December 2014
Volume 55, Issue 12
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Glaucoma  |   December 2014
A New Approach to Measure Visual Field Progression in Glaucoma Patients Using Variational Bayes Linear Regression
Author Affiliations & Notes
  • Hiroshi Murata
    Department of Ophthalmology, University of Tokyo Graduate School of Medicine, Tokyo, Japan
  • Makoto Araie
    Department of Ophthalmology, University of Tokyo Graduate School of Medicine, Tokyo, Japan
    Kanto Central Hospital, the Mutual Aid Association of Public School Teachers, Tokyo, Japan
  • Ryo Asaoka
    Department of Ophthalmology, University of Tokyo Graduate School of Medicine, Tokyo, Japan
  • Correspondence: Ryo Asaoka, Department of Ophthalmology, University of Tokyo Graduate School of Medicine, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8655 Japan; rasaoka-tky@umin.ac.jp
Investigative Ophthalmology & Visual Science December 2014, Vol.55, 8386-8392. doi:10.1167/iovs.14-14625
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      Hiroshi Murata, Makoto Araie, Ryo Asaoka; A New Approach to Measure Visual Field Progression in Glaucoma Patients Using Variational Bayes Linear Regression. Invest. Ophthalmol. Vis. Sci. 2014;55(12):8386-8392. doi: 10.1167/iovs.14-14625.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Purpose.: We generated a variational Bayes model to predict visual field (VF) progression in glaucoma patients.

Methods.: This retrospective study included VF series from 911 eyes of 547 glaucoma patients as test data, and VF series from 5049 eyes of 2858 glaucoma patients as training data. Using training data, variational Bayes linear regression (VBLR) was created to predict VF progression. The performance of VBLR was compared against ordinary least-squares linear regression (OLSLR) by predicting VFs in the test dataset. The total deviation (TD) values of test patients' 11th VFs were predicted using TD values from their second to 10th VFs (VF2-10), the root mean squared error (RMSE) associated with each approach then was calculated. Similarly, mean TD (mTD) of test patients' 11th VFs was predicted using VBLR and OLSLR, and the absolute prediction errors compared.

Results.: The RMSE resulting from VBLR averaged 3.9 ± 2.1 (SD) and 4.9 ± 2.6 dB for prediction based on the second to 10th VFs (VF2-10) and the second to fourth VFs (VF2-4), respectively. The RMSE resulting from OLSLR was 4.1 ± 2.0 (VF2-10) and 19.9 ± 12.0 (VF2-4) dB. The absolute prediction error (SD) for mTD using VBLR was 1.2 ± 1.3 (VF2-10) and 1.9 ± 2.0 (VF2-4) dB, while the prediction error resulting from OLSLR was 1.2 ± 1.3 (VF2-10) and 6.2 ± 6.6 (VF2-4) dB.

Conclusions.: The VBLR more accurately predicts future VF progression in glaucoma patients compared to conventional OLSLR, especially in short VF series.

Introduction
Glaucoma is characterized by progressive visual field (VF) damage and is the second leading cause of blindness in the world.1 Visual loss impacts on the quality of life of glaucoma patients,2 and, as glaucomatous VF damage is irreversible, it is very important to accurately measure and predict progression of VF defects so timely treatment decisions can be made. 
Research indicates that analyzing VF progression using an analysis tool is useful in the clinical management of glaucoma.3 In the Humphrey Guided Progression Analysis (GPA) software, VF progression is measured using a trend analysis, in which simple ordinary least-squares linear regression (OLSLR) is done on the mean deviation (MD) or the visual field index (VFI) over time.4 The software PROGRESSOR (Medisoft Ltd., Leeds, UK) also uses OLSLR to analyze VF progression in a pointwise manner.5 One concern with applying OLSLR is that the method relies on the hypothesis that VF progression occurs linearly. Indeed, previous reports suggest that nonlinear methods, such as the exponential model or an episodic model, can better estimate VF progression,68 although other reports give inconsistent conclusions.9,10 Thus, the most appropriate model for prediction of VF progression still is controversial and the choice may vary throughout the disease process.11 
Several recent studies have suggested that estimates of VF progression can be improved by incorporating other information, such as structural measurements12 and spatial information,13 and it is rational that VF progression models can be improved significantly by using information about the distinctive patterns associated with different types of VF defects. Glaucoma specialists know that glaucomatous VF progression often follows characteristic patterns, such as the nasal step defect and the Bjerrum scotoma.14 Moreover, patients with different types of VF damage often have distinct risk factors15 that can be observed at presentation.16 However, to date, almost all methods that measure pointwise VF progression have failed to take into account the correlation among neighboring test points.1719 
Machine learning (ML) methods are particularly useful for clustering and classifying high dimensional data.20 Many real-world complex data can be represented, using ML methods, in a low-dimensional subspace corresponding to just tens of categories to which the data belong.21 Accordingly, VF data from glaucoma patients could be clustered into groups based on different patterns of VF damage using ML methods; moreover, a VF progression prediction model could be generated that uses this clustering information. Bayesian methods are used commonly to incorporate prior information into a statistical model to improve its accuracy.12,22,23 Recently, Zhu et al.24 reported a novel approach to measure VF progression that modeled spatial correlations as well as nonstationary variability. In this study, we constructed a different type of Bayesian model to predict VF damage that uses the spatial and temporal patterns associated with glaucomatous VF progression as described in a training data set. We then investigated the usefulness of this novel “variational Bayes linear regression” (VBLR) approach over conventional OLSLR to predict VF progression in an unrelated test data set. 
Methods
This retrospective study was approved by the Research Ethics Committee of the Graduate School of Medicine and Faculty of Medicine at the University of Tokyo. Written consent was given by patients for their information to be stored in the hospital database and used for research. This study was performed according to the tenets of the Declaration of Helsinki. 
All training and test data were obtained at the University of Tokyo Hospital between 2002 and 2013. No patients overlap between test and training data. In both data sets, patients' first VF tests were excluded. 
The training dataset consists of 5049 eyes of 2858 subjects; all subjects underwent at least five VF tests (range, 5–30 VFs) over a mean (±SD) duration of 4.4 (±1.8) years (Table 2). The training data were unlabeled; therefore, patients with complications, other than glaucoma, that could affect their VF results, such as cataract or senile macular degeneration, were included. 
Table 1
 
Demographics of Patients in Training Dataset
Table 1
 
Demographics of Patients in Training Dataset
Demographics Value
Follow-up time, y, mean ± SD 4.4 ± 1.8
Age, y, mean ± SD 58.7 ± 14.2
MD of initial VF, dB, mean ± SD −7.0 ± 6.7
Table 2
 
Demographics of Patients in Test Dataset
Table 2
 
Demographics of Patients in Test Dataset
Demographics Value
Follow-up time, y, mean ± SD 5.5 ± 1.2
Age, y, mean ± SD 55.4 ± 13.6
Sex, male:female 253:294
MD of initial VF, dB, mean ± SD −7.8 ± 6.2
Type of glaucoma, POAG, NTG, ACG, PE, steroid, others 323, 458, 28, 16, 36, 50
The test data set consisted of 911 eyes of 547 subjects who had undertaken a minimum of 11 VF tests. The demographic information of patients in the test data is shown in Table 2. The mean (±SD) follow-up was 5.5 (±1.2) years. Inclusion criteria for the test data were patients with a visual acuity better than 6/12, spherical equivalent error less than 5-diopter (D) ametropia, no previous ocular surgery (except for cataract extraction, intraocular lens implantation, trabeculectomy, and trabeculotomy), and no other posterior segment eye diseases. Patients were diagnosed with primary open angle glaucoma (POAG), normal tension glaucoma (NTG), primary angle closure glaucoma (PACG), steroid glaucoma, glaucoma associated with pseudoexfoliation syndrome, or other types of glaucoma. Patients with other diseases that could affect VF sensitivity, such as diabetic mellitus retinopathy, corneal opacity, and age-related macular degeneration, were excluded from the test data. 
All VFs were recorded using the Humphrey Field Analyzer (HFA; Carl Zeiss Meditec, Dublin, CA, USA) with the 24-2 or 30-2 test pattern and the SITA standard strategy with a Goldmann size III target. Refractive errors of the patients, including ametropia, were corrected appropriately during VF tests, in accordance with the manufacturer's instructions. Test locations adjacent to the blind spot were excluded from our analyses. When a VF was measured using the 30-2 test pattern, only the 52 test points overlapping with the 24-2 test pattern were used. Reliability criteria applied were fixation losses less than 33%, false-positive responses less than 33%, and false-negative rate less than 33%. 
Statistical Modeling and Training
Here we describe the mathematical formulation of the VBLR model. Let Display FormulaImage not available = ( Display FormulaImage not available , Display FormulaImage not available ,…, Display FormulaImage not available ) represent the total deviation (TD) values of a patient's nth VF in their series; Dt is the dimension of the vector tn and is equal to 52 in this study. Let nm be the set of indices of data obtained from the mth eye, Tm denotes the set Display FormulaImage not available , while wm is the parameter vector of the mth eye (where the first half and latter half of this vector include the intercept and slope coefficients of all 52 test VF points, respectively). Next, let xn denote the interval from the first VF test of the nth data, Φ(xn) denotes a matrix defined as Φ(xn) = Display FormulaImage not available where Display FormulaImage not available is a 52-dimensional identity matrix; Dw then is the dimension of vector wm (equal to 104 in this study). Then, λm represents the scalar of the magnitude of reliability of VFs obtained from the mth eye. We assumed the data, tn, were independently drawn from a Gaussian distribution with mean vector Φ(xn)Twm and inverse of covariance matrix Display FormulaImage not available where Lm is a 52 by 52 matrix. Consequently, the likelihood is given by:    
This model differs from OLSLR, in that it can capture the correlation among VF test points, because the matrix Lm is not constrained to the diagonal. 
We assumed wm, λm, and Lm were random variables that followed a Gaussian mixture distribution, a Gamma mixture distribution, and a Wishart mixture distribution, respectively. The Gaussian mixture model25 is a type of soft clustering method and can represent the presence of subgroups in a multimodal population. Unlike hard clustering methods, such as K-means methods, the Gaussian mixture model allows the derivation of a probability density distribution, and hence, it is suitable to be integrated in a probabilistic statistical model; for instance, the certainty or reliability of the estimated VF sensitivity can be inferred. The likelihoods were given by:      and   where zmk∈{0,1}, Display FormulaImage not available = 1, ζmh∈{0,1}, Display FormulaImage not available = 1, γmg∈{0,1}, Display FormulaImage not available = 1, ηh∈[0,1], Display FormulaImage not available = 1, πk∈[0,1], Display FormulaImage not available = 1, θg∈[0,1], Display FormulaImage not available = 1; K, H, and G are the number of components in each mixture distribution, W(·|·) denotes the Wishart distribution, and G(·|·) denotes the gamma distribution. If we marginalize Equation 2 over the ζmh weighted by Equation 3, the Wishart mixture distribution  is given. Similarly,  is given by Equations 4 and 5, and  is given by Equations 6 and 7. In this model, the slopes and intercepts of eyes were clustered during the training phase because p(wm|λm) was a Gaussian mixture model. The full joint density function of the data T ≜ {T1,…,TM} and all the latent variables were given by:  where M is the number of eyes. We estimated the hyperparameters ν, W, μ, Λ, a, b, η, π, and θ, where ν ≜ {ν1,…,νH}, W ≜ {W1,…,WH}, μ ≜ {μ1,…,μK}, Λ ≜ {Λ1,…,ΛK}, a ≜ {a1,…,ag}, b ≜ {b1,…,bg}, η ≜ {η1,…,ηh}, π ≜ {π1,…,πk}, and θ ≜ {θ1,…,θg}, with the training data, using the E-M algorithm by maximizing the log-likelihood    
In the E-step, expectations were calculated approximately by mean field variational approximation since exact inference was analytically intractable. Since the E-M algorithm and the variational approximation method are both gradient algorithms, they suffer from a serious local optimal problem in practice. To overcome this problem, we applied deterministic annealing26,27 for the E-M algorithm, variational approximation in the E-step of E-M algorithm, and variational Bayes method in the prediction phase. See the Supplementary Material for the detailed updating equations. 
Prediction With Variational Bayes Linear Regression Method
Suppose we are predicting the VF of a patient in accordance with the results of the aforementioned training phase. Let tN+1 be the (N + 1)th VF of a patient at time xN+1 where N is the number of VF tests already observed. Since we have not observed tN+1 yet, tN+1 was introduced as a latent variable. Hence, the complete likelihood was:    
Then p(tN+1|T) is calculated approximately using the variational Bayes method. 
Prediction Accuracy
Prediction accuracy was compared between the novel VBLR approach and conventional OLSLR. First, OLSLR analysis was done using TD values at each of the 52 test points from the second to the fourth VFs (VF2-4) of each patient, and the TD values of the 11th VF test were predicted. The same procedure was done using the TD values in different series: VF2-5, VF2-6, VF2-7, VF2-8, VF2-9, and VF2-10, and the TD values of 11th VFs were predicted every time. Likewise, TD values of 11th VFs were predicted with the VBLR approach using the TD values of all series lengths from VF2-2 (the second VF only) to VF2-10 (all previous VFs). The predictive accuracy of each method was compared through the root mean squared error (RMSE) statistic, defined as follows:    
We also predicted the mean TD (mTD) values of the 11th VFs, by taking the average of the predicted TD values obtained with OLSLR using the series of VF2-4, VF2-5, VF2-6, VF2-7, VF2-8, VF2-9, and VF2-10; likewise, mTD was predicted with VBLR using VF series: VF2-2 to VF2-10. The predicted mTD by our method is given by:  where Display FormulaImage not available is a Dt = 52 dimensional column vector filled with ones. Predictive accuracy for mTD was compared using absolute errors.  
The prediction accuracies described above were compared between VBLR and ORSLR using a linear mixed-effects model, to overcome the problem of correlation between both eyes of a patient. 
Software
Data preparation and analyses were carried out using the statistical programming language R (available in the public domain at http://www.R-project.org/, ver. 2.14.2); VBLR was written in C++ using the Armadillo C++ linear algebra library.28 
Results
The demographic information of patients in the test data is shown in Table 2. Mean deviation at baseline was −7.8 ± 6.2 dB (mean ± SD) and initial age was 55.4 ± 13.6 years. 
Parameters of the VBLR model, K, H, and G, were set to 30, 5, and 2, respectively. Larger numbers of the clusters (K, H, and G) led to divergence of the likelihood. The best lower bound of log-likelihood obtained was −3.99e + 6, and the results shown below are computed using this solution. 
Figure 1 and Table 3 show the RMSEs associated with the VBLR approach and OLSLR. The RMSEs of OLSLR become smaller as the number of VF tests increases, though this tendency was much less obvious with VBLR. Given the same number of previous VFs, RMSEs with the VBLR approach were always smaller than those with OLSLR, and there was a significant difference between the two methods at all time points from VF2-4 through VF2-10 inclusive (linear mixed-effects model, P < 0.05, Table 3) Furthermore, the RMSE associated with the VBLR approach using only the initial two VFs (VF2-3) was smaller than that with seven VFs (VF2-8) using the OLSLR conventional method. Similar differences in absolute errors were observed for each of the 52 points in the VF (Fig. 2). 
Figure 1
 
The RMSEs obtained by predicting the TD values of 11th VF. The plot represents the RMSEs obtained by predicting the TD values of the 11th VFs using the TD values of the series of VF2-4, VF2-5, VF2-6, VF2-7, VF2-8, VF2-9, and VF2-10 with OLSLR, and those using from VF2-2 to VF2-10 with VBLR (as shown in the Table 2). Error bars indicate standard deviations.
Figure 1
 
The RMSEs obtained by predicting the TD values of 11th VF. The plot represents the RMSEs obtained by predicting the TD values of the 11th VFs using the TD values of the series of VF2-4, VF2-5, VF2-6, VF2-7, VF2-8, VF2-9, and VF2-10 with OLSLR, and those using from VF2-2 to VF2-10 with VBLR (as shown in the Table 2). Error bars indicate standard deviations.
Figure 2
 
The absolute prediction error for predicting the TD values of 11th VF. The plots represent the absolute prediction errors obtained by predicting the TD values of the 11th VFs using the TD values of the series of VF2-4, VF2-5, VF2-6, VF2-7, VF2-8, VF2-9, and VF2–10 with OLSLR, and those using from VF2-2 to VF2-10 with VBLR. The scatterplots are illustrated in the shape of the VF of the right eye. Error bars indicate standard deviations.
Figure 2
 
The absolute prediction error for predicting the TD values of 11th VF. The plots represent the absolute prediction errors obtained by predicting the TD values of the 11th VFs using the TD values of the series of VF2-4, VF2-5, VF2-6, VF2-7, VF2-8, VF2-9, and VF2–10 with OLSLR, and those using from VF2-2 to VF2-10 with VBLR. The scatterplots are illustrated in the shape of the VF of the right eye. Error bars indicate standard deviations.
Table 3
 
RMSEs of Predicting 11th VF
Table 3
 
RMSEs of Predicting 11th VF
OLSLR VBLR P Value
VF2-2 5.3 (±2.8)
VF2-3 5.0 (±2.7)
VF2-4 19.9 ± (12.0) 4.9 (±2.6) <1e-10
VF2-5 12.4 ± (7.1) 4.7 (±2.5) <1e-10
VF2-6 8.9 ± (4.6) 4.5 (±2.4) <1e-10
VF2-7 6.8 ± (3.4) 4.4 (±2.3) <1e-10
VF2-8 5.6 ± (2.8) 4.2 ± (2.2) <1e-10
VF2-9 4.7 ± (2.2) 4.1 ± (2.1) <1e-10
VF2-10 4.1 ± (2.0) 3.9 ± (2.1) 2.7e-7
Figure 3 and Table 4 show the absolute errors with the VBLR approach and OLSLR when predicting mTD. There were statistically significant differences between the methods for all comparisons except for VF2-9 and VF2-10 (linear mixed-effects model, P < 0.05). The RMSE with VBLR using only the initial two VFs (VF2-3) was smaller than that using six VFs (VF2-7) and OLSLR. 
Figure 3
 
The absolute prediction errors obtained by predicting the mean TD values of 11th VF. The plot represents the absolute prediction errors obtained by taking the average of the predicted TD values obtained with OLSLR using the series of VF2-, VF2-5, VF2-6, VF2-7, VF2-8, VF2-9, and VF2-10, and those with VBLR using the series of from VF2-2 to VF2-10 (as shown in Table 3). Error bars indicate standard deviations.
Figure 3
 
The absolute prediction errors obtained by predicting the mean TD values of 11th VF. The plot represents the absolute prediction errors obtained by taking the average of the predicted TD values obtained with OLSLR using the series of VF2-, VF2-5, VF2-6, VF2-7, VF2-8, VF2-9, and VF2-10, and those with VBLR using the series of from VF2-2 to VF2-10 (as shown in Table 3). Error bars indicate standard deviations.
Table 4
 
Absolute Errors of Predicting 11th VF Mean TD
Table 4
 
Absolute Errors of Predicting 11th VF Mean TD
OLSLR VBLR P Value
VF2-2 2.2 ± (2.2)
VF2-3 2.0 ± (2.0)
VF2-4 6.2 ± (6.6) 1.9 ± (2.0) <1e-10
VF2-5 4.0 ± (4.1) 1.8 ± (1.9) <1e-10
VF2-6 2.9 ± (3.1) 1.7 ± (1.7) <1e-10
VF2-7 2.1 ± (2.2) 1.5 ± (1.7) <1e-10
VF2-8 1.8 ± (1.9) 1.5 ± (1.6) 3.0e-7
VF2-9 1.4 ± (1.5) 1.4 ± (1.5) 0.17
VF2-10 1.3 ± (1.4) 1.2 ± (1.3) 0.13
Discussion
In the current study, a novel trend-based approach to predict VF progression was created known as VBLR. This approach incorporates information about the spatial and temporal pattern of patients' VF data. As a result, prediction accuracy of VBLR considerably surpassed that of conventional OLSLR. 
Previous studies have suggested that at least five VFs are needed to acquire accurate results from OLSLR,29 while other studies suggested that even longer series are necessary.30,31 Under the assumption that the VF is measured every six months, it would take, on average, roughly 3 to 4 years to obtain sufficiently accurate results; however, many patients could suffer detrimental VF loss in this period of time. Other reports recommend that six VF tests should be obtained in two years,32,33 but it is often unrealistic to adhere to this testing frequency in all patients34 due to limited clinical capacity, especially in regions where ophthalmic facilities are limited. This highlights the clinical usefulness of the VBLR approach: with two or even a single VF test, the VBLR approach can predict future VFs as accurately as OLSLR with a series of six VF tests. In other words, with the VBLR approach, it is possible to predict what a patient's VF will look like in 3 to 4 years' time using only their baseline VF, while OLSLR would require a further 6 to 8 VF tests to make a prediction with equivalent accuracy to VBLR. Consequently, with the VBLR approach, we could identify a patient with rapid progression much earlier than we used to using OLSLR. Therefore, clinical decisions could be made much earlier helping to save patients' vision related quality of life. In addition, earlier identification of patients with slow progression could spare the expense of unnecessary treatments. 
In Bayesian analyses, the Markov chain Monte Carlo method often is used to calculate posterior distributions or expectations. This method was intentionally avoided in the current approach, and instead, variational approximation was used; this is because variational approximation is computationally very efficient. For example, it took merely 0.2 seconds with a dual Xeon E5-2670 CPU computer to predict a patient's future VF based on nine VF tests. This efficiency would be especially beneficial in a clinical setting. 
The training data in this study likely included VFs from patients with additional ocular diseases other than glaucoma; as a consequence, the parameterization of our model may be imperfect and so the performance of our model could be improved by the exclusion of these patients. Nonetheless, it is unlikely that our results will be largely affected as the clustering procedure in the Gaussian mixture model inherently deals with different patterns of VF defects. 
We attempted to reduce the learning effect associated with VF results by discarding patients' first VF test35; however, the learning effect can persist over many VFs.36 Hence, we also compared the prediction accuracy of OLSLR and VBLR, excluding the first two or three VFs. The VBLR continued to outperform OLSLR, with a magnitude similar to the results shown in Figures 1 through 3. We also compared the prediction accuracy of OLSLR and VBLR in POAG and NTG groups separately, since previous studies have reported that the pattern of VF deterioration differs between these disease groups.3739 Our results (not shown) suggested there was no significant difference in prediction accuracy between the groups. 
Previous reports suggest that clinical information, such as type of glaucoma, is related to the pattern of VF damage observed.4042 One of the limitations of the current study is that this type of information, along with IOP, was not incorporated into the current VBLR approach. However, the model could be extended to incorporate such information by introducing a mixture of expert models—this should be carried out in a future study to attempt to further improve prediction accuracy.43 In the current analysis, VF series often belong to different clusters at different time points, which is inevitable because the clustering depends on spatial and temporal patterns. The purpose of the current study was to improve VF predictions, not to cluster VFs; however, it would be interesting to analyze the relationship between VF patterns and predictive ability; future studies are merited to shed light on this issue. Furthermore, we did not investigate how varying parameters K, H, and G in the VBLR model influenced the performance of VBLR; this was simply to avoid over-fitting. A future study could investigate this question, however, a further dataset would be require to infer the optimum values without over-fitting. 
Caprioli et al.6 recently reported that an exponential model might be more suitable than OLSLR to measure pointwise VF sensitivity decay. We initially applied exponential regression to our data, but prediction accuracy was not as good as expected, compared to that with OLSLR (results not shown). There is a possibility that the exponential model fits better in a longer series of VFs since follow-up was 8 years on average in the report of Caprioli et al.6 Russell and Crabb44 proposed Tobit regression as an alternative to the exponential model, since Tobit regression respects the floor effect of VF sensitivity data; however, the Tobit model is not suitable for the variational Bayes framework due to its nonlinear optimization and the resulting problem of local optima. The exponential model could be applied using a local variational method,45 but this would require applying further approximation. On the other hand, Bryan et al.10 recently concluded that, despite violation of its assumptions, the classical uncensored OLSLR model provides the best fit for VF data and performs best at predicting future VFs when compared to the Tobit and exponential regression models. Moreover, they stated that more advanced regression models exploring the temporal–spatial relationships of glaucomatous progression are required to reduce prediction errors to clinically meaningful levels. 
In conclusion, we have applied the VBLR approach to measure VF progression. The prediction accuracy of this approach considerably surpassed that of OLSLR, which would result in much earlier clinical decisions, and consequently, more efficient management of glaucoma related VF deterioration. 
Acknowledgments
Supported in part by Japan Science and Technology Agency (JST) CREST (RA, HM), and Grants 25861618 (HM) and 26462679 (RA) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. The authors alone are responsible for the content and writing of the paper. A patent application (PCT/JP2013/073426: Visual-field-test assistance device) has been filed and variational Bayes linear regression is part of analytical methods. 
Disclosure: H. Murata, P; M. Araie, None; R. Asaoka, P 
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Figure 1
 
The RMSEs obtained by predicting the TD values of 11th VF. The plot represents the RMSEs obtained by predicting the TD values of the 11th VFs using the TD values of the series of VF2-4, VF2-5, VF2-6, VF2-7, VF2-8, VF2-9, and VF2-10 with OLSLR, and those using from VF2-2 to VF2-10 with VBLR (as shown in the Table 2). Error bars indicate standard deviations.
Figure 1
 
The RMSEs obtained by predicting the TD values of 11th VF. The plot represents the RMSEs obtained by predicting the TD values of the 11th VFs using the TD values of the series of VF2-4, VF2-5, VF2-6, VF2-7, VF2-8, VF2-9, and VF2-10 with OLSLR, and those using from VF2-2 to VF2-10 with VBLR (as shown in the Table 2). Error bars indicate standard deviations.
Figure 2
 
The absolute prediction error for predicting the TD values of 11th VF. The plots represent the absolute prediction errors obtained by predicting the TD values of the 11th VFs using the TD values of the series of VF2-4, VF2-5, VF2-6, VF2-7, VF2-8, VF2-9, and VF2–10 with OLSLR, and those using from VF2-2 to VF2-10 with VBLR. The scatterplots are illustrated in the shape of the VF of the right eye. Error bars indicate standard deviations.
Figure 2
 
The absolute prediction error for predicting the TD values of 11th VF. The plots represent the absolute prediction errors obtained by predicting the TD values of the 11th VFs using the TD values of the series of VF2-4, VF2-5, VF2-6, VF2-7, VF2-8, VF2-9, and VF2–10 with OLSLR, and those using from VF2-2 to VF2-10 with VBLR. The scatterplots are illustrated in the shape of the VF of the right eye. Error bars indicate standard deviations.
Figure 3
 
The absolute prediction errors obtained by predicting the mean TD values of 11th VF. The plot represents the absolute prediction errors obtained by taking the average of the predicted TD values obtained with OLSLR using the series of VF2-, VF2-5, VF2-6, VF2-7, VF2-8, VF2-9, and VF2-10, and those with VBLR using the series of from VF2-2 to VF2-10 (as shown in Table 3). Error bars indicate standard deviations.
Figure 3
 
The absolute prediction errors obtained by predicting the mean TD values of 11th VF. The plot represents the absolute prediction errors obtained by taking the average of the predicted TD values obtained with OLSLR using the series of VF2-, VF2-5, VF2-6, VF2-7, VF2-8, VF2-9, and VF2-10, and those with VBLR using the series of from VF2-2 to VF2-10 (as shown in Table 3). Error bars indicate standard deviations.
Table 1
 
Demographics of Patients in Training Dataset
Table 1
 
Demographics of Patients in Training Dataset
Demographics Value
Follow-up time, y, mean ± SD 4.4 ± 1.8
Age, y, mean ± SD 58.7 ± 14.2
MD of initial VF, dB, mean ± SD −7.0 ± 6.7
Table 2
 
Demographics of Patients in Test Dataset
Table 2
 
Demographics of Patients in Test Dataset
Demographics Value
Follow-up time, y, mean ± SD 5.5 ± 1.2
Age, y, mean ± SD 55.4 ± 13.6
Sex, male:female 253:294
MD of initial VF, dB, mean ± SD −7.8 ± 6.2
Type of glaucoma, POAG, NTG, ACG, PE, steroid, others 323, 458, 28, 16, 36, 50
Table 3
 
RMSEs of Predicting 11th VF
Table 3
 
RMSEs of Predicting 11th VF
OLSLR VBLR P Value
VF2-2 5.3 (±2.8)
VF2-3 5.0 (±2.7)
VF2-4 19.9 ± (12.0) 4.9 (±2.6) <1e-10
VF2-5 12.4 ± (7.1) 4.7 (±2.5) <1e-10
VF2-6 8.9 ± (4.6) 4.5 (±2.4) <1e-10
VF2-7 6.8 ± (3.4) 4.4 (±2.3) <1e-10
VF2-8 5.6 ± (2.8) 4.2 ± (2.2) <1e-10
VF2-9 4.7 ± (2.2) 4.1 ± (2.1) <1e-10
VF2-10 4.1 ± (2.0) 3.9 ± (2.1) 2.7e-7
Table 4
 
Absolute Errors of Predicting 11th VF Mean TD
Table 4
 
Absolute Errors of Predicting 11th VF Mean TD
OLSLR VBLR P Value
VF2-2 2.2 ± (2.2)
VF2-3 2.0 ± (2.0)
VF2-4 6.2 ± (6.6) 1.9 ± (2.0) <1e-10
VF2-5 4.0 ± (4.1) 1.8 ± (1.9) <1e-10
VF2-6 2.9 ± (3.1) 1.7 ± (1.7) <1e-10
VF2-7 2.1 ± (2.2) 1.5 ± (1.7) <1e-10
VF2-8 1.8 ± (1.9) 1.5 ± (1.6) 3.0e-7
VF2-9 1.4 ± (1.5) 1.4 ± (1.5) 0.17
VF2-10 1.3 ± (1.4) 1.2 ± (1.3) 0.13
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