**Purpose**:
Accurate assessment of visual field (VF) trend may help clinicians devise the optimum treatment regimen. This study was conducted to investigate the behavior of VF sequences using pointwise and region-wise linear, exponential, and sigmoid regression models.

**Materials and Methods**:
In a retrospective cohort study, 277 eyes of 139 patients with glaucoma who had been followed for at least 7 years were investigated. Linear, exponential, and sigmoid regression models were fitted for each VF test location and Glaucoma Hemifield Test (GHT) region to model the trend of VF loss. The model with the lowest root mean square error (RMSE) was selected as the best fit.

**Results**:
The mean age (standard deviation [SD]) of the patients was 59.9 years (9.8) with a mean follow-up time of 9.3 (0.7) years. The exponential regression had the best fit based on pointwise and region-wise approaches in 39.3% and 38.1% of eyes, respectively. The results showed a better performance based on sigmoid regression in patients with initial VF sensitivity threshold greater than 22 dB (71.6% in pointwise and 62.2% in region-wise approaches). The overall RMSE of the region-wise regression model was lower than the overall RMSE of the pointwise model.

**Conclusions**:
In the current study, nonlinear regression models showed a better fit compared to the linear regression models in tracking VF loss behavior. Moreover, findings suggest region-wise analysis may provide a more appropriate approach for assessing VF deterioration.

**Translational Relevance**:
Findings may confirm a nonlinear progression of VF deterioration in patients with glaucoma.

^{1}is a progressive disorder characterized by changes in the optic nerve head accompanied by visual field (VF) loss. It typically affects subjects who are 40 to 80 years old and nearly 3.5% of this population. It is estimated that approximately 111.8 million people will develop glaucoma by 2040.

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^{5}Assessing a longitudinal series of VF enables clinicians to detect early progression of the disease to direct treatment resources optimally toward worsening cases.

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^{7}VF global measures, such as mean deviation (MD), fail to ascertain local changes.

^{8}As such, some researchers have suggested using pointwise or region-wise models to better capture the local characteristics of VF loss.

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^{16}Linear regression assumes a constant additive rate of deterioration, whereas this rate is multiplicative for exponential regression. The rate of change would be constant, fast, or slow in different parts of the same individual VF test during the time. As all VF locations do not have the same pattern, sigmoid regression may represent a better fit, particularly in cases that transit from normal to perimetric blindness.

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^{20}Dividing VF locations into regions and monitoring the progression by region-wise pattern analysis is a useful way to address this problem, which could improve prediction of deterioration.

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^{23}Therefore, we can analyze VF defects based on spatial dependency to evaluate the progression of the disease in each sector.

^{24}Demographic information, including sex, age, and baseline intraocular pressure (IOP) was collected. Individuals were tested every 6 months and at each visit, SAP was performed using Humphrey Visual Field Analyzers (Carl Zeiss Meditec) with a standard white-on-white 24-2 field with the full threshold program. Tests with greater than 33% fixation losses, 20% false-negative or false-positive error rates were excluded (a full description of data is available at http://rod-rep.com).

^{25}In this approach, the superior and inferior hemifields were divided into 10 glaucoma regions named

*Arcuate1*,

*Arcuate2*,

*Nasal*,

*Central*, and

*Paracentral*. Each sector consisted of three to five VF test locations; therefore, we calculated the mean of threshold sensitivity in each region to regress it against age in predefined models mentioned above.

*y*= α + β

*x*+ ε; y represents the threshold sensitivity (dB);

*x*indicates patients’ age (or time index equivalently), and

*α*and

*β*are the model parameters demonstrating the initial sensitivity estimation and its change by 1 year increase in age, respectively. The error term is defined by ε that represents the part not explained by the explanatory variable

*x*. Estimating the parameters was performed using the ordinary least-squares method; where the estimations of threshold sensitivity values were negative, they were censored to 0 dB.

*y*=

*e*

^{α + βx}+ ε; it treats as a logarithmic transformation of a simple linear regression model (ln

*y*= α + β

*x*+

*ln*ε). Exponential regression is used to model variables in which the increase starts slowly and then speeds up rapidly without bound, or where deterioration begins rapidly and then slows down to get closer and closer to zero (the latter is considered here). The model includes e

^{β}that represents the rate of change in sensitivity per year with increasing in age and the error term is denoted by ε. For estimating the parameters of this nonlinear model, the Newton-Raphson method was used.

*y*that is not explained with

*x*. Parameter estimation was carried out using the Newton-Raphson method.

*p*is the number of terms in the model. The best fit for both locations and sectors was determined according to the lowest RMSE amount. The Pairwise

*t*-test with Bonferroni correction was also used to compare RMSE of three models based on pointwise and region-wise approaches.

^{17}All the statistical analyses were done using Stata software version 14.2 and the level of significance was 0.05.

*P*< 0.001), however, this result might occur due to the large sample size.

*P*= 0.291), but there were meaningful differences between sigmoid regressions and the other two models (

*P*< 0.001), however, the effect of sample size on significancy should be considered.

^{26}Therefore, identifying VF worsening through investigation of longitudinal VF tests is critical to prevent or slow irreversible vision loss.

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^{28}Pathak et al. designed a study to compare the linear and nonlinear mixed effect models in tracking the rate of change in pointwise VF sensitivity over time. They concluded that the nonlinear method was preferable and provided a better fit than ordinary least squares linear model.

^{16}Similar to their findings, we also observed that nonlinear models provide a better fit. In addition, they denoted the advantage of the exponential model in a longitudinal series of MD over time as well.

^{28}Caprioli et al. examined the rate of VF progression for each VF test location through three models of linear, exponential, and quadratic that resulted in a far better fit for the exponential model.

^{27}Our findings are in agreement with Caprioli et al. in the superiority of the exponential regression for modeling VF worsening over time.

^{5}Our results are also in agreement with those reported by Otarola et al. in which the sigmoid regression outperformed in patients with the initial sensitivities in the normal range. However, their result was overall in favor of sigmoid regression.

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^{29}Due to the spatial dependency between some VF test locations, different techniques of clustering, typically consistent with retinal nerve fiber layer bundle patterns, have been proposed to evaluate the functional changes.

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^{31}We conducted this study based on predefined regions by GHT.

^{25}Accordingly, the mean values of RMSE were smaller for region-wise models compared with pointwise ones (see Tables 2, 3). Previous studies, such as the one carried out by Hirasawa et al., demonstrated a lower absolute prediction error of mean TD values in region-wise analysis compared with pointwise analysis.

^{20}The other studies also confirmed the appropriateness of region-wise analysis in terms of reduction in prediction error.

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^{34}In this study, we used a different approach demonstrating a high percentage of nonlinear decay in VFs of glaucomatous eyes with a great caution for clinicians in the case of facing normal sensitivities that may progress later in a sigmoid shape. Although, biologic underpinnings of glaucoma disease may not be represented by models accurately, it may provide valuable insights for clinicians regarding VF changes through a simplified form of progression. This would be attained by identifying the model that fits the data more truly. Our study showed the superiority of the exponential regression over the other two models and in conditions with more aggressive form of decay, sigmoid regression had a better performance. Although the pointwise approach enabled us to evaluate local changes, such models suffer from a high degree of variability.

^{35}Hereon, region-wise analysis may partially address the VF variability.

**S. Sabouri**, None;

**S. Pourahmad**, None;

**K.A. Vermeer**, Acoustic Insight and Novo Research Consultancy, Voorburg;

**H.G. Lemij**, None;

**S. Yousefi**, None

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