April 2015
Volume 56, Issue 4
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Glaucoma  |   April 2015
Applying “Lasso” Regression to Predict Future Visual Field Progression in Glaucoma Patients
Author Affiliations & Notes
  • Yuri Fujino
    Department of Ophthalmology, The University of Tokyo, Graduate School of Medicine, Tokyo, Japan
  • Hiroshi Murata
    Department of Ophthalmology, The University of Tokyo, Graduate School of Medicine, Tokyo, Japan
  • Chihiro Mayama
    Department of Ophthalmology, The University of Tokyo, Graduate School of Medicine, Tokyo, Japan
  • Ryo Asaoka
    Department of Ophthalmology, The University of Tokyo, Graduate School of Medicine, Tokyo, Japan
  • Correspondence: Ryo Asaoka, Department of Ophthalmology, The 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 April 2015, Vol.56, 2334-2339. doi:10.1167/iovs.15-16445
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      Yuri Fujino, Hiroshi Murata, Chihiro Mayama, Ryo Asaoka; Applying “Lasso” Regression to Predict Future Visual Field Progression in Glaucoma Patients. Invest. Ophthalmol. Vis. Sci. 2015;56(4):2334-2339. doi: 10.1167/iovs.15-16445.

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

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Abstract

Purpose.: We evaluated the usefulness of various regression models, including least absolute shrinkage and selection operator (Lasso) regression, to predict future visual field (VF) progression in glaucoma patients.

Methods.: Series of 10 VFs (Humphrey Field Analyzer 24-2 SITA-standard) from each of 513 eyes in 324 open-angle glaucoma patients, obtained in 4.9 ± 1.3 years (mean ± SD), were investigated. For each patient, the mean of all total deviation values (mTD) in the 10th VF was predicted using varying numbers of prior VFs (ranging from the first three VFs to all previous VFs) by applying ordinary least squares linear regression (OLSLR), M-estimator robust regression (M-robust), MM-estimator robust regression (MM-robust), skipped regression (Skipped), deepest regression (Deepest), and Lasso regression. Absolute prediction errors then were compared.

Results.: With OLSLR, prediction error varied between 5.7 ± 6.1 (using the first three VFs) and 1.2 ± 1.1 dB (using all nine previous VFs). Prediction accuracy was not significantly improved with M-robust, MM-robust, Skipped, or Deepest regression in almost all VF series; however, a significantly smaller prediction error was obtained with Lasso regression even with a small number of VFs (using first 3 VFs, 2.0 ± 2.2; using all nine previous VFs, 1.2 ± 1.1 dB).

Conclusions.: Prediction errors using OLSLR are large when only a small number of VFs are included in the regression. Lasso regression offers much more accurate predictions, especially in short VF series.

Glaucomatous visual field (VF) deterioration is irreversible, but may be delayed through appropriate IOP reduction.1–3 However, the medical and surgical IOP reduction can be associated with various ocular and general complications.4–8 Therefore, it is essential to accurately predict future VF progression when making glaucoma treatment decisions. One popular approach to assess VF deterioration and predict future damage is to apply ordinary least squares linear regression (OLSLR) to global VF indices, such as mean deviation (MD), as implemented in the Guided Progression Analysis (GPA) software on the Humphrey Visual Field Analyser (HFA; Carl Zeiss Meditec AG, Dublin, CA, USA). However, VF sensitivity fluctuates in the short-term9 and long-term,10 and the reliability of a measured VF is inherently affected by a patient's concentration. Furthermore, VF measurement noise can be very large even when reliability indices are deemed good.11,12 
It has been reported that clinicians should acquire a considerable number of VFs to accurately forecast future progression, such as five or eight VFs.1315,17 Accumulating this number of VFs can take years in many clinics; hence, various attempts have been made to develop better models to predict VF progression. For instance, Caprioli et al.16 recently suggested that an exponential method offers more accurate predictions, especially when VF sensitivity approaches the floor level (0 dB).17,18 Very recent research has developed dedicated VF regression models, that take into account nonstationary variability and spatial correlations using a Bayesian approach19 or spatial/temporal patterns of glaucomatous VF progression also using a Bayesian approach.20 There also has been a renewed interest in applying alternative “ready-made” regression models as an alternative to OLSLR. A number of regression models, such as M-estimator robust regression (M-robust), MM-estimator robust regression (MM-robust), skipped regression (Skipped), and deepest regression (Deepest), have been developed to overcome the sensitivity of OLSLR to outliers.21 In addition, others have proposed a shrinkage method for OLSLR in which the sum of the absolute values of the regression coefficients is constrained or penalized, known as least absolute shrinkage and selection operator regression (Lasso).22,23 The most important merit of using Lasso is the optimum penalty can be decided using the actual clinical information of other patients; how the regression model should be shrunk to accurately predict the future VF. On the contrary, M-robust, MM-robust, Skipped, and Deepest obtain the robustness using only an individual's data. Lasso regression has been used in many different fields, including the analysis of human perception and genetic analysis.24,25 Indeed, we recently applied the method to predict the MD values in the 10-2 HFA VFs from 24-2 HFA VFs.26 In this study, several different robust regression models as well as Lasso regression were applied to the mean of total deviation values (mTD) in patients' VF series, and the performance of the different methods for predicting future progression was compared. 
Method
The study was approved by the Research Ethics Committee of the Graduate School of Medicine and Faculty of Medicine at the University of Tokyo. Informed consent was obtained from all subjects. This study was performed according to the tenets of the Declaration of Helsinki. 
Subjects and VFs
The VF data were retrospectively obtained from a total of 513 eyes in 324 patients with normal tension glaucoma, primary open-angle glaucoma, or exfoliation glaucoma. Patients were followed up in the glaucoma clinic at the University of Tokyo Hospital between 2002 and 2013; all of the patients had their VFs measured at least 12 times. Glaucoma was diagnosed when the following findings were present: (1) presence of typical glaucomatous changes in the optic nerve head, such as a rim notch with a rim width ≤ 0.1 disc diameters or a vertical cup-to-disc ratio of >0.7 and/or a retinal nerve fiber layer defect with its edge at the optic nerve head margin greater than a major retinal vessel, diverging in an arcuate or wedge shape confirmed by a panel of glaucoma specialists (HM and RA) after inspection of stereo-fundus photographs, and (2) a glaucomatous VF defined following the criteria27 of a cluster of ≥3 points in the pattern deviation plot in a single hemifield (superior/inferior) with P < 0.05, one of which must have been P < 0.01, excluding the outermost test point of Humphrey Field Analyzer 30-2 program; glaucoma hemifield test (GHT) result outside of normal limits; or abnormal pattern standard deviation (PSD) with P < 0.05. The VF measurements were performed using the HFA with either the 30-2 or 24-2 program and the Swedish Interactive Threshold Algorithm Standard. When VFs were obtained with the 30-2 test pattern, only the 52 test locations overlapping with the 24-2 test pattern were used in the analysis and for the calculation of mTD. Patients' first two VFs were excluded from the analysis. Other inclusion criteria in this study were best corrected visual acuity better than 6/12, refraction within ± 6 diopter (D) ametropia, no previous ocular surgery except for cataract extraction, and intraocular lens implantation, and no other anterior and posterior segment of the eye disease that could affect the VF, including cataract other than clinically insignificant senile cataract. Reliability criteria for VFs were applied: fixation losses less than 20% and false-positive responses less than 15%; the false-negative rate was not applied as a reliability criterion based on a previous report.28 The VF of a left eye was mirror-imaged to that of a right eye for statistical analyses. 
Prediction Accuracy
Prediction accuracy was compared between conventional OLSLR, M-robust, MM-robust, Skipped, Deepest, and Lasso regression. For each method, regression was done using mTD values from the first to the third VFs (VF1–3) of each patient, and the mTD values of the 10th VF (VF10) were predicted. The same procedure was carried out using the TD values in different series: VF1–4, VF1–5, VF1–6, VF1–7, VF1–8, and VF1–9, and the mTD values of VF10 were predicted every time. The predictive accuracy of each method was compared using absolute errors. As a subanalysis, predictive accuracy also was compared in eyes with an mTD progression rate < −0.25 dB/y (based on a patient's first 10 VFs), which commonly is considered to be deterioration by pressure-independent damaging factors.29–31 
Statistical Models
The following regression models were performed. 
For the OLSR method:    
For the M-robust regression method:  for the ith of n observations, the general M-estimator minimizes the objective function:  where the function ρ gives the contribution of each residual to the objective function.32 In short, M-robust regression, due to the formula above, is much less affected by outliers than OLSLR. In the current study, M-statistics were calculated using Huber's method.32  
The MM-robust regression method obtains further robustness to outliers by combining M estimation and high breakdown value estimation, as introduced by Yohai33 in 1987. 
The Skipped regression method is identical to the Theil–Sen estimator. This method also gains robustness to outliers by choosing the median slope among all lines constructed by all pairs of sample points.34 
The Deepest regression method gains robustness to outliers as well, by identifying the regression line with largest “depth” (as opposed to weighting residuals), which is defined as how well the data are balanced by the model.35–38 
The Lasso regression is a shrinkage method for OLSLR in which the sum of the absolute values of the regression coefficients is constrained or penalized. More precisely, let x ∈ Rp denote the variables and let y ∈ R denote the response (please note xij are normalized and y has mean zero). The Lasso algorithm solves the following problem:      
In Equation 1, min ( β 0 , β ) R p + 1 [ ( 1 / 2 N ) i = 1 N ( y i β 0 x i Y β ) 2 ] MathType@MTEF@5@4@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6AE0@ is identical to OLSLR and λPα(β) is the penalty term for the shrinkage.22,23 Thus the λ (Lambda) value represents the degree of penalty in Lasso. Equation 1 is Lasso regression when α = 1 and Ridge regression when α = 0; however, this discrimination is not applicable to the current study, because there is only one variable (mTD).  
Statistical Analysis
Absolute prediction accuracy was calculated as the absolute value of the difference between the model-predicted and the observed mTD value. For Lasso regression, the prediction error was calculated using leave-one-out cross validation. In this validation method, the data from a single patient (one or two eyes) were used as a testing dataset and all other data were used as training data; this procedure was repeated so that each patient was used only once as the testing dataset. In other words, for each individual, only the data from all other subjects (n = 323 in 324) were used to produce a diagnosis. An optimum λ value was identified for each iteration (patient), and the prediction error was calculated. 
All statistical analyses were done using the statistical programming language R (ver. 2.15.0; The R Foundation for Statistical Computing, Vienna, Austria). The M-robust, MM-robust, Skipped, Deepest, and Lasso regressions were calculated using the R packages “WRS” and “glmnet.” Absolute prediction errors were compared using the repeated measures ANOVA. Benjamini's method was used to correct P values for the problem of multiple testing.39 A linear mixed model was used to analyze the relationship between two values, whereby patients were treated as a “random effect.” 
Results
Characteristics of the study population are summarized in the Table. The mTD at baseline was −6.9 ± 6.2 dB (mean ± SD) and initial patient age was 54.2 ± 12.3 years. The progression rate of mTD was −0.24 ± 0.63 dB/y (Fig. 1). 
Table.
 
Demographic Characteristics
Table.
 
Demographic Characteristics
Figure 1
 
Histogram of progression of mTD progression rate. The progression rate of mTD was −0.24 ± 0.63 dB/y.
Figure 1
 
Histogram of progression of mTD progression rate. The progression rate of mTD was −0.24 ± 0.63 dB/y.
Figure 2 shows the absolute prediction errors associated with OLSLR, M-robust, MM-robust, Skipped, Deepest, and Lasso regression. The prediction error for MM-robust regression with VF1–3 could not be calculated because a leverage point could not be calculated. Absolute prediction errors became smaller as the number of VF tests included in the regression increased. There was no significant improvement in error by applying M-robust, MM-robust, Skipped, and Deepest, compared to using the OLSLR at any time point (P > 0.05, repeated ANOVA with Benjamini's correction for multiple testing39), except for M-robust with VF1–8 (P = 0.028, repeated ANOVA with Benjamini's correction for multiple testing39). The absolute prediction errors with the Lasso model were significantly better than OLSLR when VF1–3 to VF1–8 were used for prediction (P <0.0001, <0.0001, <0.0001, <0.0001, <0.0001, 0.0017, 0.016, repeated ANOVA with Benjamini's correction for multiple testing39). Among the 513 eyes, 234 eyes showed progression faster than −0.25 dB/y. Interestingly, as shown in Figure 3, prediction accuracy tended to be large in eyes with mTD progression rate < −0.25 dB/y. A significant improvement was observed when applying Lasso, compared to OLSLR, when the initial one or two VFs were used to predict (P = 0.007, 0.035, repeated ANOVA with Benjamini's correction for multiple testing39). 
Figure 2
 
Mean prediction errors for predicting the 10th VF using the initial 3 to 9 VFs, with various regression models (all subjects, N = 513). The absolute prediction error of OLSLR became small as the number of VF tests increased. There was no significant improvement in the absolute errors by applying M-robust, MM-robust, Skipped, and Deepest regression, compared to using OLSLR at any time point (P > 0.05, repeated ANOVA with Benjamini's correction for multiple testing39), except for M-robust with VF1–8 (P = 0.028, repeated ANOVA with Benjamini's correction for multiple testing39). The absolute prediction errors with the Lasso model were significantly better than OLSLR when VF1–3 to VF1–8 were used for prediction (P <0.0001, <0.0001, <0.0001, <0.0001, <0.0001, 0.0017, 0.016, repeated ANOVA with Benjamini's correction for multiple testing39). Error bar: represents 95% confidence interval of mean value. **P < 0.05 in the comparison between OLSLR and Lasso with Benjamini's correction for multiple testing.39
Figure 2
 
Mean prediction errors for predicting the 10th VF using the initial 3 to 9 VFs, with various regression models (all subjects, N = 513). The absolute prediction error of OLSLR became small as the number of VF tests increased. There was no significant improvement in the absolute errors by applying M-robust, MM-robust, Skipped, and Deepest regression, compared to using OLSLR at any time point (P > 0.05, repeated ANOVA with Benjamini's correction for multiple testing39), except for M-robust with VF1–8 (P = 0.028, repeated ANOVA with Benjamini's correction for multiple testing39). The absolute prediction errors with the Lasso model were significantly better than OLSLR when VF1–3 to VF1–8 were used for prediction (P <0.0001, <0.0001, <0.0001, <0.0001, <0.0001, 0.0017, 0.016, repeated ANOVA with Benjamini's correction for multiple testing39). Error bar: represents 95% confidence interval of mean value. **P < 0.05 in the comparison between OLSLR and Lasso with Benjamini's correction for multiple testing.39
Figure 3
 
Mean prediction errors for predicting the 10th VF using the initial 3 to 9 VFs, with various regression models (progression rate faster than −0.25 dB/y, N = 234). The absolute prediction error in eyes with progression rate faster than −0.25 dB/y (N = 234). A significant improvement was observed for Lasso regression only when the initial one or two VFs were used to predict (P = 0.007 and 0.035, repeated ANOVA with Benjamini's correction for multiple testing39). **P < 0.05 in the comparison between OLSLR and Lasso with Benjamini's correction for multiple testing.39
Figure 3
 
Mean prediction errors for predicting the 10th VF using the initial 3 to 9 VFs, with various regression models (progression rate faster than −0.25 dB/y, N = 234). The absolute prediction error in eyes with progression rate faster than −0.25 dB/y (N = 234). A significant improvement was observed for Lasso regression only when the initial one or two VFs were used to predict (P = 0.007 and 0.035, repeated ANOVA with Benjamini's correction for multiple testing39). **P < 0.05 in the comparison between OLSLR and Lasso with Benjamini's correction for multiple testing.39
Figure 4 shows the optimum λ value derived in relationship to the number of VFs used for prediction. The λ value decreased as the number of VFs used in the prediction increased. 
Figure 4
 
The relationship between the optimum λ value and the number of VFs used for prediction. The mean optimum λ value decreased with an increase in the number of VFs used in the prediction. Error bar represents 95% confidence interval of mean value.
Figure 4
 
The relationship between the optimum λ value and the number of VFs used for prediction. The mean optimum λ value decreased with an increase in the number of VFs used in the prediction. Error bar represents 95% confidence interval of mean value.
As shown in Figure 5, there was no significant relationship between the optimum λ value derived for VF1–3, VF1–4, and VF1–5, and the mTD value of VF1 (P = 0.18, 0.12, and 0.31, respectively, linear mixed model). As shown in Figure 6, there was no significant relationship between the optimum λ value derived for VF1–3, VF1–4, and VF1–5, and the difference between mTDs in VF1 and VF10 (P = 0.18, 0.33, and 0.94, respectively, linear mixed model). Figures 5 and 6 are smoothed scatter plots (plotted using the R package “graphics”), which better differentiate dense regions of points. 
Figure 5
 
The relationship between the optimum λ value and the mean total deviation value. There was no significant relationship between the optimum λ value with VF1–3, VF1–4, and VF1–5, and the mTD value in the VF1 (P > 0.05, linear mixed model). Figure 4 is plotted as a smoothed scatter plot.
Figure 5
 
The relationship between the optimum λ value and the mean total deviation value. There was no significant relationship between the optimum λ value with VF1–3, VF1–4, and VF1–5, and the mTD value in the VF1 (P > 0.05, linear mixed model). Figure 4 is plotted as a smoothed scatter plot.
Figure 6
 
The relationship between the optimum λ value and the difference of the mean total deviation value in the initial and last VF. There was no significant relationship between the optimum λ value with VF1–3, VF1–4, and VF1–5, and the difference of mean total deviation value in the initial and last VF (P > 0.05, linear mixed model). Figure 5 is plotted as a smoothed scatter plot.
Figure 6
 
The relationship between the optimum λ value and the difference of the mean total deviation value in the initial and last VF. There was no significant relationship between the optimum λ value with VF1–3, VF1–4, and VF1–5, and the difference of mean total deviation value in the initial and last VF (P > 0.05, linear mixed model). Figure 5 is plotted as a smoothed scatter plot.
Discussion
In this study, the prediction performance of various robust regression models and Lasso regression were compared against OLSLR. As a result, a considerable number of VFs were needed to obtain an accurate prediction with OLSLR; it also was shown that there was no significant improvement in error by applying robust regression (M-robust, Skipped, and Deepest regression methods). On the contrary, a significantly smaller prediction error was observed with Lasso regression even when the number of VFs included was just three long. This difference should be attributed to the algorithms; in Lasso regression, the model is optimized using other patients' information (i.e., the degree of shrinkage is decided by optimizing the penalty value), while in M-robust, MM-robust, Skipped, and Deepest regression method only data from each individual eye were used. 
The VF variability significantly hampers the usefulness of VF trend analyses,40 and consequently, the minimum number of VFs required to obtain reliable VF trend analysis results has been widely discussed in previous studies, with research suggesting that at least five or eight VFs, or even higher are required.1315,17 Indeed, prediction accuracy associated with OLSLR was poor when a small number of VFs were used in the current study. This poor prediction accuracy was not improved by applying a number of different robust regression methods. On the other hand, Lasso regression performed much better due to the fact that the method uses a penalty term, which helps to reduce prediction errors. In other words, the coefficient terms in Lasso regression are adjusted by the optimum penalty (λ), which is obtained from real data (from other patients). As a result, Lasso mTD trend analysis becomes much more robust to measurement noise and, consequently, prediction accuracy was dramatically improved. This is different to M-robust, MM-robust, Skipped, and Deepest regression, which attempt to improve robustness using only an individual's data. 
In the current study, leave-one-out cross validation was used to evaluate the performance of Lasso regression. As described in the Methods section, the original dataset was divided into a testing dataset (one patient) and training data (all other patients), and the λ value was calculated using only the training dataset; this process was repeated so that each patient was used as a testing dataset once. This is identical to the clinical situation—a new patient can be classified according to the predetermined optimum λ value. Furthermore, in clinical practice, a λ value could be calculated continuously by adding the data of new patients to an ever-growing database of patient data, which would further improve prediction accuracy. In addition, the Lasso regression performed in this study was built using free statistical software and packages, specifically “R” (ver. 3.1.0; The R Foundation for Statistical Computing). 
As shown in Figure 4, a large optimum λ value was observed when a small number of VFs was used for prediction and it decreased as the number of VFs used for prediction increased. This suggested that any mTD trend analysis should be penalized according to the number of VFs used. It is well-known that test–retest reproducibility varies according to the level of VF sensitivity,19,41 and recently Zhu et al.19 reported a novel approach to measure VF progression that modeled nonstationary variability.19 Interestingly, as shown in Figures 4 and 5, there was no significant relationship between the initial mTD value (and also the difference of mTD values in VF10 and VF1) and the optimum λ value; this suggests that the same penalty should be given to the MD trend analysis regardless of disease level. This could suggest that the variability of measured VF sensitivity is not merely due to inherent test–retest variability associated with the sensitivity level, but also other elements, such as patients' concentration. However, the information obtained from a small number of VFs is fundamentally limited compared to the large variability and clinicians always should be careful when interpreting trend analyses with a small number of VFs.1315,17 In agreement, the optimum λ value shrank when the number of VFs used in the prediction was large. 
In the subanalysis, in which only progressing eyes were analyzed, the advantage of using Lasso regression was observed when a small numbers of VFs were used for prediction. However, no advantage was observed when a larger numbers of VFs were used. As shown in Figure 1, more than half of eyes in the complete sample did not progress, but the Lasso λ value was decided based on all eyes. Recent studies have improved prediction accuracy by clustering eyes with different progression patterns.20,42–44 Thus, it may be advantageous to optimize Lasso regression (the λ value) based on a patient's progression pattern; that is, prediction accuracy may be further improved by using a clustering approach in combination with Lasso regression; this should be carried out in a future study. 
It has been suggested using a VF progression analysis tool, such as “PROGRESSOR”45 (Medisoft, Inc., London, UK) improves clinicians' decisions regarding VF progression.46 A possible caveat of the current results is that the prediction with the Lasso regression is not readily useable at the clinical setting. Therefore, it would be clinically beneficial to develop software/support tools to predict VF progression, as introduced in this study, similar to PROGRESSOR.45 In particular, only standard data are needed to apply the current methodology in the clinical setting; having the record of MD values with the date of VF measurements of a patient, since the optimum penalty (λ) value can be calculated from other patients. 
In conclusion, prediction accuracy of VF progression is poor when OLSLR is used with a small number of VFs. Prediction performance was not improved by applying M-robust, MM-robust, Skipped, and Deepest regressions; however, considerable improvement was observed by applying the Lasso regression technique. 
Acknowledgments
Supported in part by Japan Science and Technology Agency (JST)-CREST and Grant 26462679 from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. 
Disclosure: Y. Fujino, None; H. Murata, None; C. Mayama, None; R. Asaoka, None 
References
Danias J, Podos SM. Comparison of glaucomatous progression between untreated patients with normal-tension glaucoma and patients with therapeutically reduced intraocular pressures. The effectiveness of intraocular pressure reduction in the treatment of normal-tension glaucoma. Am J Ophthalmol. 1999; 127: 623–625.
Leske MC, Hyman L, Hussein M, et al. Comparison of glaucomatous progression between untreated patients with normal-tension glaucoma and patients with therapeutically reduced intraocular pressures. The effectiveness of intraocular pressure reduction in the treatment of normal-tension glaucoma. Am J Ophthalmol. 1999; 127: 625–626.
Jampel HD. Target pressure in glaucoma therapy. J Glaucoma. 1997; 6: 133–138.
Shields MB. Textbook of Glaucoma. Baltimore, MD: William & Wilkins; 1997.
Aihara M, Shirato S, Sakata R. Incidence of deepening of the upper eyelid sulcus after switching from latanoprost to bimatoprost. Jpn J Ophthalmol. 2011; 55: 600–604.
Sakata R, Shirato S, Miyata K, et al. Incidence of deepening of the upper eyelid sulcus on treatment with a tafluprost ophthalmic solution. Jpn J Ophthalmol. 2014; 58: 212–217.
Yamada Y, Takayanagi R, Tsuchiya K, et al. Assessment of systemic adverse reactions induced by ophthalmic beta-adrenergic receptor antagonists. J Ocul Pharmacol Ther. 2001; 17: 235–248.
Yamamoto T, Sawada A, Mayama C, et al. The 5-year incidence of bleb-related infection and its risk factors after filtering surgeries with adjunctive mitomycin C: collaborative bleb-related infection incidence and treatment study 2. Ophthalmology. 2014; 121: 1001–1006.
Flammer J, Drance SM, Fankhauser F, et al. Differential light threshold in automated static perimetry. Factors influencing short-term fluctuation. Arch Ophthalmol. 1984; 102: 876–879.
Flammer J, Drance SM, Zulauf M. Differential light threshold. Short- and long-term fluctuation in patients with glaucoma, normal controls, and patients with suspected glaucoma. Arch Ophthalmol. 1984; 102: 704–706.
Bengtsson B, Heijl A. False-negative responses in glaucoma perimetry: indicators of patient performance or test reliability? Invest Ophthalmol Vis Sci. 2000; 41: 2201–2204.
Henson DB, Evans J, Chauhan BC, et al. Influence of fixation accuracy on threshold variability in patients with open angle glaucoma. Invest Ophthalmol Vis Sci. 1996; 37: 444–450.
Krakau CE. A statistical trap in the evaluation of visual field decay. Acta Ophthalmol. 1985; 173 (suppl): 19–21.
Holmin C, Krakau CE. Regression analysis of the central visual field in chronic glaucoma cases. A follow-up study using automatic perimetry. Acta Ophthalmol (Copenh). 1982; 60: 267–274.
Spry PG, Bates AB, Johnson CA, et al. Simulation of longitudinal threshold visual field data. Invest Ophthalmol Vis Sci. 2000; 41: 2192–2200.
Caprioli J, Mock D, Bitrian E, et al. A method to measure and predict rates of regional visual field decay in glaucoma. Invest Ophthalmol Vis Sci. 2011; 52: 4765–4773.
Bengtsson B, Patella VM, Heijl A. Prediction of glaucomatous visual field loss by extrapolation of linear trends. Arch Ophthalmol. 2009; 127: 1610–1615.
Russell RA, Crabb DP. On alternative methods for measuring visual field decay: Tobit linear regression. Invest Ophthalmol Vis Sci. 2011; 52: 9539–9540.
Zhu H, Russell RA, Saunders LJ, et al. Detecting changes in retinal function: analysis with Non-Stationary Weibull Error Regression and Spatial enhancement (ANSWERS). PLoS One. 2014; 9: e85654.
Murata H, Araie M, Asaoka R. A new approach to measure visual field progression in glaucoma patients using Variational Bayes linear regression. Invest Ophthalmol Vis Sci. 2014; 55: 8386–8392.
Wilcox R. Introduction to Robust Estimation and Hypothesis Testing. Amsterdam, The Netherlands: Elsevier, 2011.
Tibshirani R. Regression shrinkage and selection via the lasso. J R Stat Soc Series B. 1996; 58: 267–288.
Friedman J, Hastie T, Tibshirani R. Regularization paths for generalized linear models via coordinate descent. J Stat Softw. 2010; 33: 1–22.
Barbosa MS, Bubna-Litic A, Maddess T. Locally countable properties and the perceptual salience of textures. J Opt Soc Am A Opt Image Sci Vis. 2013; 30: 1687–1697.
Akutekwe A, Seker H. A hybrid dynamic Bayesian network approach for modelling temporal associations of gene expressions for hypertension diagnosis. Conf Proc IEEE Eng Med and Biol Soc. 2014; 2014: 804–807.
Asaoka R. Measuring visual field progression in the central 10 degrees using additional information from central 24 degrees visual fields and ‘lasso regression'. PLoS One. 2013; 8: e72199.
Anderson D. Automated Static Perimetry. St. Louis, MO: C.V. Mosby Co.; 1992.
Bengtsson B, Heijl A. False-negative responses in glaucoma perimetry: indicators of patient performance or test reliability? Am J Ophthalmol. 2000; 130: 689.
Anderson DR, Drance SM, Schulzer M, et al. Natural history of normal-tension glaucoma. Ophthalmology. 2001; 108: 247–253.
Araie M. Basic and clinical studies of pressure-independent damaging factors of open angle glaucoma. Nippon Ganka Gakkai zasshi. 2011; 115: 213–236, discussion 237.
Koseki N, Araie M, Tomidokoro A, et al. A placebo-controlled 3-year study of a calcium blocker on visual field and ocular circulation in glaucoma with low-normal pressure. Ophthalmology. 2008; 115: 2049–2057.
Huber PJ. 1972 Wald Memorial Lectures - Robust Regression - Asymptotics, Conjectures and Monte-Carlo. Ann Stat. 1973; 1: 799–821.
Yohai VJ. High breakdown-point and high-efficiency robust estimates for regression. Ann Stat. 1987; 15: 642–656.
Theil H. A rank-invariant method of linear and polynomial regression analysis. I, II, III. Nederl Akad Wetensch Proc. 1950; 53: 386–392, 521–525, 1397–1412.
Rousseeuw PJ, Hubert H. Regression depth. J Am Stat Assoc. 1999; 94: 388–402.
Bai Z, He X. Asymptotic distributions of the maximal depth estimators for regression and multivariate location. Ann Stat. 1999; 27: 1616–1637.
Van Aelst S, Rousseeuw PJ. Robustness of deepest regression. J Multivariate Anal. 2000; 73: 83–106.
Van Aelst S, Rousseeuw PJ, Hubert M, et al. The deepest regression method. J Multivariate Anal. 2002; 81: 138–166.
Benjamini Y, Hochberg Y. Controlling the false discovery rate: a practical and powerful approach to multiple testing. J Roy Stat Soc Series B. 1995; 57: 289–300.
Jansonius NM. On the accuracy of measuring rates of visual field change in glaucoma. Br J Ophthalmol. 2010; 94: 1404–1405.
Artes PH, Iwase A, Ohno Y, et al. Properties of perimetric threshold estimates from Full Threshold, SITA Standard, and SITA Fast strategies. Invest Ophthalmol Vis Sci. 2002; 43: 2654–2659.
Liang ZH, Tomioka R, Murata H, et al. Quantitative prediction of visual field loss due to glaucoma from few measurements. Proc IEEE Int Conf Data Mining (ICDM2013). 2013; 1121–1126.
Sample PA, Chan K, Boden C, et al. Using unsupervised learning with variational bayesian mixture of factor analysis to identify patterns of glaucomatous visual field defects. Invest Ophthalmol Vis Sci. 2004; 45: 2596–2605.
Goldbaum MH, Lee I, Jang G, et al. Progression of patterns (POP): a machine classifier algorithm to identify glaucoma progression in visual fields. Invest Ophthalmol Vis Sci. 2012; 53: 6557–6567.
Fitzke FW, Hitchings RA, Poinoosawmy D, et al. Analysis of visual field progression in glaucoma. Br J Ophthalmol. 1996; 80: 40–48.
Viswanathan AC, Crabb DP, McNaught AI, et al. Interobserver agreement on visual field progression in glaucoma: a comparison of methods. Br J Ophthalmol. 2003; 87: 726–730.
Figure 1
 
Histogram of progression of mTD progression rate. The progression rate of mTD was −0.24 ± 0.63 dB/y.
Figure 1
 
Histogram of progression of mTD progression rate. The progression rate of mTD was −0.24 ± 0.63 dB/y.
Figure 2
 
Mean prediction errors for predicting the 10th VF using the initial 3 to 9 VFs, with various regression models (all subjects, N = 513). The absolute prediction error of OLSLR became small as the number of VF tests increased. There was no significant improvement in the absolute errors by applying M-robust, MM-robust, Skipped, and Deepest regression, compared to using OLSLR at any time point (P > 0.05, repeated ANOVA with Benjamini's correction for multiple testing39), except for M-robust with VF1–8 (P = 0.028, repeated ANOVA with Benjamini's correction for multiple testing39). The absolute prediction errors with the Lasso model were significantly better than OLSLR when VF1–3 to VF1–8 were used for prediction (P <0.0001, <0.0001, <0.0001, <0.0001, <0.0001, 0.0017, 0.016, repeated ANOVA with Benjamini's correction for multiple testing39). Error bar: represents 95% confidence interval of mean value. **P < 0.05 in the comparison between OLSLR and Lasso with Benjamini's correction for multiple testing.39
Figure 2
 
Mean prediction errors for predicting the 10th VF using the initial 3 to 9 VFs, with various regression models (all subjects, N = 513). The absolute prediction error of OLSLR became small as the number of VF tests increased. There was no significant improvement in the absolute errors by applying M-robust, MM-robust, Skipped, and Deepest regression, compared to using OLSLR at any time point (P > 0.05, repeated ANOVA with Benjamini's correction for multiple testing39), except for M-robust with VF1–8 (P = 0.028, repeated ANOVA with Benjamini's correction for multiple testing39). The absolute prediction errors with the Lasso model were significantly better than OLSLR when VF1–3 to VF1–8 were used for prediction (P <0.0001, <0.0001, <0.0001, <0.0001, <0.0001, 0.0017, 0.016, repeated ANOVA with Benjamini's correction for multiple testing39). Error bar: represents 95% confidence interval of mean value. **P < 0.05 in the comparison between OLSLR and Lasso with Benjamini's correction for multiple testing.39
Figure 3
 
Mean prediction errors for predicting the 10th VF using the initial 3 to 9 VFs, with various regression models (progression rate faster than −0.25 dB/y, N = 234). The absolute prediction error in eyes with progression rate faster than −0.25 dB/y (N = 234). A significant improvement was observed for Lasso regression only when the initial one or two VFs were used to predict (P = 0.007 and 0.035, repeated ANOVA with Benjamini's correction for multiple testing39). **P < 0.05 in the comparison between OLSLR and Lasso with Benjamini's correction for multiple testing.39
Figure 3
 
Mean prediction errors for predicting the 10th VF using the initial 3 to 9 VFs, with various regression models (progression rate faster than −0.25 dB/y, N = 234). The absolute prediction error in eyes with progression rate faster than −0.25 dB/y (N = 234). A significant improvement was observed for Lasso regression only when the initial one or two VFs were used to predict (P = 0.007 and 0.035, repeated ANOVA with Benjamini's correction for multiple testing39). **P < 0.05 in the comparison between OLSLR and Lasso with Benjamini's correction for multiple testing.39
Figure 4
 
The relationship between the optimum λ value and the number of VFs used for prediction. The mean optimum λ value decreased with an increase in the number of VFs used in the prediction. Error bar represents 95% confidence interval of mean value.
Figure 4
 
The relationship between the optimum λ value and the number of VFs used for prediction. The mean optimum λ value decreased with an increase in the number of VFs used in the prediction. Error bar represents 95% confidence interval of mean value.
Figure 5
 
The relationship between the optimum λ value and the mean total deviation value. There was no significant relationship between the optimum λ value with VF1–3, VF1–4, and VF1–5, and the mTD value in the VF1 (P > 0.05, linear mixed model). Figure 4 is plotted as a smoothed scatter plot.
Figure 5
 
The relationship between the optimum λ value and the mean total deviation value. There was no significant relationship between the optimum λ value with VF1–3, VF1–4, and VF1–5, and the mTD value in the VF1 (P > 0.05, linear mixed model). Figure 4 is plotted as a smoothed scatter plot.
Figure 6
 
The relationship between the optimum λ value and the difference of the mean total deviation value in the initial and last VF. There was no significant relationship between the optimum λ value with VF1–3, VF1–4, and VF1–5, and the difference of mean total deviation value in the initial and last VF (P > 0.05, linear mixed model). Figure 5 is plotted as a smoothed scatter plot.
Figure 6
 
The relationship between the optimum λ value and the difference of the mean total deviation value in the initial and last VF. There was no significant relationship between the optimum λ value with VF1–3, VF1–4, and VF1–5, and the difference of mean total deviation value in the initial and last VF (P > 0.05, linear mixed model). Figure 5 is plotted as a smoothed scatter plot.
Table.
 
Demographic Characteristics
Table.
 
Demographic Characteristics
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