A Joint Bayesian Longitudinal Model for Macular Structure–Function Correlations in Glaucoma

Abstract

Purpose

To investigate global longitudinal structure–function (SF) relationships between macular ganglion cell complex (GCC) thickness and central visual field (VF) mean deviation (MD) rates of change using a Bayesian joint bivariate longitudinal model.

Design

Prospective cohort study.

Participants

One hundred seventeen eyes from 117 patients with glaucoma with central damage or moderate to advanced glaucoma were included. Eligible patients had at least 4 visits over a follow-up period of 2 years or longer.

Methods

Longitudinal GCC thickness was assessed using optical coherence tomography, and central VF MD was measured with 10-2 standard automated perimetry. A Bayesian joint bivariate longitudinal model was used to estimate random intercepts, slopes, and residual standard deviations (SDs) for structural and functional measures and their correlations. A simulation study compared the Bayesian model (BM)'s performance against simple linear regression (SLR) in estimating these correlations.

Main Outcome Measures

Correlations between GCC and MD intercepts, slopes, and residual errors.

Results

The mean baseline MD was −8.3 (SD: 5.2) decibels, with an average follow-up period of 5.0 (SD: 0.9) years. The mean correlation was 0.47 (95% credible interval: 0.32 to 0.61) for GCC-MD intercepts (baseline values), 0.29 (95% credible interval: 0.04 to 0.52) for GCC-MD slopes (rates of change), 0.20 (95% credible interval: −0.06 to 0.44) for GCC-MD log residual SDs, and 0.060 (95% credible interval: −0.013, 0.132) for the observation-level GCC-MD residual correlation. The BM consistently demonstrated a smaller root mean squared error than SLR in estimating GCC-MD slope correlations in all simulated scenarios where GCC-MD residual correlation differed from GCC-MD slope correlation.

Conclusions

The Bayesian joint model improved accuracy and reduced uncertainty in estimating SF relationships compared to SLR. Correlations between global SF rates of change were significantly positive, although weaker than random intercept correlations. This model represents a key step towards developing local longitudinal SF models to enhance glaucoma progression monitoring.

Financial Disclosure(s)

Proprietary or commercial disclosure may be found in the Footnotes and Disclosures at the end of this article.

Glaucoma is a progressive optic neuropathy and a leading cause of blindness worldwide^1,2; hence, proper monitoring of disease progression is crucial for patient management. Traditionally, both structural and functional assessments are utilized to monitor patients with glaucoma.1, 2, 3, 4, 5, 6 It is widely accepted that evaluating structural and functional measures together is critical for timely detection of glaucoma progression and optimizing management.

Understanding the structure–function (SF) relationship, which refers to the correlation between structural damage and functional deterioration, is essential for early detection, assessing treatment effectiveness, and gaining a deeper understanding of glaucoma progression.7, 8, 9, 10, 11, 12, 13 Many previous studies investigating SF relationships in glaucoma have relied on simple linear regression (SLR) models.^8,14, 15, 16, 17 However, SLR has several limitations, such as analyzing outcomes independently and ignoring correlations between them.^18,19 Simple linear regression also fails to connect baseline measurements (intercepts) and rates of change (slopes), which is important for understanding progression.²⁰ Another shortcoming of SLR is that it fails to leverage valuable population-level information from the cohort to refine individual estimates of rates of change.²¹

To overcome these limitations, Bayesian joint bivariate longitudinal models offer a more advanced approach by jointly modeling multiple correlated outcomes, such as ganglion cell complex (GCC) thickness and central visual field (VF) mean deviation (MD), over time.22, 23, 24 These models account for individual variability through random effects and allow for estimating correlations between GCC and MD intercepts, slopes, and regression residuals. This means that these models can show how structural measurements and functional outcomes evolve together, which is crucial for understanding glaucoma progression. Moreover, Bayesian models (BMs) incorporate prior knowledge, can handle heteroscedasticity through random residual variances, and provide more accurate predictions of future observations.25, 26, 27

This study aimed to investigate the global longitudinal SF relationships between macular GCC thickness and central VF MD using a Bayesian joint bivariate longitudinal model. This model can properly capture correlations between GCC and MD intercepts, slopes, and regression residuals. The performance of this model was then compared with that of SLR, particularly in estimating SF correlations and predicting future observations. Additionally, a simulation study assessed the model's robustness across different scenarios, highlighting its potential advantages over SLR.

Methods

We analyzed data from 117 eyes (117 patients) from the Advanced Glaucoma Progression Study, a longitudinal study at the Stein Eye Institute, University of California, Los Angeles. The study was approved by the University of California, Los Angeles's Human Research Protection Program, adhered to the tenets of the Declaration of Helsinki, and conformed to the Health Insurance Portability and Accountability Act policies. All patients provided written informed consent at the time of enrollment in the study.

The inclusion criteria included (1) clinical diagnosis of primary open-angle glaucoma, pseudoexfoliative glaucoma, pigmentary glaucoma, or primary angle-closure glaucoma and (2) evidence of either central damage on 24-2 VF, defined as ≥2 points within the central 10° with P < 0.05 on the pattern deviation plot or 24-2 VF MD worse than −6 decibels (dB). The exclusion criteria were baseline age <40 years or >80 years, best-corrected visual acuity <20/50, refractive error exceeding 8 diopters (D) of sphere or 3D of cylinder, or any significant retinal or neurologic disease potentially affecting OCT measurements. We included eyes with at least 4 visits and 2 years of follow-up time.

Macular Imaging and VF Testing

We used Spectralis spectral-domain OCT (Heidelberg Engineering) to obtain macular volume scans. The Posterior Pole Algorithm of the Spectralis OCT acquires 30° × 25° volume scans of the macula (61 B-scans spaced approximately 120 μm apart) centered on the fovea. The Glaucoma Module Premium Edition software was used to segment the central 24° × 24° of the volume scan and obtain thickness measurements for individual retinal layers in an 8 × 8 array of 3° × 3° superpixels. Ganglion cell complex thickness was calculated by summing up the thickness measurements from the macular retinal nerve fiber layer (RNFL), ganglion cell layer (GCL), and inner plexiform layer. We used GCC thickness instead of GCL since GCC provides a more comprehensive structural assessment by incorporating the macular RNFL, GCL, and inner plexiform layer. Our previous studies showed that GCC was the optimal macular measure for detecting disease progression in moderate to advanced glaucoma, as it captured glaucomatous changes more effectively than the ganglion cell-inner plexiform layer or the GCL.^20,28 Global macular GCC measurements were estimated by averaging the GCC values across the central 7×7 superpixels after excluding the most inferior row and most nasal column of superpixels of the grid due to the high noise and frequent misreads in those peripheral superpixels. Before the averaging, 416 per 54 918 (0.8%) of the GCC observations were removed as outliers using a semiautomated algorithm.²⁹

We used the 10-2 pattern on the Humphrey Field Analyzer (Zeiss Meditec) to measure the VF sensitivity with the standard Swedish Interactive Thresholding Algorithm. Sensitivity measurements were obtained at 68 test locations within the central 10° of the VF. We excluded VFs with a false-positive rate >15%. The 10-2 VF MD value was used as the functional global measure.

Statistical Modeling

We fit our Bayesian joint bivariate longitudinal model with correlated random intercepts, slopes, residual standard deviations (SDs), and residuals to the global VF MD and GCC measures. The model estimates the correlations between structural and functional intercepts, slopes, residual SDs, and residuals while estimating each measure's eye-specific intercepts, slopes, and residual SDs. Letting y_1ij and y_2ij denote the global GCC thickness (μ m) and MD (dB) measurement, respectively, for a single eye i assessed at time t_ij, the model is

$$y_{1ij}={\beta }_{11i}+{\beta }_{12i}{t}_{ij}+{ϵ}_{1ij}$$

$$y_{2ij}={\beta }_{21i}+{\beta }_{22i}{t}_{ij}+{ϵ}_{2ij}$$

$$\left(\begin{array}{l}{ϵ}_{1ij}\\ {ϵ}_{2ij}\end{array}\right)\sim N\left(\left(\begin{array}{l}0\\ 0\end{array}\right),\left(\begin{array}{cc}{\sigma }_{1i}^2& \rho {\sigma }_{1i}{\sigma }_{2i}\\ \rho {\sigma }_{1i}{\sigma }_{2i}& {\sigma }_{2i}^2\end{array}\right)\right)\text{,}$$

where β_11i and β_21i are the eye-specific intercepts for GCC and MD, respectively; β_12i and β_22i are the eye-specific slopes for GCC and MD, respectively; ϵ_1ij and ϵ_2ij are random errors at time t_ij for GCC and MD, respectively; ${\sigma }_{1i}^2$ and ${\sigma }_{2i}^2$ are eye-specific residual variances for GCC and MD, and ρ is the correlation between the GCC-MD residuals. For each eye i, we model the 6 random effects ${\mathit{\beta }}_{\mathit{i}}={\left({\beta }_{11i},{\beta }_{21i},{\beta }_{12i},{\beta }_{22i},\log {\sigma }_{1i},\log {\sigma }_{2i}\right)}^{\prime }$ using a multivariate normal prior with mean $\mathit{\mu }={\left({\alpha }_{11},{\alpha }_{21},{\alpha }_{12},{\alpha }_{22},{\alpha }_{13},{\alpha }_{23}\right)}^{\prime }$ and an unstructured covariance matrix Σ, which can be decomposed into marginal variances and correlations between the elements of β_i. Here, α₁₁, α₁₂, and α₁₃ are the global intercept, slope, and log residual SD of GCC, respectively, and α₂₁, α₂₂, and α₂₃ are the global intercept, slope, and log residual SD of MD, respectively. We obtain correlations between the GCC-MD intercepts, slopes, and residual SDs from the posterior of Σ. If the interest lies in the changes in GCC (or MD) from the beginning to the end of follow-up, this change equals a multiple of the subject slope for GCC (MD), where the subject slope is equal to the sum of the population slope and the subject's random slope.

We compared the BM to a simpler method used in practice where the SF correlations are measured from estimated parameters from SLR fit separately on each eye and measure. Of course, SLR does not have natural built-in parameters to estimate the correlations, but this ad hoc procedure of correlating estimates has been used previously, although it is biased. We compared the estimated correlations for intercepts, slopes, residual SDs, and residuals between GCC and MD. Log residual SDs were used to better meet the assumptions of normality. To assess the practical usefulness of the BM, we evaluated the predictive performance by leaving out the last observation for each eye and fitted the Bayesian and SLR models to predict the last observation. We used root mean squared error (RMSE) to assess model prediction accuracy of the true observed value. We identified eye-specific rates of change as significantly negative or positive when the upper or lower limit of the 95% credible interval was less than or greater than 0, respectively. All statistical analyses were performed using R version 4.3.1,³⁰ and BMs were implemented with the R package NIMBLE.^31,32

Simulation Study

We conducted a simulation study to compare the performance of the BM to the SLR method. By varying the true SF slope correlations and SF residual correlations, we can test the validity of the models under a range of scenarios. We simulated global SF data for 117 subjects under the BM, using posterior means for the parameters from fitting the model on cohort data. We set the observed follow-up time to be 5 years for all subjects, with testing done at 0.5-year intervals. We simulated 200 data sets for each of 18 scenarios where the SF residual correlation was set as either −0.5, 0, or 0.5, SF slope correlation was either −0.5, 0, or 0.5, and the random slope variances were either set as the posterior means from the model fit or double of the posterior means (3 × 3 × 2 = 18 scenarios). We modified the residual correlation between GCC and MD measurements while keeping the residual variances constant. Similarly, we varied the correlation between random slopes while keeping the random slope variances constant. All other correlations and global parameters were set at the posterior means from the model fit. Table S1 (available at www.ophthalmologyscience.org) provides a summary of the 18 scenarios.

We compared the models across the 18 scenarios regarding bias, RMSE, and proportion of significant negative and positive correlations. Bias was calculated as the average estimate across 200 simulation data sets minus the truth, which measures how closely the model estimates the true parameter value.

Results

We included 1171 pairs of GCC-MD observations from 117 eyes (117 subjects). The mean baseline 10-2 MD was −8.3 (SD: 5.2) dB with a mean follow-time of 5.0 (SD: 0.9) years. Table S2 (available at www.ophthalmologyscience.org) provides the demographics of the cohort.

Simulation Results

Table 1 summarizes the RMSE for the BM and SLR across the 18 simulation scenarios. The BM consistently outperformed SLR in all scenarios for the GCC-MD residual correlation with RMSE >3 times smaller than the SLR RMSE; the only exception was that when the true residual correlation was 0, SLR performed comparably to the BM. For the GCC-MD slope correlation, the BM performed better than SLR whenever the true residual and true slope correlations differed. Simple linear regression performed better, although only modestly, than the BM when the true residual and slope correlations were the same. The RMSEs of SLR varied largely across the scenarios for a given true slope correlation, whereas the BM had a much smaller range of values. For example, in scenarios 1, 2, and 3, where the true slope correlation was −0.5, the RMSE for SLR varied from 0.074 to 0.235, whereas the RMSE only ranged from 0.090 to 0.105 for the BM. The SLR method performed especially poorly when the true residual and slope correlations had opposite signs (−0.5 and 0.5).

Table 1.

Summary of the RMSE of the Estimated Residual Correlation and Slope Correlation between GCC and MD across the 18 Simulation Scenarios for SLR and the BM

Scenario	GCC-MD Residual Correlation			GCC-MD Slope Correlation
	Truth	SLR RMSE	BM RMSE	Truth	SLR RMSE	BM RMSE
1	−0.5	0.090	0.024	−0.5	0.074	0.090
2	0.0	0.032	0.032	−0.5	0.143	0.096
3	0.5	0.097	0.024	−0.5	0.235	0.105
4	−0.5	0.090	0.024	0.0	0.146	0.111
5	0.0	0.032	0.032	0.0	0.097	0.112
6	0.5	0.096	0.024	0.0	0.131	0.113
7	−0.5	0.090	0.024	0.5	0.250	0.104
8	0.0	0.032	0.032	0.5	0.157	0.099
9	0.5	0.095	0.024	0.5	0.079	0.094
10	−0.5	0.090	0.024	−0.5	0.071	0.078
11	0.0	0.032	0.032	−0.5	0.101	0.081
12	0.5	0.097	0.024	−0.5	0.147	0.084
13	−0.5	0.090	0.024	0.0	0.114	0.098
14	0.0	0.032	0.032	0.0	0.094	0.099
15	0.5	0.096	0.024	0.0	0.105	0.099
16	−0.5	0.090	0.024	0.5	0.159	0.087
17	0.0	0.032	0.031	0.5	0.109	0.084
18	0.5	0.095	0.024	0.5	0.072	0.081

BM = Bayesian model; GCC = ganglion cell complex; MD = mean deviation; RMSE = root mean squared error; SLR = simple linear regression.

A smaller RMSE indicates better performance. The BM estimated residual correlations better than SLR; SLR estimated slope correlations better than the BM when the true residual correlation and slope correlation are the same, but SLR performed more poorly when the true correlations differed.

Tables S3 to S5 (available at www.ophthalmologyscience.org) present more detailed results from the simulation study, including bias and proportion of significant correlations for the GCC-MD intercepts, slopes, and residuals. The BM and SLR performed well in estimating the correlation for the GCC-MD intercepts. The true GCC-MD intercept correlation was 0.474, and both models identified the correlation as significantly positive 100% of the time. For GCC-MD slopes, SLR tended to have a higher bias when the true slope correlation and residual correlation had opposite signs. When the true slope correlation was 0, SLR would still identify a large proportion of slope correlations as significantly negative (positive) because the true residual correlation was −0.5 (0.5). For GCC-MD residual correlation, SLR tended to have a lot of bias whenever the true residual correlation was not 0, whereas the BM had similar bias across all scenarios.

Cohort Results

Table 2 gives the posterior means of the intercepts, slope, and residual SD for GCC and MD and the GCC-MD correlations from the BM. The correlation between GCC-MD intercepts, slopes, and residual SDs was 0.47 (95% credible interval: 0.32 to 0.61), 0.29 (95% credible interval: 0.04 to 0.52), and 0.20 (95% credible interval: −0.06 to 0.44), respectively. In addition, the residual correlation between GCC and MD was 0.060 (95% credible interval: −0.013 to 0.132). In comparison, the SLR method found correlations for GCC-MD intercepts, slopes, and residual SDs to be 0.49 (95% confidence interval: 0.34 to 0.62), 0.22 (95% confidence intervals 0.04 to 0.39), and 0.17 (95% confidence interval: −0.01 to 0.34), respectively.

Table 2.

Summary of the Posterior Means (95% Credible Intervals) of the Global Intercept, Slope, and Residual SD for GCC and 10-2 Visual Field MD; the Correlations between GCC-MD Random Intercepts, Slopes, and Log Residual SDs (Last Column); and the Observation-Level Correlation between GCC-MD Residuals (Last Row or Last Column)

Parameter	GCC (μ m)	MD (dB)	GCC-MD Correlation
Intercept	73.2 (71.4, 75.0)	−8.7 (−9.7 to −7.7)	0.47 (0.32 to 0.61)
Slope (/year)	−0.42 (−0.52 to −0.32)	−0.24 (−0.34 to −0.14)	0.29 (0.04 to 0.52)
Residual SD	1.00 (0.88 to 1.12)	1.23 (1.13 to 1.34)	0.20 (−0.06 to 0.44)
Residuals			0.060 (−0.013 to 0.132)

dB = decibel; GCC = ganglion cell complex; MD = mean deviation; SD = standard deviation.

Intercepts and slopes are significantly positively correlated between structural and functional measures.

The GCC-MD intercepts and GCC-MD slope correlations were significantly positive using either method. Figure 1A, B plot the GCC and MD posterior mean intercepts and slopes from the BM. Figure 1C, D display the GCC slopes vs intercepts and MD slopes vs intercepts, respectively. The model found a significant negative correlation of −0.25 (95% credible interval: −0.44 to −0.04) between GCC intercepts and slopes, but a nonsignificant correlation of 0.14 (95% credible interval: −0.07 to 0.35) between MD intercepts and slopes. Table S6 (available at www.ophthalmologyscience.org) provides additional pairwise correlations between random intercepts, slopes, and log residual SDs of GCC and MD. There was a significant positive correlation of 0.28 (95% credible interval: 0.08 to 0.46) between GCC intercepts and log residual SDs and a significant negative correlation of −0.48 (95% credible interval: −0.71 to −0.20) between MD slopes and log residual SDs. Also, the residuals did not vary as a function of the underlying GCC or MD intercept estimates (Fig S1, available at www.ophthalmologyscience.org).

Figure 1.

Scatter plots of estimated intercepts and slopes from the Bayesian model: A, GCC intercepts against MD intercepts; B, GCC slopes against MD slopes; C, GCC slopes against GCC intercepts; and D, MD slopes against MD intercepts. The estimated correlation (95% credible interval) is labeled in red, and the blue line is the linear regression fit. GCC = ganglion cell complex; MD = mean deviation.

Table 3 gives the number of eyes identified as having significantly negative slopes (rates of change) on either GCC or MD, both GCC and MD, or neither measure. There were 44 (37.6%) eyes with significant negative slopes on GCC but stable on MD. A higher proportion of eyes were progressing for GCC than MD (GCC 51.3% vs. MD 23.9%; P < 0.001), whereas a similar proportion of eyes had significantly positive slopes (GCC 0.9% vs. MD 1.7%; P = 0.56). For predicting future observations, the BM reduced prediction error by 9% for both GCC and MD compared to the SLR method. The RMSE for the BM was 2.02 μm and 1.76 dB for GCC and MD, respectively, whereas they were 2.19 μm and 1.92 dB, respectively, for SLR.

Table 3.

The Number (%) of Eyes Identified as Progressing (Significantly Negative) on GCC, MD, Both Measures, or Neither Measure

GCC	MD
	Stable	Progressing
Stable	45 (38.5%)	12 (10.3%)
Progressing	44 (37.6%)	16 (13.7%)

GCC = ganglion cell complex; MD = mean deviation.

A higher proportion of eyes demonstrated significant structural progression as compared to functional deterioration.

Discussion

We applied a Bayesian joint bivariate longitudinal model to examine longitudinal SF relationships between global macular GCC thickness and central VF MD in patients with moderate to advanced glaucoma. The model outperformed the SLR model, particularly in capturing the correlation between structural and functional changes. These findings emphasize the benefits of this approach for better understanding the progression of structural and functional damage and may enhance clinical monitoring strategies in glaucoma.

Our BM effectively accounted for the correlations between estimated GCC thickness and MD at baseline (intercepts) and their rates of change (slopes), providing a more comprehensive understanding of the relationship between longitudinal changes in structural and functional measures.^29,33, 34, 35, 36 Significant positive correlations between GCC and MD intercepts and slopes were observed, indicating that patients with thinner baseline GCC tended to have worse initial MD values; more importantly, eyes with faster structural thinning also exhibited more rapid functional decline, although the observed longitudinal correlation was modest. The correlation between GCC and MD observation-level residuals was not significant, suggesting that short-term fluctuations in GCC and MD measurements do not strongly covary at each visit. This finding aligns with previous studies that reported only modest SF associations at the global level,^10,11,37,38 potentially because of the fact that localized glaucomatous changes that are not well captured by global measures.

Our Bayesian joint bivariate longitudinal model builds upon previous work in SF modeling but introduces several key advancements. Medeiros et al^23,39 employed a Bayesian hierarchical framework to predict functional loss in glaucoma but did not explicitly model residual variances as random effects, limiting the ability to capture individual variability in measurement noise. Similarly, Hu et al²⁴ introduced a dynamic SF model that utilized velocity-based modeling to estimate progression; this was effective for short-term predictions but lacked the hierarchical structure needed for robust long-term tracking. Our model extends these methodologies by explicitly modeling individual intercepts, slopes, and residual SDs as random effects within a hierarchical Bayesian framework. This approach enhances the estimation of the correlations between baseline structure and function, their rates of change, and measurement variability, providing a more comprehensive understanding of SF relationships. As a result, it could allow for a more personalized and probabilistic assessment of glaucoma progression.

Previous longitudinal studies found that rapid RNFL or GCC thickness thinning is associated with a faster decline in MD.40, 41, 42, 43 However, few studies have directly compared the rates of change between structural and functional measures. Mohammadzadeh et al¹¹ reported weak to fair correlations between the rates of change in GCC thickness at individual macular superpixels and total deviation values at central VF test locations. The weaker longitudinal correlation in our study may stem from our global modeling approach, which averaged structural and functional measurements prior to modeling rather than capturing localized variations.^44,45 Our study's lower longitudinal SF correlation compared to Mohammadzadeh et al⁴⁶ may be because we only modeled global SF correlations. Given this limitation, future studies could employ localized-level analyses (macular superpixels or regions vs. VF locations or clusters) to better capture milder glaucomatous changes that global metrics may overlook.⁴⁶

In both simulations and cohort analyses, our BM exhibited lower RMSE for estimating SF slope correlations than SLR, particularly when the true residual and slope correlations differed. This suggests the Bayesian approach is robust under different conditions and can more accurately estimate the underlying relationships between GCC thickness and MD. The joint modeling framework allows structural and functional random effects to be estimated simultaneously, improving the estimated relationship between GCC and MD progression accuracy. The SLR method, in contrast, exhibited higher bias and variability, especially when the residual and slope correlations had opposite signs. These SLR limitations likely arise from its inability to model joint distributions or incorporate random effects and shared variability, resulting in increased variance in correlation estimates.

In the cohort analysis, the BM improved prediction accuracy, reducing error by 9% for GCC and MD compared to SLR. Although this improvement alone might appear modest, the true strength of our BM is its ability to jointly estimate structural and functional parameters while explicitly accounting for measurement variability, which could lead to earlier and more reliable identification of rapidly progressing eyes. The SLR method is an unbiased slope estimator for separate longitudinal analyses of GCC and MD; conversely, Bayesian hierarchical models inherently introduce shrinkage toward the population mean, resulting in some bias. However, based on the simulation scenario analysis, the substantial reduction in variance achieved by the Bayesian approach more than compensates for this bias, leading to a lower overall RMSE. Importantly, even within the clinically relevant slope range from 0 to −1 dB (or μm)/year, the bias itself remained modest (Fig S2, available at www.ophthalmologyscience.org). Further, the additional variability of SLR means that the estimates of correlations using correlations of SLR estimates are biased as compared to the Bayesian bivariate longitudinal model. These findings align with our previous reports, which demonstrated that a smaller estimator variance induced the decreased RMSE in the BM.^33,36 Furthermore, our prior work has shown that Bayesian hierarchical modeling yields lower RMSE for slope estimates, even in extreme cases, than SLR, suggesting improved clinical reliability.³³ Notably, the BM found more eyes with significant GCC progression without functional progression than those with functional progression without structural progression. This disagreement is not surprising and partially related to the tests' ability to identify progression at various disease stages.^47,48 It also stems from different variability in structural and functional measurements49, 50, 51, 52, 53, 54 and the differing sensitivities of structural and functional assessments at various disease stages.^13,55,56 These findings highlight the potential utility of macular OCT measures for earlier detection of disease deterioration in the macular region, paving the way for timely treatment escalation when indicated. Consistent with previous studies, our results further emphasize that structural changes in glaucoma may precede functional changes.^11,57,58

Our findings also showed a significant negative correlation between GCC intercepts and slopes (r = −0.25), suggesting that eyes with thinner baseline GCC tend to experience slower rates of structural thinning over time. This could be attributed to a floor effect; structure cannot continue to decrease as the floor is approached.59, 60, 61 Conversely, no significant correlation was found between baseline MD (intercepts) and MD rates of change (slopes), indicating that baseline functional status did not predict the rate of functional decline; this was likely due to the noisier nature of VF testing, where global MD measures are affected by variability unrelated to true progression.^59,62 The significant positive correlation between estimated baseline GCC thickness (intercepts) and log residual SDs implies that eyes with thinner baseline GCC exhibit greater variability in measurements, which may reflect increased measurement noise with advancing structural damage.^63,64

Although the Bayesian joint bivariate longitudinal model offers significant advantages, it also has limitations. First, the population-dependent model estimates subject-level parameters by shrinking toward population-level priors. This dependency makes direct application to individual eyes challenging in clinical practice. Although using population-derived estimates as informative priors for single-eye modeling is a promising solution, additional validation is necessary. Future research should explore methods to adapt the model for individualized assessments, ensuring robust performance across diverse clinical populations. Second, the model assumes that residuals follow a normal distribution, which may not adequately capture the full range of variability in clinical data, especially in the presence of outliers. Although we identified and excluded obvious outliers, future extensions of this model could incorporate robust methods for outlier detection or non-Gaussian distributions to improve performance. Similarly, our model assumed the joint normality of random intercepts and slopes for GCC and MD. This normality assumption seems reasonable for GCC but may not be the best assumption for MD intercepts. Third, the model's computational demands are higher than traditional methods, requiring more sophisticated statistical software and longer run times. However, these challenges can be mitigated with ongoing improvements in computational power and Bayesian software. Fourth, the linearity of glaucoma progression is assumed, which may not hold true over longer periods. Although this is a reasonable approximation over short to medium follow-up periods and especially with global measures, glaucoma progression may not be strictly linear over longer periods. Gardiner et al⁶⁵ suggested that the estimation of VF rates of change be limited to shorter timeframes (8–10 tests) to avoid potential nonlinearities in VF progression rates. Additionally, Montesano et al⁶⁶ proposed taking the logarithm of the structural metrics to address this issue. However, our cohort had relatively preserved GCC thickness and intermediate follow-up duration (mean ∼5 years), during which most eyes did not approach the structural floor. Therefore, we believe that a linear approximation in the original scales remains appropriate for this study. Furthermore, from a clinical standpoint, estimating and predicting the course of the disease may not be clinically relevant beyond a 3- to 5-year timeframe. The superior performance of our BM should be evaluated over shorter timeframes, such as 2 to 3 years, which aligns more closely with practical clinical decision-making.

In conclusion, our study demonstrates that a Bayesian joint bivariate longitudinal model provides a more accurate and consistent estimation of longitudinal central SF correlations in glaucoma compared to the traditional SLR method. The Bayesian approach effectively captures structural and functional progression jointly; the significant positive correlation between GCC and MD rates of change emphasizes the interdependence of structural and functional deterioration. Our model offers a potential tool for clinicians and researchers to better understand glaucoma progression and allow the development of personalized monitoring and treatment strategies.

Manuscript no. XOPS-D-24-00567.