**Purpose.**:
To evaluate the minimum number of visual field (VF) tests required to precisely predict future VF results using ordinary least squares linear regression (OLSLR), quadratic regression, exponential regression, logistic regression, and M-estimator robust regression model.

**Methods.**:
Series of 15 VFs (Humphrey Field Analyzer 24-2 SITA standard) were analyzed from 247 eyes of 155 open-angle glaucoma patients. Future point-wise (PW) VF results and mean VF sensitivities were predicted with varying numbers of VFs in each regression method.

**Results.**:
In PW-OLSLR, as expected, the minimum absolute prediction error was obtained using the maximum number of VFs in the regression (14 VFs); mean absolute prediction error was equal to 2.4 ± 0.9 dB. Ten VFs were required to reach the 95% confidence interval (CI) of the minimum absolute prediction error. Prediction errors associated with the exponential and quadratic regression models were significantly larger than those from PW-OLSLR, whereas errors from logistic regression were not significantly smaller than those from PW-OLSLR; however, the absolute prediction error from the M-estimator robust regression model was significantly smaller than those associated with PW-OLSLR (*P* < 0.01, paired Wilcoxon test). Like PW-OLSLR, 10 VFs were needed to obtain the minimum absolute prediction error of mean VF sensitivity, but there were no significant differences in errors using the different regression methods.

**Conclusions.**:
Approximately 10 VFs, are needed to achieve an accurate prediction of PW VF sensitivity and mean sensitivity. Prediction error of PW VF sensitivity can be significantly minimized using the M-estimator robust regression model compared with conventional OLSLR.

^{1}As glaucoma can lead to irreversible blindness, it is essential to accurately measure visual field (VF) progression so that physicians can treat patients with glaucoma accordingly, especially because medical and surgical IOP reduction interventions can be accompanied with a number of ocular and general complications.

^{2–6}It is therefore imperative to accurately predict future VF progression when making glaucoma treatment decisions. Visual field trend analyses can be used to measure the progression of global indices, such as mean deviation (MD) and the visual field index (VFI), or point-wise (PW) sensitivity using ordinary least square linear regression (OLSLR). This type of analysis is widely used in glaucoma clinics using clinical support tools such as PROGRESSOR (Medisoft, Ltd., London, UK).

^{7}

^{8}and long-term.

^{9}The reliability of VF measurements is inherently affected by a patient's concentration, but previous reports have suggested that measurement noise is considerable even with good reliability indices.

^{10,11}The ability of VF trend analyses, in particular PW linear regression (PLR), to detect progression will be significantly impeded by VF variability.

^{12}Consequently, the minimum number of VFs required to obtain reliable PLR results has been widely discussed in previous studies.

^{13–16}

^{14}reported that omitting VFs helps to identify progression, whereas Hirasawa et al.

^{17}suggested that the accuracy of PW-OLSLR can be improved by applying it to the average sensitivities of small sectors. Alternatively, robust linear regression models can be applied to mitigate the effects of outliers and improve the detection of progression. In the robust model, the weight of each data point in the regression is dependent on the size of its residual; thus, the model is more “robust” to the influence of outliers.

^{18}

^{19,20}and the remaining 15 VFs were obtained over 7.5 ± 2.1 (mean ± SD) years of follow-up. Inclusion criteria for this study were patients with a visual acuity better than or equal to 6/12, refraction less than 5 diopters ametropia, no previous ocular surgery (except for cataract extraction), and no other posterior segment eye diseases. Patients with other ocular diseases that could affect VF sensitivity, such as diabetes mellitus retinopathy, corneal opacity, and AMD were excluded. Patients with cataract other than clinically insignificant senile cataract were excluded. All VFs were recorded using the Humphrey Field Analyzer (HFA; Carl-Zeiss Meditec, CA, USA), with the 24-2 or 30-2 test pattern and the SITA standard strategy with a Goldmann size III target. When the VF was measured using the 30-2 test pattern, only the 52 test points overlapping with the 24-2 test pattern were used for the analysis.

_{6}), seventh VF (VF

_{7}), and eighth VF (VF

_{8}) were predicted from the first five VFs (VF

_{1–5}). Absolute prediction accuracy was calculated as the absolute value of the difference between the predicted and the observed PW sensitivities. In addition, the absolute prediction errors using the exponential, quadratic, logistic, and M-robust methods were also calculated. The process was iterated to predict the PW sensitivities of the seventh, eighth and ninth VFs (first, second, third future VF: VF

_{7}, VF

_{8}, and VF

_{9}) using VF

_{1–6}, the PW sensitivities of VF

_{8}, VF

_{9}, and VF

_{10}using VF

_{1–7}, and so on, up to prediction of the PW sensitivities of VF

_{13,}VF

_{14}, and VF

_{15}using VF

_{1–12}. The minimum prediction error was calculated for each regression method and the minimum number of VFs to reach the 95% confidence interval (CI) was identified. These analyses were also performed using the mean sensitivity of the entire VF. As a subanalysis, predictive accuracy was compared in eyes with early-stage glaucoma (initial mTD [mean of total deviation of all 52 total deviation values in the 24-2 HFA VF] larger than −6 dB) and eyes with moderate to advanced-stage glaucoma (initial mTD ≤ −6 dB). In addition, accuracy was evaluated in eyes with an mTD progression rate smaller than −0.25 dB per year and the remaining eyes, which is commonly considered to be the deterioration by pressure-independent damaging factors.

^{21–23}

*y*=

*ax*+

*b*

*y*=

*e*

^{ax+b}

*y*=

*ax*

^{2}+

*bx*+

*c*

^{25,26}was used to correct

*P*values for the problem of multiple testing.

_{1}was −6.6 ± 5.3 (range, 1.1 to −25.8) dB and −8.7 ± 6.3 (range, 1.3 to −27.0) dB in VF

_{15}. The mean ± SD interval between the first, second, and third future VFs was 5.9 ± 0.2 months, 11.9 ± 0.2 months, and 17.8 ± 0.2 months, respectively.

**Table 1**

_{1–14}(2.4 ± 0.9 [range, 0.6–5.9] dB: see Fig. 1A), VF

_{1–13}(2.6 ± 1.0 [range, 0.6–6.5] dB: Fig. 1B), and VF

_{1–12}(2.8 ± 1.2 [range, 0.6–9.1] dB: Fig. 1C), respectively. The absolute prediction errors associated with the exponential and quadratic regressions were significantly larger then OLSLR in all comparisons (paired Wilcoxon test,

*P*< 0.05 after correction of

*P*values for multiple testing using Holm's method

^{25,26}). No significant improvement was observed in the prediction error by applying logistic regression compared with OLSLR (paired Wilcoxon test,

*P*> 0.05 after correction of

*P*values for multiple testing using Holm's method

^{25,26}). The absolute prediction errors (mean ± SD) associated with the M-estimator robust regression were 2.3 ± 0.9 (range, 0.6–5.8) dB using VF

_{1–14}, 2.5 ± 0.9 (range, 0.6–6.2) dB using VF

_{1–13}, and 2.7 ± 1.1 (range, 0.6–9.0) dB using VF

_{1–12}. These values were significantly smaller than those in other models, in all comparisons for first future VF prediction (paired Wilcoxon test,

*P*< 0.001 except for VF

_{1–5}:

*P*= 0.010, after correction of

*P*values for multiple testing using Holm's method

^{25,26}), except for VF

_{1–5}for second future VF prediction (

*P*< 0.001, except for VF

_{1–6}:

*P*= 0.022, VF

_{1–12}:

*P*= 0.0047), and except for VF

_{1–5}for third future VF prediction (

*P*< 0.001 except for VF

_{1–11}:

*P*= 0.0018). In PW-OLSLR, the absolute prediction error for predicting the first, second, and third future VFs reached 95% CI of the minimum prediction error using VF

_{1–11}, VF

_{1–10}, and VF

_{1–10}, respectively.

**Figure 1**

**Figure 1**

_{1–10}, VF

_{1–11}, and VF

_{1–13}(

*P*= 0.0071, 0.025, and 0.010, paired Wilcoxon test, Fig. 2A), and when predicting the second future VF using VF

_{1–9}, V

_{1–10}, and VF

_{1–11}(

*P*= 0.015, 0.044, and 0.044, paired Wilcoxon test, Fig. 2B). No significant difference was observed in predicting the third future VF using VF

_{1–5}to VF

_{1–12}(

*P*> 0.05, paired Wilcoxon test, Fig. 2C). In eyes with more advanced glaucoma (mTD ≤ −6 dB; 121 eyes, Table 2), significantly smaller prediction errors were observed when predicting the first future VF using VF

_{1–5}to VF

_{1–14}(VF

_{1–5}: 0.025, VF

_{1–6}:

*P*= 0.0010, VF

_{1–7}:

*P*= 0.0014, and VF

_{1–8}to VF

_{1–14}:

*P*< 0.001, paired Wilcoxon test, Fig. 3A), predicting the second future VF using VF

_{1–6}to VF

_{1–13}(VF

_{1–6}: 0.001, VF

_{1–7}to VF

_{1–13}:

*P*< 0.001, paired Wilcoxon test, Fig. 3B), but not for predicting the third future VF using VF

_{1–5}to VF

_{1–12}(

*P*> 0.05, paired Wilcoxon test, Fig. 3C).

**Table 2**

**Figure 2**

**Figure 2**

**Figure 3**

**Figure 3**

_{1–6}and VF

_{1–8}to VF

_{1–12}(

*P*= 0.028, 0.0080, 0.029, <0.001, <0.001, 0.0020, paired Wilcoxon test, Fig. 4A), predicting the second future VF using VF

_{1–7}to V

_{1–11}and VF

_{1–13}(

*P*< 0.001 except for VF

_{1–8}:

*P*= 0.0013 and VF

_{1–13}:

*P*= 0.0068, paired Wilcoxon test, paired Wilcoxon test, Fig. 4B), and predicting the third future VF using from VF

_{1–7}to VF

_{1–12}(

*P*= 0.029, 0.044, 0.031, 0.0085, 0.049, 0.013, paired Wilcoxon test, Fig. 4C). In the remaining eyes with mTD progression rate equal to or slower than −0.25 dB per year (130 eyes, Table 2), significantly smaller prediction errors were observed when predicting the first future VF using VF

_{1–7}to VF

_{1–14}(VF

_{1–7}:

*P*= 0.0015, VF

_{1–8}: 0.013, VF

_{1–9}: 0.014, VF

_{1–7}to VF

_{1–14}:

*P*< 0.001, paired Wilcoxon test, Fig. 5A), predicting the second future VF using VF

_{1–9}to VF

_{1–13}(

*P*values were <0.001, 0.0024, 0.0025, 0.0015, and <0.001, respectively, paired Wilcoxon test, Fig. 5B). No significant difference was observed in predicting the third future VF using VF

_{1–5}to VF

_{1–12}(

*P*> 0.05, paired Wilcoxon test, Fig. 5C).

**Figure 4**

**Figure 4**

**Figure 5**

**Figure 5**

_{1–14}(2.4 ± 0.9 dB [0.004–5.3]) for predicting the first future VF, using VF

_{1–13}(2.6 ± 1.0 dB [0.002–5.2]) for predicting the second future VF, and using VF

_{1–12}(2.8 ± 1.2 dB [0.008–9.1]) for predicting the third future VF. There were no significant differences in the absolute errors associated with OLSLR, exponential regression, and M-estimator robust regression models at any time point. The absolute prediction errors with the quadratic regression model were significantly worse than the OLSLR method (paired Wilcoxon test,

*P*> 0.05 after correction of

*P*values for multiple testing using Holm's method

^{25,26}).

**Figure 6**

**Figure 6**

^{13,27–29}

^{28}recently fitted the exponential regression model, the quadratic regression model, and the OLSLR model to series of VFs and suggested that the exponential model was the best-fitting because it obtained the smallest Akaike's Information Criterion (AIC); the authors argue that the exponential method better models VF sensitivity as it approaches the floor level of VF sensitivity (0 dB).

^{29,30}Nevertheless, the model with the smallest AIC does not guarantee it is always the best model for prediction accuracy.

^{31}Indeed, in the current study, the exponential regression model was significantly worse in terms of prediction accuracy, compared with OLSLR in the PW analysis, in agreement with a previous study.

^{32}One possible reason for this apparently contradictory result may be the difference in datasets analyzed. In our study, the average MDs of the initial and last VFs were −6.6 and −8.7 dB, respectively, whereas in the Caprioli et al.

^{28}article, the MDs were much worse: −10.9 and −12.9 dB, respectively. As a result, many more VFs will tend to reach the floor level in the previous study. It is possible that the “best” model depends on the length of the observation period and the stage of glaucoma. Nonetheless, the clinical usefulness of applying the exponential regression model and also the logistic regression model

^{33}instead of OLSLR, may be limited because it requires approximately 7 or 8 years to obtain 15 VFs under an assumption that VF measurements are carried out every 6 months.

^{34}

^{14}recommended applying a “Three-Omitting” rule in which

*the last*VF is omitted and instead

*two future*VFs are used to detect and confirm progression. Over 6 years of follow-up, they showed that the standard PW-OLSLR method was sensitive for detecting VF progression with a sensitivity value equal to 97.1%; however, the specificity of the method was very low (25.4%). The “Three-Omitting” method, on the other hand, achieved a sensitivity of 65.7% with a specificity of 87.4%. Thus, the proposed method affords a significant improvement in specificity, but nevertheless is accompanied with a notable loss in sensitivity. It should be pointed out, however, that the “Three-Omitting” approach is not a method to improve the regression model itself and, furthermore, future VFs are always needed to diagnose progression; this is in contrast to the robust regression method used in the current study.

^{29}investigated the usefulness of the initial five VFs for predicting future progression, using 11 VFs captured over 8.2 years on average. They showed that linear extrapolation of the five initial VF results offered a reliable predictor of future field loss in most patients. Until now, however, it has not been investigated how many VFs are needed to saturate the prediction accuracy. In the current study, longer series of VFs were investigated and it was shown that a considerable number (approximately 10 VFs taken over 5.1 ± 1.6 years) are required for this purpose.

^{35}and so the merit of reducing the effect of outliers in the M-estimator robust regression is greater. On the other hand, reducing the effect of outliers could be argued to affect the ability of robust regression to identify fast-progressing eyes. The current results, however, suggest that the prediction accuracy of M-estimator robust regression tended to be significantly better than those with OLSLR in such eyes (Figs. 4, 5). When predicting mean VF sensitivity, no significant differences were observed in prediction errors among the OLSLR, exponential, quadratic, and robust regression models. This is probably because outliers are much less frequent in mean sensitivity measurement compared with PW sensitivity. Indeed, there was no significant difference between the prediction errors associated with M-estimator robust regression and OLSLR when predicting mean VF sensitivity (Fig. 6).

^{36}This may have contributed to improve prediction accuracy in the robust regression method. Nonetheless, the improvement of the prediction accuracy is relatively small and it might have only marginal benefit at the clinical settings. There are recent articles that succeeded to improve the prediction accuracy by implementing clinical knowledge into the prediction model, such as clustering eyes with the progression patterns,

^{37–40}setting a penalty term in the regression model,

^{41}and dividing VF into small sectors.

^{17}A future study should be carried out to further improve the diagnostic accuracy by combining the merits of these approaches.

**Y. Taketani**, None;

**H. Murata**, None;

**Y. Fujino**, None;

**C. Mayama**, None;

**R. Asaoka**, None

*Br J Ophthalmol*. 2006; 90: 262–267.

*Shields' Textbook of Glaucoma*. 6th ed. Philadelphia PA: Lippincott Williams & Wilkins; 2010.

*Jpn J Ophthalmol*. 2011; 55: 600–604.

*Jpn J Ophthalmol*. 2014; 58: 212–217.

*J Ocul Pharmacol Ther*. 2001; 17: 235–248.

*Acta Ophthalmol*. 2013; 91: 619–624.

*Br J Ophthalmol*. 1996; 80: 40–48.

*Arch Ophthalmol*. 1984; 102: 876–879.

*Arch Ophthalmol*. 1984; 102: 704–706.

*Invest Ophthalmol Vis Sci*. 2000; 41: 2201–2204.

*Invest Ophthalmol Vis Sci*. 1996; 37: 444–450.

*Br J Ophthalmol*. 2010; 94: 1404–1405.

*Invest Ophthalmol Vis Sci*. 2004; 45: 4346–4351.

*Invest Ophthalmol Vis Sci*. 2002; 43: 1400–1407.

*Acta Ophthalmol Suppl*. 1985; 173: 19–21.

*Invest Ophthalmol Vis Sci*. 2000; 41: 2192–2200.

*Invest Ophthalmol Vis Sci*. 2014; 55: 7681–7685.

*Introduction to Robust Estimation and Hypothesis Testing*. Amsterdam: Elsevier/Academic Press; 2012.

*Arch Ophthalmol*. 1996; 114: 19–22.

*Acta Ophthalmol (Copenh)*. 1989; 67: 537–545.

*Ophthalmology*. 2001; 108: 247–253.

*Nihon Ganka Gakkai Zasshi*. 2011; 115: 213–236; discussion 237.

*Ophthalmology*. 2008; 115: 2049–2057.

*Ann Math Statist*. 1964; 35: 73–101.

*Scand J Statist*. 1979; 6: 65–70.

*Am J Public Health*. 1996; 86: 726–728.

*Graefes Arch Clin Exp Ophthalmol*. 1995; 233: 750–755.

*Invest Ophthalmol Vis Sci*. 2011; 52: 4765–4773.

*Arch Ophthalmol*. 2009; 127: 1610–1615.

*Invest Ophthalmol Vis Sci*. 2011; 52: 9539–9540.

*Proc Inst Stat Math*. 1999; 47: 3–27.

*Invest Ophthalmol Vis Sci*. 2013; 54: 6694–6700.

*Invest Ophthalmol Vis Sci*. 2014; 55: 7881–7887.

*BMJ Open*. 2013; 3:e002067.

*Invest Ophthalmol Vis Sci*. 2002; 43: 2654–2659.

*PLoS One*. 2014; 9: e85654.

*Invest Ophthalmol Vis Sci*. 2014; 55: 8386–8392.

*Proc IEEE Int Conf Data Min*. 2013; 1121–1126.

*Invest Ophthalmol Vis Sci*. 2004; 45: 2596–2605.

*Invest Ophthalmol Vis Sci*. 2012; 53: 6557–6567.

*Invest Ophthalmol Vis Sci*. 2015; 56: 2334–2339.