**Purpose**:
To develop a method to measure the rate of glaucomatous visual field (VF) deterioration and to identify fast progressors.

**Methods**:
Retrospective, longitudinal, observational study of 8486 eyes of 4610 glaucomatous patients with ≥6 VFs and ≥3 years of follow-up. A Glaucoma Rate Index (GRI) was calculated. VF locations were partitioned into exponential decay or exponential improvement models. A pointwise rate of change (PRC) was estimated with an exponential fit and expressed as the percent/year change of the age- and location-matched normal perimetric range, presented as a spatially conserved VF map. PRCs were summed and normalized with boundary rates set by simulated decaying and improving VF series on a scale of −100 to +100, respectively.

**Results**:
A total of 89,704 VF examinations with 425,039 test location series was used. Median follow-up and number of VFs/eye were 9.7 years and 9 VFs, respectively. Initial and final mean deviations (±SD) were −4.2 (±5.2) and −5.7 (±6.4) dB. The proportions of test locations designated as decayed, improved, and unchanged were 13%, 4%, and 83%, respectively. Mean PRCs for decay, improvement, and no change were −3.7 (±4.7)%/y, 2.5 (±2.6)%/y, and −0.5 (±2.1)%/y, respectively. The number of eyes with negative and positive GRIs was 5802 (68%) and 2390 eyes (28%), respectively. The proportion of eyes defined as fast progressors was 6.8%.

**Conclusions**:
GRI provides a robust measure of glaucomatous VF change, operates without discontinuity over the entire perimetric range, and can be used to identify fast progressors.

**Translational Relevance**:
This study describes a novel method that can help the clinician to determine VF progression.

^{1}

^{2}Critics argue that the linear core of the regression may not be best suited for biological systems; linear models assume a constant additive rate of VF deterioration, but as the VF deteriorates, the relative changes that can occur become more clinically significant and are restricted by a measurement floor (0 decibels [dB]).

^{3}An alternative trend-based approach uses an exponential regression to better model glaucomatous VF measurements.

^{4–6}Although global trend analysis can be used to determine the overall rate of change in VF series, it does not model regional deterioration, which is common in early to moderate glaucoma.

^{3,7}Existing indices are designed to measure abnormalities, not rates of change. It has been repeatedly shown that the global index mean deviation (MD) is nonspecific for glaucoma and insensitive to localized VF changes, and pattern standard deviation (PSD) decreases as damage advances in moderate to advanced disease.

^{8}Visual field index (VFI) has its own problems with discontinuity at more advanced stages of loss.

^{9}VFI can also miss early diffuse VF damage due to a ceiling effect.

^{9}Event-based analyses are binary algorithms that depend on defined criteria for progression and alert the user when these criteria are met. Guided progression analysis (GPA) of the Humphrey Field Analyzer (Carl Zeiss Meditec, Inc., Dublin, CA) is the most widely used event-based analysis; this proprietary tool identifies significant pointwise progression based on statistical probabilities estimated from at least three sequential VFs. Although it has been shown to be slightly more sensitive than PLR or the use of Advanced Glaucoma Intervention Study (AGIS) scores, it lacks information on rates and therefore is not easily used to predict future change.

^{10}Regardless of which model is used, the algorithm should take into account the appropriate age- and location-matched normal values and aging rates of change.

^{11,12}

^{1}Glaucomatous VF deterioration is neither linear nor constant over time, and psychophysical measurements are notoriously noisy. These inherent properties can limit the detection of VF progression. Different regions of the same VF can show fast worsening, slow worsening, no change, or even improvement.

^{5}One can certainly measure the rate of change of these indices, but the usefulness of these estimated rates are hindered by their demonstrated limitations.

^{13}

- An a priori screening was used to categorize each pointwise series as either decaying or improving, according to whether its linear trend was negative or positive, respectively.
- Pointwise exponential regression (PER) of each series was performed. For VF locations with a negative trend, the decay PER model was applied, expressed as
*y*=*e*^{(}^{a}^{+}^{bx}^{)}, where*y*= sensitivity in decibels,*a*= constant,*b*= regression coefficient (slope), and*x*= time in years. This model had an asymptote as a floor (0 dB). The estimates of*a*and*b*were obtained by regressing ln(*y*) on*x*. For VF points with a positive trend, we used a mirror image approach, where the exponential function had an asymptote as a ceiling (the normal age-matched value for that particular test location plus twice its standard deviation). The function of the improvement model is expressed as*Y*−*y*=*e*^{(}^{a}^{+}^{bx}^{)}, where*y*= sensitivity (decibels) at time*x*(years),*Y*= the normal age-matched threshold sensitivity + 2 SD,*a*= constant, and*b*= regression coefficient (slope). - Outliers were removed on the basis of their Cook's distance
^{14}and the Studentized residual test^{15}(Fig. 1). All threshold values in a pointwise series for a test location were sequentially removed with Cook's distance >1 and a |Studentized residual| >3. The former method removes influential points at the extremities able to leverage the regression slope, while the latter method removes points with high root mean squared error away from the ends of the series, which can affect the y-intercept and the width of the confidence interval (CI). According to Weisberg,^{16}these cutoffs are recommended to balance removing too many versus too few influential points. - The pointwise rate of change (PRC) of each series, expressed as a percentage of the entire normal perimetric range, corrected for age and location, was calculated.
^{11,12}PRC was calculated as the change between the initial (*y*(0)) and final (*y*(*t*)) values of the PER model fit divided by the age- and location-matched dynamic range of threshold sensitivity (in dB) for each test location in the VF. Thus, the rates of change are expressed as the proportion (%) change per year of the entire perimetric range, corrected for location and age. Locations that met the criteria for change and had a positive PRC were marked as improving. Locations that met the criteria for change and had a negative PRC were marked as decaying. Decaying PRC locations were further divided into slow and fast decay based on the distribution of PRC values: the test locations in the fastest (most negative) quartile of decaying locations were categorized as “fast decay.” The PRCs are displayed graphically to show the spatial relationships of the pointwise rates of change (Fig. 2) and are used to calculate the GRI for each eye. - The 90% CI of the exponential regression line was used to determine whether a VF location was categorized as no change, decay, or improvement. If the rate defined by the slope of the line joining the points from initial bottom to final top of the CI bands (Fig. 3) is negative and the rate exceeded the 95th percentile of the normal aging rate, then the rate was counted as significant and was used to define the decay rate at that location. In the case of an improving PER, if the rate defined by the line joining the points from initial top of the CI band to final bottom of the CI band was positive, then the rate was counted as significant and was used to define the improving rate at that location (Fig. 3). This approach required the points to be within a sufficiently tight fit and to have a sufficiently negative or positive trend in order to be considered either decaying or improving. Test locations that did not meet either of these criteria were designated as “no change.” These are presented as a spatially conserved VF map, which indicates the status of each test location, whether worsening, improving, or without change, as shown in Figure 2.
- A GRI score for each eye was calculated by summing the PRCs for all test locations that met the above criteria for change. If none of the 54 test locations analyzed in a VF series had significant change, then this eye was assigned a GRI value of “0.” GRI scores were normalized from −100 to +100, where −100 represents an extreme rate of decay and +100 represents an extreme rate of improvement. To construct the extreme decay model, we started with location-matched normal eyes at 60 years of age. Over a span of 20 years, each starting decibel value was decreased at a constant (linear) rate so that the final decibel value would be 0. The PER model was then fit to this series, and the PRCs were summed. This value was used for the decay normalization and was set to −100. The inverse was performed for improvement, and the sum of the PRCs was normalized to +100 (Fig. 4). Eyes within the fastest decaying decile of GRI values were designated as fast progressors.

^{17}Briefly, we randomly selected

*n*VF exams, where

*n*is the total number of examinations for the eye with replacement (a VF may appear multiple times in the same sample). A new GRI was calculated based on the randomly selected

*n*exams and subtracted from the original GRI of the eye. This was performed 1000 times, and a 90% CI of GRI for each eye was calculated. Decaying eyes (GRI < 0) were grouped by GRI values to the nearest integer. Within each group, we calculated the percentage of eyes that had a 90% CI upper limit < 0. The GRI value of the group having 50% of eyes meeting this criterion was set as the decaying cutoff. To better estimate the GRI at the 50% cutoff, the locally weighted scatterplot smoothing (LOWESS) fit was applied to the scatterplot.

^{18}The same process also was employed for GRI > 0 to estimate a cutoff for improvement.

^{6}compared the behavior of linear, exponential, and logistic models in serial VF examinations and found that although logistic models fit glaucomatous VF behavior well over a long period of time, an exponential model provided the best overall predictions. Azarbod et al.

^{4}showed that exponential models are effective across a wide range of severities and can predict future change better than can linear models. Exponential approaches appear to be better than either linear or other nonlinear models in patients with moderate to severe glaucomatous damage because values typically approach 0 dB in an asymptotic fashion, a property that cannot be accounted for with linear techniques without discontinuities.

^{6,19,20}Although it is based on PERs, GRI uses a linear trend as a first step to categorize each location as improving or decaying. The nonlinear exponential regression model exists in two different, but symmetrical and specular forms: decaying and improving (Fig. 3). The a priori categorization with a linear trend is required to select the appropriate model for a given VF series.

^{21}Conversely, pointwise approaches can retain their perimetric spatial relationships and allow the measurement of the rate of progression at every test location. Katz and colleagues

^{22}compared regression of global indices, clusters of locations, and single locations to identify perimetric progression. In their study, regression of global indices failed to recognize localized damage with a rate of <1 dB/y and an annual frequency of VF testing. On the other hand, global indices are generally more specific than pointwise approaches, since they require a higher magnitude of change to detect progression.

^{22,23}The GRI includes only those test locations with significant change, as it is defined by the method. Other indices, including permutation analyses of pointwise linear regression (PoPLR) and GPA, also consider only locations significantly deteriorating. Global rates of VF progression, conversely, take into consideration all test locations, including those points that are stable throughout the VF series and points with absolute defects at baseline. Inclusion of these data, however, has the potential to obscure the signal of clinically important localized changes.

^{5}The method presented here allows the user to choose any number or any group of consecutive VFs for analysis, so long as there are at least six. For example, one may choose to analyze the VFs only after a major change in treatment and compare this with the GRI before the change if a sufficient number of VFs are available. A caveat is that the fewer number of VFs included in a series, the more the potential error in its estimation. This variability is reasonable with eight or more VF tests (1 SD is approximately 2%–10% of the range of GRI); thus for best results, eight or more VF tests should be used to improve the accuracy of the index. Although it can always be mathematically calculated, we recommend GRI calculation based on an adequate number of test locations. We suggest a minimum of 26 locations, which corresponds to half of the VF test locations. This requirement might not be satisfied in case of far-advanced glaucoma, where multiple locations can be entirely excluded in case of a threshold sensitivity value of 0 dB in two out of the first three tests. This was, however, an uncommon event in this very large database of all comers and occurred only in 0.8% of all eyes.

^{24}

^{25}In the graphical presentation of GRI, color codes are based on the PRC values. Specifically, we require serial sensitivity measurements for each test location to be within a sufficiently tight fit and to have a sufficiently negative or positive trend to be categorized as decaying or improving. Those test locations not meeting either of these criteria were designated as “no change.” Test locations decaying at a rate faster than 5%/y were categorized as fast decay. Thus, all test locations were categorized as either (1) unmeasurable, (2) fast decay, (3) slow decay, (4) no change, or (5) improving.

^{26}measured rates of glaucoma progression in a large clinical population under routine clinical care and found that 5.8% of patients were fast progressors. In another study by Baril et al.

^{27}the proportion of fast progressing eyes was 3.9% and 9.4% for eyes receiving medical and surgical glaucoma treatments, respectively. Prevalence of fast progressors, however, may vary across different populations since it is dependent on many factors, such as age, stage of disease, treatment, and subtype of glaucoma.

^{28}In addition, the way to define and measure fast progression remains arbitrary. Most of the previous studies defined fast progression as the rate of MD decline, usually ranging from −1 dB/y to −2 dB/y. As discussed earlier, however, MD rate is based on the entire VF, and it could underestimate profound but localized changes, delaying the recognition of fast progressors at risk. VFI rate is similar in this regard and may additionally suffer from a ceiling effect and discontinuity in severe stages.

^{29}Event-based approaches do not provide rates of progression. Methods based on PLR could be employed; however, they have not been investigated for this purpose, and it is unclear which criteria should be used to identify fast progressing eyes. Medeiros and colleagues

^{30}proposed a model to evaluate rates of progression with an empirical Bayes estimate of rates of change and clustered VF progression into four groups, based on similarities of their trajectories. The authors claimed that their method was better than simple linear regression, especially in identifying fast progressors. In this study, we propose a method for identifying fast progressors. Based on the results of previous studies regarding the prevalence of fast progressors, we defined the fastest decile of negative GRI values as those belonging to fast progressors.

^{26–28,30}The fastest decile of negative MD rate corresponded to a value of −0.97 dB/y, which resembles the cutoff of −1 dB/y adopted by other studies.

^{26,27}The GRI value for fast progressors is largely arbitrary and is presented as an example of how this identification might be accomplished. Where one draws the cutoff ultimately depends on many factors, including patient-related factors (other risk factors, longevity, patients' desires, etc.), public health concerns, and resource availability.

^{5,31–34}We have included the possibility of improvement in our method by applying an exponential fit to VF test locations with an a priori improving trend. An inverted exponential decay fit with a ceiling asymptote of the age-matched normal +2 SD value was applied, as opposed to applying a constant multiplicative (linear) rate of improvement to a longitudinal VF series, which seems to not be physiologically appropriate.

^{5}In addition, other trend-based and event-based methods (except for the current version of GPA) could be used to detect improvement.

^{35}Uncorrected refractive error, patient motivation and instruction, fatigue, technician experience, time of day, season, race, cognitive level, and percentage of false-positive responses have been recognized as additional sources of variability.

^{36–40}Large fluctuations are an obstacle to the proper modeling of VF data, and several strategies have been employed to reduce the noise.

^{41}A first measure relies on controlling known causes, such as repetition of examinations scored as poorly reliable and training and motivation of both patient and technical staff. Trend-based methods based on regression of global indices (i.e., MD, VFI) are less influenced by pointwise variability since they are averaged measures. Bryan et al.

^{41}proposed a model called global visit effects, which takes into account known and unknown factors affecting all the locations of the same examination. When applied to longitudinal data, correction of global visit effects improved the estimation of true rates of progression and increased the ability to predict future changes. Interestingly, the global visit effects model performed better than the simple correction for established factors of variability, suggesting a contribution by other, unknown variables.

*y*-intercept and the width of the CI. About 3% of all measurements were removed with this approach. A previous study showed a difference in results obtained before and after Cook's correction.

^{5}In the current study, exclusion of ≥26 points (more than half of VF test locations) from a single examination occurred in only 1.1% of all examinations, suggesting that outlier removal performed by GRI can omit a large portion or even an entire examination. This can mitigate the effect of the aforementioned factors, inducing variability at every test location at the same visit, and could represent an alternative to other proposed methods, such as proposed by the global visit model.

^{41}Our data indicate, however, that removal of a large part of an examination is an uncommon event, and the majority of visits (74.8%) had no points excluded at all.

^{42}This represents an assumption that is not unique to GRI (it is also found with GPA, algorithms based on PLR, and PoPLR).

^{2,43,44}Data derived from pointwise linear or exponential regressions can be postprocessed with spatial filters, and this can improve the prediction ability of both models.

^{7}Spatial correlation among test locations is taken into account by very few algorithms, such as ANSWERS, recently proposed by Zhu and colleagues.

^{45}Although such an index seems to perform better than MD regression and PoPLR, further testing is required for validation. GRI is based on PER, and therefore it assumes that VF decay follows an exponential model over time. As linear regression is vulnerable to both left and right censoring effects, exponential regression is vulnerable to left censoring effects since the backward extrapolation of a series could lead to unfeasibly high or low sensitivities in the case of decay or growth exponential models, respectively. In this regard, GRI assumes that there is a time at which a discontinuity in the series occurred and glaucomatous VF damage “began” for an eye. Backward extrapolation, however, is not clinically relevant, whereas forward extrapolation is useful for prediction. GRI treats locations with a threshold sensitivity of 0 dB in any two of the initial three examinations as locations of perimetric blindness, and progression or improvement cannot be established at such locations. This assumption is not unique to GRI since other approaches (e.g., GPA) omit locations with low sensitivities at baseline.

^{46}Computer-simulated VF sequences with predetermined rates and patterns of progression may help in this regard. GRI may miss early diffuse loss from glaucoma, since GRI includes only test locations that meet criteria for change. While early glaucoma may cause a mild generalized reduction of whole-field sensitivity, such changes are not common or specific for glaucoma. Application of the method requires extra time and resources to export data and apply external calculations and display. This could be remedied easily by incorporation of the algorithm into standard VF software. The cutoff for fast progressors was set arbitrarily. This could be modified, depending on the goals of recognizing and treating such patients in the particular population being treated and the treatment resources available to that population. The utility of GRI in clinical research regarding progression and treatment effects would need further study. Studies are underway to test this method in a separate database and to evaluate its predictive results compared to those of other established indices.

^{5}We believe that these properties are complementary and that their presence in a single model would be useful for clinicians and investigators. Finally, GRI has the ability to flag eyes as fast progressors and can be used to help guide the appropriate use of resources to improve the outcomes of patients at high risk for visual disability.

**J. Caprioli**, None;

**L. Mohamed**, None;

**E. Morales**, None;

**A. Rabiolo**, None;

**N. Sears**, None;

**H. Pradtana**, None;

**R. Alizadeh**, None;

**F. Yu**; None;

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

**A.L. Coleman**, None;

**K. Nouri-Mahdavi**, None

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