represent the coefficients associated with each one of the*

_{k}*k*th predictive factors and were obtained from step 1.

^{ 28 }For each eye

*j*, the likelihood for the MD slope indicates the range of values of β

*that are most consistent with the data available for that eye. The likelihood is more informative as the sample size increases, that is, the larger the number of MD measurements available for a particular eye, the more informative the likelihood will be. The prior distribution conveys information about the distribution of the MD slopes (β*

_{j}*s) among the eyes in the population with similar values on the predictive factors. The posterior distribution is centered at a point between the maximum likelihood estimate and the maximum of the prior distribution—a weighted average of the likelihood and prior estimates—falling closer to the prior when the number of measurements per eye is small or have a large variance, and closer to the likelihood when the number of measurements is large or there is a small variance. Therefore, the influence of the prior is greater when there is less information to estimate the slope, based on the fact that only a few MD measurements are available over time or there is large variability. When there are many measurements available over time for a particular eye, its slope of change can usually be estimated with great precision and, therefore, the prior exerts less influence.*

_{j}**Figure 1.**

**Figure 1.**

^{ 29 }However, fitting a more complex model with eyes nested within patient did not provide any overall improvement in our model and, therefore, only the results of the simpler model were reported here.

^{ 30 }Ten thousand iterations were used after discarding the initial 5000 iterations for burn-in. Convergence of the generated samples was assessed by standard tools in WinBUGS (trace plots, autocorrelation function plots) as well as Gelman-Rubin convergence diagnostics. After the posterior distributions were estimated, summary measures were calculated, such as mean and credible intervals. For the Bayesian MD slope, it was considered that progression had occurred if the upper limit of the 95% credible interval for the slope was less than zero.

*optic disc progression × follow-up time*in Table 1. The shorter the duration of follow-up until detection of optic disc change, the faster was the associated rate of glaucomatous visual field loss. Higher mean intraocular pressure during follow-up was also associated with faster visual field progression; however, the effect of IOP showed a quadratic relationship with rates of field loss as shown in Figure 2. For values of mean IOP above approximately 30 mm Hg, visual field losses accelerated at a faster rate than below this number. Thin corneas were also associated with faster rates of MD change, with a 0.21 dB/year faster rate for each 100 μm thinner cornea.

**Table 1.**

**Table 1.**

Predictive Factor | Coefficient | Estimate | 95% CI | P |

Mean IOP* | γ_{1} | −0.028 | −0.045 to −0.011 | <0.001 |

(Mean IOP)^{∧}2 | γ_{2} | −0.0035 | −0.005 to −0.002 | <0.001 |

CCT,† per 100 μm | γ_{3} | 0.21 | 0.04 to 0.38 | 0.015 |

Optic disc progression‡ | γ_{4} | −0.35 | −0.51 to −0.19 | <0.001 |

Optic disc progression × follow-up time§ | γ_{5} | 0.083 | 0.029 to 0.136 | 0.003 |

Constant | γ_{0} | 0.08 | −0.001 to 0.162 | 0.054 |

**Figure 2.**

**Figure 2.**

*P*= 0.042). However, when included along with evidence of progressive optic disc damage on stereophotographs, in a multivariable analysis, optic disc hemorrhages lost their significance (

*P*= 0.788). Age at baseline did not significantly influence rates of progressive visual field loss in the studied population (

*P*= 0.239). Owing to their lack of significance in the multivariable model, both age and optic disc hemorrhages were not included as factors influencing the estimation of individual slopes of change in the Bayesian method.

**Figure 3.**

**Figure 3.**

*P*< 0.001). Figure 4 shows a scatterplot of the difference between the absolute residuals for each method versus time. In this plot, the initial time in the horizontal axis was set to zero at the first visual field available to check predictions. The plot shows that residuals for OLS estimates tended to be larger than those for the Bayesian model, that is, predictions from OLS regression were in general worse than those based on the Bayesian model.

**Figure 4.**

**Figure 4.**

*P*< 0.001), indicating that Bayesian slopes were in general more precise than those obtained by OLS regression. Figure 6 shows a scatterplot of the standard errors of the MD slopes obtained by the Bayesian and OLS methods.

**Figure 5.**

**Figure 5.**

**Figure 6.**

**Figure 6.**

**Figure 7.**

**Figure 7.**

^{ 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.

^{ 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.

^{ 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.

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