**Purpose.**:
To present and evaluate a new methodology for combining longitudinal information from structural and functional tests to improve detection of glaucoma progression and estimation of rates of change.

**Methods.**:
This observational cohort study included 434 eyes of 257 participants observed for an average of 4.2 ± 1.1 years and recruited from the Diagnostic Innovations in Glaucoma Study (DIGS). The subjects were examined annually with standard automated perimetry, optic disc stereophotographs, and scanning laser polarimetry with enhanced corneal compensation. Rates of change over time were measured using the visual field index (VFI) and average retinal nerve fiber layer thickness (TSNIT average). A Bayesian hierarchical model was built to integrate information from the longitudinal measures and classify individual eyes as progressing or not. Estimates of sensitivity and specificity of the Bayesian method were compared with those obtained by the conventional approach of ordinary least-squares (OLS) regression.

**Results.**:
The Bayesian method identified a significantly higher proportion of the 405 glaucomatous and suspect eyes as having progressed when compared with the OLS method (22.7% vs. 12.8%; *P* < 0.001), while having the same specificity of 100% in 29 healthy eyes. In addition, the Bayesian method identified a significantly higher proportion of eyes with progression by optic disc stereophotographs compared with the OLS method (74% vs. 37%; *P* = 0.001).

**Conclusions.**:
A Bayesian hierarchical modeling approach for combining functional and structural tests performed significantly better than the OLS method for detection of glaucoma progression. (ClinicalTrials.gov number, NCT00221897.)

^{ 1 –9 }On the other hand, several patients show evidence of functional deterioration without measurable changes in currently available structural tests.

^{ 4,5,10 }The disagreement between structural and functional methods for detecting progression could be related to the different algorithms used to assess change, the variability of measurements over time, or the different scales used to assess structure and function.

^{ 8,10 –19 }Whatever the reason might be, it is likely that a combination of structural and functional measurements would improve detection of clinically significant disease progression compared with either method used alone.

^{ 9,20 –23 }Although most patients progress relatively slowly, others have aggressive disease with fast deterioration that can eventually result in blindness or substantial impairment unless appropriate interventions are made. The elucidation of the longitudinal relationship between structural and functional tests and their rates of change over time is essential to determine the relative utility of these tests in monitoring the disease.

^{ 24 }In brief, the VFI represents the percent of normal age-corrected visual function, and it is intended for use in calculating rates of progression and staging glaucomatous functional damage. Evaluation of rates of functional loss in glaucomatous eyes with the VFI has been proposed to be less susceptible than the mean deviation (MD) to the effects of cataract or diffuse media opacities.

^{ 24 –26 }The VFI can range from 100% (normal visual field) to 0% (perimetrically blind field). The current Humphrey software analyzes the rate of VFI change over time using ordinary least-squares (OLS) linear regression, and the printout shows a message indicating whether the slope of the regression line is statistically significant.

^{ 27 –30 }Assessment of SLP image quality was performed by an experienced examiner masked to the subject's identity and the results of the other tests. To be classified as good quality, an image required a focused and evenly illuminated reflectance image with a centered optic disc. The image quality score had to be greater than or equal to 7. Image quality was evaluated by masked trained technicians at a reading center.

^{ 31 }

^{ 32 –35 }Linear mixed models were used to evaluate the evolution of each response over time. In these models, the average evolution of a specific response is described using some function of time, and subject-specific deviations from this average evolution are introduced by random intercepts and random slopes, allowing for different baseline values and different rates of change for each patient. Linear mixed models are a natural extension of Bayesian hierarchical models, where the first level of hierarchy corresponds to within-patient variation and the second level to between-patient variation. The Bayesian framework, however, allows more flexible specification of the model assumptions and easier implementation of the joint modeling approach, as described below.

^{ 36 –42 }

^{ 32 }

*Y*

_{1}and

*Y*

_{2}represent the longitudinal measurements of SLP and SAP, respectively, for a subject

*i*taken at time

*t*, and

*m*

_{1}(

*t*) and

*m*

_{2}(

*t*) represent the average linear evolution of each response over time, that is, the average rate of change of SLP measurements and the average rate of change in SAP sensitivity. Both response trajectories are tied together through a multivariate distribution of the random effects, where

*a*

_{1}and

*b*

_{1}correspond to random intercepts and random slopes, respectively, for process 1 (SLP), and

*a*

_{2}and

*b*

_{2}correspond to random intercepts and random slopes for process 2 (SAP). By adding an additional level for eye nested within patient, it is also possible to use information from both eyes of the same patient taking into account the correlation between them.

^{43,44}In the present application, heterogeneity is to be expected, as only a proportion of eyes will show progression over time. Further, in the progressing group, only a small proportion is expected to have relatively fast progression. This situation can induce considerable nonnormality, or skewness, in the random effects distribution. To address the problem, we used a multivariate skew

*t*distribution to model random effects. Both the

*t*and skew

*t*distributions allow greater flexibility in the distribution of random effects compared with the normal distribution and have been successfully applied to model nonnormal random effects.

^{41,45}The probability density function for a random effect

*b*that is

_{j}*t*distributed

*t*(μ,

*v*,

*k*) is where μ is the mean,

*v*the scale parameter, and

*k*the degrees of freedom determining the weight of the tails, giving variance σ

^{2}= [

*k*/(

*k*− 2)]

*v*. The

*t*distribution can be extended to accommodate the multivariate case.

^{46}To allow for skewing, Fernandez and Steel

^{47}proposed the introduction of skewness to any unimodal distribution symmetric ∼0, using a scale factor γ on each side of 0. For a random variable −∞ <

*x*< ∞, this method gives: where

*I*

_{[c,d]}is the indicator function for

*x*, being between

*c*and

*d*, and γ controls the mass on each side of 0, representing the skewness.

*t*distribution. This multivariate distribution acted as a “prior” for the estimation of the intercepts and slopes for each eye and the parameters of the multivariate distribution were themselves estimated from the data. However, to perform Bayesian inference, we also need to specify priors for the parameters of the multivariate distribution (i.e., the hyperparameters). Unless there are strong prior beliefs in the hyperparameter values, it is usually desirable to use noninformative priors. Therefore, we used normal (0,1000) for μ and uniform (0,100) for σ.

^{ 48 }An exponential (0.1) prior was used for

*k*in the

*t*distribution,

^{ 47 }but restricted to

*k*> 2.5 to prevent problems with undefined variance when

*k*≤ 2. For the skew parameter γ, we used a γ(0.5, 0.318) prior for φ = γ

^{2}. These are general priors used in previous applications of Bayesian hierarchical models using

*t*and skew

*t*distributions.

^{ 45 }

^{ 49 }We used 10,000 iterations after discarding the initial 5,000 iterations for burn-in. Convergence of the generated samples was assessed by standard tools in WinBUGS (trace plots and autocorrelation function [ACF] 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 a specific test, we considered that progression had occurred if the upper limit of the 95% credible interval for the slope was less than 0.

*n*− 1 eyes and applied to obtain the estimates of progression in the

*n*th eye. We compared our algorithm for detection of progression to standard approaches reported on the printouts of the instruments and used in previous studies.

^{ 8,10,25,50 –53 }The standard approach is based on OLS linear regression of measurements over time, so that an eye is considered to have progressed if a negative OLS regression slope is significantly different from 0, with

*P*< 0.05. OLS regressions were performed separately for each eye and for each test (i.e., SAP VFI and SLP TSNIT average).

*P*< 0.001). However, eyes that progressed only by TSNIT average tended to have less severe baseline disease than did those that progressed by both methods, as indicated by differences in average baseline values for VFI (94% ± 11% vs. 87% ± 12%;

*P*= 0.004), respectively.

PGON (n = 38) | Healthy (n = 29) | P † | AUC (SE) | |
---|---|---|---|---|

Bayesian VFI slope, %/y | −0.82 ± 1.34 | 0.01 ± 0.08 | <0.001 | 0.94 (0.03) |

Bayesian TSNIT slope, μm/y | −1.04 ± 0.80 | −0.24 ± 0.20 | <0.001 | 0.90 (0.04) |

OLS VFI slope, %/y | −1.11 ± 1.67 | 0.10 ± 0.48 | <0.001 | 0.77 (0.06) |

OLS TSNIT slope, μm/y | −1.27 ± 1.46 | 0.47 ± 1.74 | <0.001 | 0.81 (0.05) |

*P*= 0.04).

*P*< 0.001), while showing the same specificity in healthy eyes. In addition, the Bayesian method identified as progressing a significantly higher proportion of eyes that had progression on optic disc stereophotographs compared to the OLS method (74% vs. 37%;

*P*= 0.001). The ROC curve areas to discriminate healthy eyes from those with PGON on stereophotographs were significantly larger for the Bayesian method compared with OLS regression for VFI slopes (0.94 vs. 0.77;

*P*< 0.001) and for TSNIT average slopes (0.90 vs. 0.79;

*P*= 0.004), respectively.

Progression Only by the Bayesian Method (n = 65) | Progression Only by the OLS Method (n = 25) | P † | |
---|---|---|---|

VFI | |||

Average Bayesian slope | −0.74 ± 1.23 | −0.08 ± 0.11 | <0.001 |

Average standard error of the Bayesian slopes | 0.46 ± 0.26 | 0.31 ± 0.04 | <0.001 |

Average OLS slope | −0.90 ± 1.55 | −0.11 ± 0.39 | <0.001 |

Average standard error of the OLS slopes | 1.06 ± 1.20 | 0.34 ± 0.30 | <0.001 |

TSNIT Average | |||

Average Bayesian slope | −1.04 ± 0.62 | −0.40 ± 0.32 | <0.001 |

Average standard error of the Bayesian slopes | 0.58 ± 0.24 | 0.42 ± 0.11 | <0.001 |

Average OLS slope | −1.44 ± 0.98 | −0.99 ± 0.84 | 0.033 |

Average standard error of the OLS slopes | 0.84 ± 0.75 | 0.28 ± 0.38 | <0.001 |

^{ 1 }and, therefore, is a suitable, although still imperfect, reference method for detection of progression.

^{ 54 }In the subjects with PGON by optic disc photos, the Bayesian method also largely outperformed the OLS regression approach with sensitivity of 74% compared to 37%, respectively.

^{ 4,5 }That is, optic disc photos have limited sensitivity. Therefore, if we rely on optic disc stereophotographs as the only reference standard, many eyes with true progression will not be detected, which is likely to make the test under investigation appear to have a high number of false positives. If we had estimated specificity on the basis of the number of the 405 glaucomatous and suspect eyes that had no change on optic disc stereophotographs, the specificities would be 82.6% and 89.6% for the combined Bayesian and OLS methods, respectively. It is likely that these specificities would be underestimated because of the inability of optic disc stereophotographs to detect all cases of glaucoma progression. In fact, the very high specificity of the combined Bayesian method when applied to healthy eyes suggests that the significant changes detected in the glaucoma and suspect population were indeed representative of true disease deterioration. It should be noted, however, that estimation of specificities in a group of healthy eyes is not without problems. In clinical practice, visual fields and imaging instruments are applied to detect and monitor disease in diseased eyes or those with suspected glaucoma. Healthy eyes may have different characteristics from the eyes followed in clinical practice such as visual field variability, for example, and therefore estimates of specificity obtained from healthy eyes may be different from those in the clinically relevant population. For this reason, some authors

^{ 8 }have used as a surrogate for specificity the proportion of eyes showing positive slopes on the clinical tests, based on the assumption that real improvement does not occur in glaucoma. Using this approach, the Bayesian and OLS methods would have specificities of 100% and 97.7%, respectively, in the 405 eyes of with diagnosed and suspected glaucoma.

^{ 55 }In addition, the correlation between results of both methods is formally taken into account in the model decision framework, which helps solve potentially conflicting results. Using the combined Bayesian method, a visual field change that would otherwise be declared nonstatistically significant by analysis of visual field data alone was declared significant after taking into consideration the structural changes occurring in the same eye and vice versa. In fact, many eyes with relatively fast rates of visual field loss were declared as nonprogressing by the OLS method in our study due to the variability of measurements over time (i.e., large SE of the OLS regression slope), as shown in Table 2 and Figure 6. In these cases, one frequently has to obtain more tests to attempt to more precisely estimate the slope of OLS regression. 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. The current Humphrey Field Analyzer printout requires a minimum of five visual field tests to calculate the OLS slope. Therefore, we reanalyzed our results on eyes with a minimum of five visual fields. From the group of eyes with progressive optic disc damage, 27 eyes had at least five visual fields. The Bayesian approach was able to detect 22 (81%) of these eyes versus only 11 (41%) of the OLS approach. Therefore, there was a benefit of the Bayesian approach, even when only eyes with a greater number of tests were considered. Although different techniques have been used in an attempt to decrease the impact of variability in the detection of visual field progression, to our knowledge, no previous method has been reported that combines structural and functional change.

^{ 32 –35,56 }It is important to note, however, that even with the combined Bayes method, some eyes were identified as progressing by structure or by function only. Such a level of disagreement is not surprising, and several previous studies have also documented disagreement on detection of glaucoma progression by different tests. The disagreement could be related to the ability of the tests in identifying progression at different stages of the disease. Previous studies have suggested that SLP may be better suited for detection of progression at relatively early stages of disease, whereas the technology may fail to detect change in cases with advanced damage.

^{ 9,15,19,23,26,57 }On the other hand, the logarithmic scaling of clinical perimetric data may favor detection of change in later stages of disease with SAP.

^{ 2,58 }Also, VFI values only show decrease after a certain threshold of abnormality on the pattern deviation plot has been exceeded.

^{ 24 }In fact, the eyes that progressed only by SLP in our study had significantly less severe baseline disease than did the eyes that progressed by visual fields. The presence of disagreement actually reinforces the need for a combined approach for detection of glaucoma progression to allow effective monitoring of the disease at all stages of damage. It is important to emphasize that, for the combined Bayes approach, the presence of clear change in function was still declared as significant, even when not occurring concomitantly to a change in structure, and vice versa. By joint modeling random intercepts and slopes for SLP and SAP, the Bayesian approach can also take into consideration the influence of different stages of disease severity on the slopes of change.

*r*= 0.93).

^{ 59 }Although rates of change in glaucoma have traditionally been estimated using OLS linear regression, the true rate of change, however, is actually a latent or 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. OLS estimates are obtained taking into account only the measurements of an individual patient, without considering the influence of the population where the patient comes from. The Bayesian hierarchical modeling approach, however, improves the precision of an individual patient's estimate of slope of change by using previously longitudinally collected data from other patients. For example, it is reasonable to assume that the best estimator of the rate of change in a patient in whom we do not have any measurements collected over time is the average rate of change in the overall population of which the individual is a part. As measurements are acquired for this patient, however, the rate of change will most likely deviate from the population average. For patients with fewer measurements, the precision of the estimates can be increased by “borrowing strength” from the population, whereas for patients with large number of measurements, precise estimates can be obtained relying almost only on the individual data and the need to borrow strength from the population decreases. It is also worth emphasizing that the combined structure–function approach, as reported in this study, is still advantageous compared with Bayesian slopes of change obtained by applying the Bayesian methodology to each test separately. For example, 23% of the eyes detected as progressing by the VFI using the combined approach would not be detected as progressing if structural information was not included in the model. Conversely, 18% of the eyes would not be detected by TSNIT slopes if the Bayesian model did not include functional information.

^{ 2,60 –66 }However, the assumption of linear change is probably a reasonable one when evaluating change in periods of short- to medium-term follow-up, as performed in clinical practice. It should be noted, however, that extensions of our methodology to incorporate nonlinear change are also possible, but the evaluation is likely to require populations with longer follow-up times.

*Arch Ophthalmol*. 2009;127(10):1250–1256. [CrossRef] [PubMed]

*Prog Retin Eye Res*. 2007;26(6):688–710. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2004;45(9):3152–3160. [CrossRef] [PubMed]

*Arch Ophthalmol*. 2002;120(6):701–713, 2002; discussion 829–730. [CrossRef] [PubMed]

*Ophthalmology*. 2005;112(3):366–375. [CrossRef] [PubMed]

*Arch Ophthalmol*. 2005;123(4):464–470. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2006;47(7):2904–2910. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2010;51(1):217–222. [CrossRef] [PubMed]

*Ophthalmology*. 2009;116(6):1125–1133, e1121–1123. [CrossRef] [PubMed]

*Prog Retin Eye Res*. 2005;24(3):333–354. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2009;50(4):1682–1691. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2003;44(9):3873–3879. [CrossRef] [PubMed]

*Arch Ophthalmol*. 2001;119(10):1492–1499. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2011;52:519–525. [CrossRef] [PubMed]

*Ophthalmology*. 2010;117(3):462–470. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2009;50(12):5741–5748. [CrossRef] [PubMed]

*Br J Ophthalmol*. 2005;89(11):1427–1432. [CrossRef] [PubMed]

*Am J Ophthalmol*. 2009;148(1):155–163, e151. [CrossRef] [PubMed]

*Br J Ophthalmol*. 2011;95:502–508. [CrossRef] [PubMed]

*Lancet*. 2004;363(9422):1711–1720. [CrossRef] [PubMed]

*Ophthalmology*. 2001;108(2):247–253. [CrossRef] [PubMed]

*Ophthalmology*. 2009;116(12):2271–2276. [CrossRef] [PubMed]

*Am J Ophthalmol*. 2010;149(6):908–915. [CrossRef] [PubMed]

*Am J Ophthalmol*. 2008;145(2):343–353. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2009;50(8):3737–3742. [CrossRef] [PubMed]

*Br J Ophthalmol*. Published online September 10, 2010.

*Am J Ophthalmol*. 1995;119(5):627–636. [CrossRef] [PubMed]

*Arch Ophthalmol*. 1990;108(4):557–560. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2002;43(7):2221–2228. [PubMed]

*Invest Ophthalmol Vis Sci*. 2006;47(9):3870–3877. [CrossRef] [PubMed]

*Arch Ophthalmol*. 2009;127(9):1136–1145. [CrossRef] [PubMed]

*Stat Med*. 2004;23(2):231–239. [CrossRef] [PubMed]

*Stat Methods Med Res*. 2007;16(5):387–397. [CrossRef] [PubMed]

*Stat Med*. 2003;22(9):1457–1464. [CrossRef] [PubMed]

*Biometrics*. 1995;51(2):413–424. [CrossRef] [PubMed]

*Stat Med*. 2008;27(3):418–434. [CrossRef] [PubMed]

*Stat Med*. 2007;26(6):1255–1267. [CrossRef] [PubMed]

*Stat Med*. 2010;29(25):2643–2655. [CrossRef] [PubMed]

*Stat Med*. 2010;29(23):2384–2398. [PubMed]

*Biometrics*. May 10 2010.

*Comput Stat Data Anal*. 2008;52(11):5033–5045. [CrossRef]

*Comput Stat Data Anal*. 2008;52(3):1347–1361. [CrossRef]

*J Am Stat Assoc*. 1996;91:217–221. [CrossRef]

*Comput Stat Data Anal*. 1997;23:541–556. [CrossRef]

*Stat Med*. 2008;10 27(3):418–434. [CrossRef]

*Multivariate Distributions and Their Applications*. Cambridge, UK: Cambridge University Press; 2004.

*J Am Stat Assoc*. 1998;93(441):359–371.

*Bayesian Anal*. 2006;1(3):515–533.

*Stat Comput*. Oct 2000:10(4):325–337. [CrossRef]

*Ophthalmology*. 2011;118:763–767. [CrossRef] [PubMed]

*Arch Ophthalmol*. 2010;128(5):560–568. [CrossRef] [PubMed]

*Ophthalmology*. 2009;116(5):840–847. [CrossRef] [PubMed]

*J Glaucoma*. 2011;20:223–227. [CrossRef] [PubMed]

*Am J Ophthalmol*. 2005;139(6):1010–1018. [CrossRef] [PubMed]

*Biometrics*. 2006;62(2):424–431. [CrossRef] [PubMed]

*Biometrics*. 1982;38(4):963–974. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2009;50(4):1675–1681. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2000;41(7):1774–1782. [PubMed]

*J Glaucoma*. Published online March 16, 2011.

*Ophthalmology*. 1988;95(6):723–727. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2007;48(4):1651–1658. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2007;48(1):258–263. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2005;(11):4182–4188.

*Invest Ophthalmol Vis Sci*. 2004;45(6):1823–1829. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2006;47(12):5356–5362. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2002;43(7):2213–2220. [PubMed]