**Purpose.**:
To evaluate and compare the ability of pointwise linear, exponential, and logistic functions, and combinations of functions, to model the longitudinal behavior of visual field (VF) series and predict future VF loss in patients with glaucoma.

**Methods.**:
Visual field series from 782 eyes (572 patients) with open-angle glaucoma had greater than 6 years of follow-up and 12 VFs performed. Threshold sensitivities from the first 5 years at each location were regressed with linear, exponential, and logistic functions to estimate model parameters. A multiple-model approach applied the model with the lowest root mean square error (RMSE) at each location as the preferred model for future predictions. Predictions for each model were compared at 1, 2, 3, and 5 years after the last VF used to determine model parameters.

**Results.**:
There were no clinically important differences between any of the models tested for fit; however, the logistic function had the lowest average RMSE (*P* < 0.001). For predictions, the exponential model consistently had the lowest average prediction RMSE for all time intervals (*P* < 0.001); the multiple-model approach did not perform better than the exponential model (*P* < 0.001).

**Conclusions.**:
While the logistic model best fit glaucomatous VF behavior over a long time period, the exponential model provided the best average predictions. A multiple-model approach for VF predictions was associated with a greater prediction error than with the best-performing single-model approach. A model's goodness of fit is not indicative of its predictive ability for measurements of glaucomatous VFs.

^{1}Event-based techniques as well as rule-based criteria from major clinical trials have been used to measure VF deterioration.

^{2–5}However, these approaches do not provide robust information about perimetric rates of change. Knowledge of the rate at which glaucomatous damage is worsening allows for appropriate and timely intervention in patients with faster rates of change to minimize visual loss and to optimally use treatment resources.

^{6,7}Pointwise regression of individual VF test locations has the advantage of providing spatial information about localized rates of change and increases the detection sensitivity of VF worsening compared to the use of global indices.

^{8}

^{9–12}The linear model assumes a constant additive rate of VF deterioration, while the exponential model assumes a constant multiplicative rate of change, that is, the change at any point in time is a constant percentage of the threshold sensitivity at that point. These assumptions may not necessarily apply to all locations of the VF, especially those that are initially normal.

^{9}For example, VF locations may have normal threshold sensitivities for a period of time before beginning to deteriorate. The VF plotted over time for these locations would have a plateau appearance before the decline in sensitivity measurements. For these locations, neither the linear nor the exponential regression models would be able to account for the transition between “normal” measurements and deterioration. The logistic function is a nonlinear model with an inverse sigmoidal shape, which may better depict the course of glaucomatous damage from normal vision to perimetric blindness as measured with clinical perimetry because it can model asymptotes at the beginning and end of data series. Just as not all locations of the VF will progress at the same rate, all VF locations will not necessarily follow the same pattern of deterioration. We hypothesized that a combination of different models may better account for the perimetric behavior at various test locations across the VF.

^{10}Figures 1A through 1C show examples of each model. The models are mathematically defined as follows:

**Figure 1**

**Figure 1**

- Linear:
Display Formula (*1*) - Exponential:
Display Formula (*2*) - Logistic:
Display Formula (*3*)

*y*represents the threshold sensitivities in decibels;

*x*represents time;

*α*,

*β*, and

*ζ*are model parameters to be estimated. The linear and exponential models were estimated with the least squares method, while the logistic model was estimated with the Newton-Raphson method.

^{13}If the regression for the linear model resulted in a line that dropped below zero, the negative values were censored at zero, given the arbitrary “floor” of 0 dB due to the dynamic range of the testing device. Both the exponential and the logistic models do not create negative values so no such censoring was required for those two models.

*t*-test with a Holm-Bonferroni correction. The calculated initial sensitivity was also assessed for at each location by using estimated model parameters and then compared to the observed initial sensitivity.

**Figure 2**

**Figure 2**

*P*< 0.001; Table 1). The logistic model had the best fit at 50.1% of locations, the exponential model in 41.4%, and the linear model in 8.5%. Figure 3 shows the RMSE for each model stratified by the initial sensitivity. There were no clinically important differences in the goodness of fits as measured by the RMSE between any of the models; however, the logistic model consistently performed the best when stratified by the initial sensitivity. Calculated initial sensitivities differed from observed initial sensitivity by an average of 2.21 (±2.42), 2.16 (±2.37), and 2.08 (±2.34) for the linear, exponential, and logistic function, respectively.

**Figure 3**

**Figure 3**

**Table 1**

**Table 1**

Model | Average RMSE | Count |

Linear | 3.057 | 3,420 |

Exponential | 3.056 | 16,707 |

Logistic | 2.925 | 20,214 |

*P*< 0.001).

*P*< 0.001 for all comparisons; Fig. 4). The number of times each model was chosen for the multiple-model approach is depicted in Figure 5. The multiple-model approach did not perform better than the best performing single-model approach, the exponential model (

*P*< 0.001 for all time points; Table 2). Figure 6 shows the 3-year prediction average RMSEs stratified by the initial sensitivity for each model. As the time interval between predictions and the last VF measurement increased, prediction RMSEs increased for all models. In the secondary analysis, which uses 7 years' worth of data to predict VF sensitivities 3 years later, the exponential model still resulted in the lowest prediction errors, although by a smaller margin (results not shown).

**Figure 4**

**Figure 4**

**Figure 5**

**Figure 5**

**Figure 6**

**Figure 6**

**Table 2**

**Table 2**

Years Predicted, y | Linear | Exponential | Logistic | Multiple Model |

1 | 4.766 | 4.521 | 5.146 | 5.150 |

2 | 5.200 | 4.891 | 5.518 | 5.526 |

3 | 5.969 | 5.497 | 6.148 | 6.131 |

5 | 7.561 | 6.720 | 7.406 | 7.328 |

8 | 8.840 | 7.395 | 8.391 | 8.251 |

*P*< 0.001 for all comparisons).

^{9–12,14–17}The linear model, which uses an ordinary least squares approach to minimize error, is an easy model to use, but applying it to VF data violates certain assumptions as described by Pathak et al.,

^{9}who also have showed that a nonlinear model better fits VF series.

^{18}It is for this reason that the predictions from the multiple-model approach likely do not perform as well as those of single-model approaches.

^{14}have explored the usefulness of a Tobit model (censored linear model) to better fit and predict VF data, given the floor effect imposed by the limitations of the measurements. They found that the Tobit model performed worse than an uncensored model. In our article, we used a linear model instead of a Tobit model, but imposed a floor for predictions below zero. Censoring the linear model in this fashion vastly improved the linear model's predictive ability. One speculation to explain the difference in the performance of these two models is that as a test location's sensitivity approaches perimetric blindness, the apparent rate of decline slows because of the increased noise at lower sensitivities as well as the arbitrary floor imposed by measurement. This in turn “slows” the rate of decline as measured by the machine, which allows for the censored linear model to outperform the Tobit model. Alternatively, the rate of decline may actually physiologically slow down as glaucomatous damage becomes severe. Whether or not the apparent slowing of VF decay toward perimetric blindness is due to measurement artifact or a physiologic phenomenon, an exponential or logistic function would better fit the data because of its asymptotic trend toward zero.

^{14}lies in the treatment of locations that were modeled as improving. The latter limited predicted sensitivities to a maximum of 40 dB. In our data series 42% of locations were found to have a positive slope or were modeled as improving. For these locations, the predictions were set at the average of the first four measured sensitivities, which was a compromise between allowing for improvement and filtering out noise (false positives). If the perimetric sensitivity in some eyes actually improves after treatment, a component of glaucomatous damage still persists, and the VF may likely continue to decay but from a higher sensitivity. In these cases, it is not unrealistic to assume that the VF progression would be best represented by a function with an increasing segment followed by a decreasing segment. These cases are a limitation of the monotonic functions used in this study and warrant further investigation.

^{14}who have found that the uncensored linear model predictions outperformed those of the exponential model. In their study, a higher percentage of locations had an initial sensitivity of zero dB, and different treatment of these points for regression analysis could affect the performance of the models tested.

^{10,14,18}None of the models used in this study address the issue of patients receiving treatment during the course of the VF series. Multimodal functions may depict the VF progression in patients undergoing treatments or procedures that may stop damage to, or even improve, visual function. Also, a low signal-to-noise ratio associated with standard automated perimetry, due to both human and equipment factors, often makes test results difficult to interpret, and certainly negatively affects the regression of VFs. Because all eyes in this study had a diagnosis of glaucoma, the results may be biased against a model that assumes an initial period of decline and make the logistic model appear to not perform as well as the exponential model. Lastly, calculations were performed on a logarithmic scale of VF data, since that is the format used clinically. The RMSE differences will be larger when compared to studies that model VF data on a linear scale.

**A. Chen**, None;

**K. Nouri-Mahdavi**, None;

**F.J. Otarola**, None;

**F. Yu**, None;

**A.A. Afifi**, None;

**J. Caprioli**, None

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