**Purpose.**:
To assess whether neuroretinal rim area (RA) measurements of the optic disc could be used to improve the estimate of the rate of change in visual field (VF) mean sensitivity in patients with ocular hypertension (OHT) using a Bayesian linear regression (BLR), compared to a standard ordinary least squares linear regression (OLSLR) of mean sensitivity (MS) measurements alone.

**Methods.**:
MS and RA measurements were analyzed from a longitudinal series of 179 patients with OHT visiting Moorfields Eye Hospital between 1992 and 2000. For each patient, linear regression of RA was computed after an appropriate transformation to “scale” RA with MS measurements, and the slope coefficient from this regression was used as a prior for BLR of MS. The BLR then was compared with the OLSLR approach by evaluating how accurately each regression technique predicted future MS measurements.

**Results.**:
On average, BLR was significantly more accurate than OLSLR for series up to 8 measurements long (root-mean-square prediction error [RMSPE] was 0.14 decibels [dB] smaller with BLR than OLSLR; *P* < 0.001, Wilcoxon signed-rank test), with OLSLR of VF data alone being more accurate for longer series (RMSPE was 0.06 dB smaller with OLSLR than BLR).

**Conclusions.**:
BLR provides a significantly more accurate estimate of the rate of change in MS than the standard OLSLR approach, especially in short time series, suggesting that structural measurements can be used successfully in statistical models to assist clinicians monitoring VF progression in patients with OHT. Further studies are necessary to validate the method in glaucoma patients.

^{ 1–3 }At present, glaucomatous eyes generally are identified and monitored using either structural or functional change.

^{ 3–5 }Change in both modalities may occur, though not always identified at the same time within the relatively short time span of typical longitudinal studies. One explanation for this dissociation is that VF sensitivity, optic nerve head (ONH) topographic parameters, and retinal nerve fiber layer (RNFL) thickness measurements all are affected by measurement variability,

^{ 6–17 }which may confound attempts to identify progression when it is present. Structural changes not linked directly to retinal ganglion cell (RGC) loss, such as conformational changes in the lamina cribrosa or in glial support tissue, and changes in function not linked directly to RGC loss, such as the development of media opacity or RGC dysfunction, are further possible explanations for structure-function dissociation.

^{ 3,4,18 }Such models not only are quantitative and objective, but also can reduce the effects of measurement variability. Bayesian approaches are useful for integrating data in this manner, yet there has been little application of Bayesian methodology to integrate data in different domains in glaucoma research.

^{ 4,18,19 }Bayesian statistics are based on the notion that

*prior*beliefs about the data under investigation should be considered in the statistical analysis to modify

*posterior*beliefs. Bayesian analysis specifies this modification statistically using a formula known as

*Bayes' theorem*. This theorem expresses the posterior probability of a given hypothesis in terms of the probability of the hypothesis (before integrating new evidence), the prior probability of the data under all possible hypotheses, and the conditional probability of observing the data given the specified hypothesis is true (known as the

*likelihood function*).

*P*value) of change.

^{ 16,20,21 }The objective of our study was to assess whether Bayesian normal error linear regression (BLR), which integrates imaging measurements of global neuroretinal rim area (RA), as an informative prior, could be used to detect more accurately a change in perimetric mean sensitivity (MS) over time compared to a standard OLSLR of mean sensitivity measurements alone in patients with ocular hypertension (OHT).

^{ 22 }Briefly, OHT was defined as a previously untreated intraocular pressure >22 mmHg and <35 mmHg on two or more occasions within a 2-week period, and a baseline mean Advanced Glaucoma Intervention Study (AGIS) VF score of 0.

^{ 23 }Patients had visual acuity at recruitment of 6/12 or better, with no coexistent ocular or neurologic disease. Patients who met the inclusion criteria and gave their informed consent were enrolled. The study adhered to the tenets of the Declaration of Helsinki and had local ethics committee approval.

^{ 24 }

*μ*m were excluded to best reflect clinical practice. Images also were excluded if the contour lines exported from the baseline mean topography could not be aligned satisfactorily in the follow-up mean topography. In total, 8 (from 1467) mean topographies were excluded for this reason. Single topographies were imported as HRTport files into a beta version of Heidelberg Retina Tomograph Explorer Version 3.1.2.0, and RA was calculated using the Moorfields reference plane, which has been shown to improve repeatability and the detection of progression compared to the “standard” and 320-

*μ*m reference planes.

^{ 25,26 }

^{ 27 }R code from the “MCMCregress” function in the “MCMCpack” package was used to carry out BLR.

^{ 28 }

**Table 1.**

**Table 1.**

Measurement | Median (Inter-Quartile Range) |

Age (y) | 60.0 (53.4, 67.7) |

Arithmetic mean sensitivity (dB) | 29.4 (28.5, 30.5) |

Global rim area (mm^{2}) | 1.19 (1.04, 1.37) |

Follow-up period (y) | 5.8 (4.8, 6.4) |

CLST tests (n) | 8 (7, 10) |

VF tests (n) | 8 (7, 10) |

**Figure 1.**

**Figure 1.**

*n*

_{1}=

*n*

_{2}= 179,

*P*< 0.001, Wilcoxon signed-rank test). OLSLR of VF data alone was more accurate than BLR for longer series (mean difference in RMSPEs, OLSLR minus BLR, was −0.06 dB;

*n*

_{1}=

*n*

_{2}= 179,

*P*> 0.1, Wilcoxon signed-rank test), although the difference in RMSPEs was not significant. While it is not recommended to deduce a trend from as few as three VFs, a Bayesian regression analysis including three MS/RA measurement pairs results in a median prediction error smaller than that from OLSLR of a series of five or six MS measurements; however, as seen in Figure 2, the range of prediction error is greater for BLR of three MS/RA measurements than OLSLR of longer series of MS measurements. BLR of a series of four MS/RA pairs leads to a median absolute prediction error approximately equivalent to that from OLSLR of a series of seven MS measurements, as well as a narrower error range (see dashed lines in Fig. 2).

**Figure 2.**

**Figure 2.**

*V*= 12,556,

*n*

_{1}=

*n*

_{2}= 179,

*P*<0.001). This suggests that, overall, BLR better predicts mean sensitivity and, therefore, more accurately estimates the rate of change in MS. Furthermore, the RMSPE for the BLR was smaller than that for the OLSLR in 134 out of 179 patients (74.9%). Figure 3A is a scatterplot (with marginal histograms) of the difference in RMSPE (over all series lengths) for the two regression approaches (BLR minus OLSLR) against the OLSLR slope derived from each patient's full series of MS measurements. Negative values in the Y-axis indicate that the RMSPE for BLR is smaller than the corresponding error for OLSLR; hence, 74.9% of the points lie below the

*y*= 0 line. Across all patients, the mean difference between the RMSPEs for the two linear regression approaches was −0.13 dB (shown by the dashed line in Fig. 3A), indicating that, on average, prediction error for OLSR was 0.13 dB greater than for BLR. Furthermore, the mean absolute difference in predicted MS using BLR as opposed to OLSLR was 0.74 dB.

**Figure 3.**

**Figure 3.**

*W*= 1334,

*n*

_{1}=

*n*

_{2}= 45,

*P*< 0.01), suggesting that VF variability was much lower in these patients. There was no statistical difference in the mean pixel height standard deviation between these two groups.

**Figure 4.**

**Figure 4.**

**Figure 5.**

**Figure 5.**

**Figure 6.**

**Figure 6.**

^{ 29–31 }This has been intensified by the growing availability of Bayesian statistical software and the speed of modern computers. Bayesian approaches offer the opportunity to add more variables to an inference model. In monitoring glaucomatous progression, clinicians are interested in a range of clinical measurements. New research evaluated a Bayesian hierarchical model for combining structural (average retinal nerve fiber layer thickness measurements from scanning laser polarimetry) and functional (visual field index [VFI] measurements from SAP) information, and found that the method improved detection of glaucoma progression compared with standard OLSLR of VFI measurements alone.

^{ 4 }The Bayesian hierarchical model is based a cohort of subjects and uses information from the whole sample to update the posterior estimate in each patient. Conversely, in our study, the BLR approach is not based on a hierarchical model and, instead, uses each patient's prior (structural) information separately. The BLR method was compared with the OLSLR approach, and the results suggested that visual field progression rates are more estimated accurately with the Bayesian method.

^{ 32 }Supplementing perimetry with imaging measurements allows clinicians to explore a different manifestation of progression, and is relatively quick to carry out. Images can be acquired on the same occasion as a VF test, whereas it is difficult in clinical practice to undertake two VF tests on a single visit; in addition, imaging generally is preferred by patients.

^{ 33 }

^{ 32 }There is a significant, growing need for clinically relevant automated, objective tools to define progression based jointly on structural and functional information together. At present, analysis software tends to judge structure and function test data separately in a binary “progression present” or “absent” manner. The approach outlined in this study provides an objective, quantitative framework for integrating structural and functional measurements that gives an output in the domain of greatest interest—the rate of visual field loss.

^{2}) units.

^{ 5 }However, studies of structure-function associations almost invariably report a non-linear relationship between visual function scaled in decibels and linearly scaled structural parameters.

^{ 34 }Thus, we investigated transforming RA in linear units into dB units, since the associations become more linear when structure and function are scaled in the same units.

^{34–36}Furthermore, the relationship between visual field sensitivity and ganglion cell number is linear when both are expressed in decibels.

^{ 37 }Interestingly, the results largely were unchanged, which suggests that scaling is less important over the short stage of the disease process examined in our study. It also is assumed that MS declines linearly over time; linear regression models have proved superior and practical compared to other more complex models for predicting VF sensitivity in time series.

^{ 38 }

^{ 16 }while RA (subsequently used to derive a prior for Bayesian analysis) is not age-corrected or weighted. Furthermore, research into structure-function models suggests arithmetic MS approximates better to retinal ganglion cell counts than indices based on a geometric mean, such as MD.

^{ 24 }This hypothesis is supported by analyzing the correlation between MD and RA for the data under investigation, which indicates that there is no significant correlation between these two variables (Kendall's

*τ*test:

*τ*= 0.004,

*Z*= 0.257,

*n*

_{1}=

*n*

_{2}= 1467,

*P*= 0.797), while MS and RA do correlate significantly (see Appendix). In addition, recent research suggests that MS may predict better future glaucomatous functional change compared with MD, and the authors suggest that this may be due to imperfect age-correction implicit in the calculation of MD.

^{39,40}

^{ 25 }Nevertheless, CLST rim area measurements are subject to errors when used to track glaucomatous progression. Errors include the incorporation of blood vessels, and other structures that may be unrelated to axon loss, into the RA measurement. Pulsations of retinal arteries in the optic nerve head also can produce variability in CLST stereoscopic measurements.

^{ 41,42 }An interesting extension to the current study would be to use other structural measurements to derive a prior for BLR that are likely to be related better to VF MS, such as RNFL thickness. Furthermore, patients often exhibit focal, rather than diffuse rim loss during the course of progressive disease. There is an unfortunate trade-off between the detection of shallow localized defects and variability. Global indices, such as MS and RA, have less variability than point-wise or sectoral measurements, but are more likely to overlook focal damage. Conversely, point-wise analyses may detect better subtle abnormalities compared with global indices, but are associated with greater variability. It is possible that structural information about focal damage could be used as a prior to reduce the large variability associated with functional analyses of localized defects, and overcome this trade-off. It remains to be ascertained whether the BLR approach can be applied successfully to RA sectors (or sectorial RNFL thickness measurements) as well as global structural measures; however, a preliminary analysis was performed using BLR on a sector-by-sector basis, and the results were very encouraging suggesting that the BLR approach offered similar improvements in accuracy and precision compared to the global approach.

^{ 17 }consequently, integrating structural information may prove even more valuable in patients with glaucoma. Moreover, patients with OHT represent a very important demographic, comprising a large proportion of patients in glaucoma clinics. Thus, methods to identify patients with OHT who have glaucoma as accurately as possible are understandably very important. The fewer number of visits, for equivalent prediction accuracy and precision, required for joint structure and function analysis would relieve a significant part of this burden of care.

^{ 43 }However, a great strength of the BLR method is that the Bayesian posterior precision is equal to the prior precision plus the precision of the likelihood. In early testing, the Bayesian confidence intervals for the slope and prediction of MS are markedly smaller than the corresponding confidence intervals derived from OLSLR of the VF data alone (see Fig. 4).

*P*< 0.05) from 27 patients compared to 71 slopes from 26 patients using OLSLR. Moreover, BLR resulted in only 8 significantly positive slopes from 7 patients compared to 29 slopes from 13 patients using OLSLR (rate greater than 0.25dB/year,

*P*< 0.05). In total, 8 patients were flagged as having progression using both methods. In these patients, on average, BLR flagged progression 0.45 years earlier than OLSLR. In those 19 patients in whom only BLR was significant, the mean time to flag progression was 2.5 years, while the corresponding time for OLSLR (15 patients) was 4.1 years. These results suggest that BLR detects glaucomatous progression sooner and with fewer false positives than OLSLR. Examining a rate of loss greater than 0.5dB/year, BLR resulted in 14 significantly negative slopes from 11 patients compared to 49 slopes from 17 patients using OLSLR. However, the greater number of progressing outcomes using OLSLR may be a result of false positives, and this is supported by examining the number of significant positive slopes from the two methods; BLR resulted in only 4 significantly positive slopes from 3 patients compared to 20 slopes from 8 patients using OLSLR.

^{ 4 }provide compelling evidence that structural measurements are informative about visual function, and moreover, that integrated structure-function statistical models can be useful tools to assist clinicians monitoring glaucomatous progression. Further research is necessary to develop such statistical methodologies for application in the clinic. However, preliminary results demonstrate clearly the feasibility and usefulness of these models.

^{ 28 }In addition, the R

^{ 27 }code to carry out BLR, as performed in our study, is available freely online as the ‘MCMCregress' function in the ‘MCMCpack' package;

^{ 28 }default settings for the function were used unless otherwise specified.

*y*is the response (outcome) variable,

*x*is the predictor variable,

*β*is a vector of the intercept term and the slope coefficient,

*e*is the error term and

*i*refers to the

*i*observation in a sample of

^{th}*n*observations (

*i*= 1, 2, … ,

*n*). In this model, the error term stipulates that the relationship between the response variable and the predictor variable is not exact, and generally is assumed to be Gaussian distributed with mean zero and constant variance (

*σ*

^{2}):

*β*and

*e*. In our study, the BLR model was implemented using semi-conjugate priors; a multivariate Gaussian prior was used for the intercept and slope coefficients and an inverse Gamma prior was used on the conditional error variance: where

_{i}*b*

_{0}is the vector of prior means and

*β,*while

*c*

_{0}is the shape parameter and

*d*

_{0}is the scale parameter for the inverse Gamma prior on

*σ*

^{2}. In our study, the mean and precision of the prior for the slope coefficient were derived from OLSLR of the patient's corresponding RA measurements over time. A non-informative prior was used for the intercept term (precision equal to zero). This assigns equal likelihood to all possible values of the intercept, and so has no impact on the posterior. The prior on the conditional error variance was specified using standard non-informative parameters −

*c*

_{0}= 0.001 and

*d*

_{0}= 0.001.

^{ 44 }Conversely, if the likelihood probability distribution is strong, that is the observed dependent variable (MS) displays a strong relationship with the independent variable (follow-up time) or the prior probability distribution is concentrated weakly, then the posterior probability distribution is largely subjugated by the likelihood. This is particularly pertinent to the application of BLR for monitoring glaucoma patients, since structure-function measurements are highly variable, and progression may be characterized better by one or the other quantity — whichever has least variability.

^{ 45 }The procedure requires that there is a significant, positive correlation using Kendall's

*τ*test between the two variables, which we demonstrate for the two variables here (Kendall's

*τ*test:

*τ*= 0.12,

*Z*= 6.79,

*n*

_{1}=

*n*

_{2}= 1544,

*P*<0.001).

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**Figure A1.**

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