**Purpose**:
To validate Gaussian mixture-model with expectation maximization (GEM) and variational Bayesian independent component analysis mixture-models (VIM) for detecting glaucomatous progression along visual field (VF) defect patterns (GEM–progression of patterns (POP) and VIM-POP). To compare GEM-POP and VIM-POP with other methods.

**Methods**:
GEM and VIM models separated cross-sectional abnormal VFs from 859 eyes and normal VFs from 1117 eyes into abnormal and normal clusters. Clusters were decomposed into independent axes. The confidence limit (CL) of stability was established for each axis with a set of 84 stable eyes. Sensitivity for detecting progression was assessed in a sample of 83 eyes with known progressive glaucomatous optic neuropathy (PGON). Eyes were classified as progressed if any defect pattern progressed beyond the CL of stability. Performance of GEM-POP and VIM-POP was compared to point-wise linear regression (PLR), permutation analysis of PLR (PoPLR), and linear regression (LR) of mean deviation (MD), and visual field index (VFI).

**Results**:
Sensitivity and specificity for detecting glaucomatous VFs were 89.9% and 93.8%, respectively, for GEM and 93.0% and 97.0%, respectively, for VIM. Receiver operating characteristic (ROC) curve areas for classifying progressed eyes were 0.82 for VIM-POP, 0.86 for GEM-POP, 0.81 for PoPLR, 0.69 for LR of MD, and 0.76 for LR of VFI.

**Conclusions**:
GEM-POP was significantly more sensitive to PGON than PoPLR and linear regression of MD and VFI in our sample, while providing localized progression information.

**Translational Relevance**:
Detection of glaucomatous progression can be improved by assessing longitudinal changes in localized patterns of glaucomatous defect identified by unsupervised machine learning.

^{1–4}Detection of glaucomatous visual function defects and detection of their progression are critical for management of the disease. Identifying patterns of visual function defects and tracking their change over time likely is a promising approach for clinical management of glaucoma.

^{5}

^{6}Because MD and VFI are global indices, these methods may not be ideal for progression detection, in that they include visual field locations that have little impact on the VF progression.

*IOVS*1990;31;ARVO Abstract 503), studies have used supervised machine-learning classifiers successfully to separate healthy from glaucomatous eyes based on VF and optical imaging measurements and to predict conversion to glaucoma in glaucoma suspect eyes.

^{7–23}More recently, we have effectively employed unsupervised machine-learning techniques to discern how VF data are organized into patterns. We found it useful to represent the structure of VFs by clusters of healthy eyes, early glaucoma eyes, and advanced glaucoma eyes, and to represent the structure within each cluster by axes obtained by independent component analysis. The estimation of the best structure representation was accomplished with post hoc assessment of the MD of the clusters, and visual inspection of the patterns of defect within the observed clusters.

^{24–28}We aimed to diminish the effects of human bias by designing a process for detecting change over time along mathematically determined glaucomatous patterns obtained by unsupervised learning techniques without human intervention, and we aimed to improve effectiveness by eliminating noncontributing data and concentrating on the data that are changing.

^{29–31}

^{24,27,32}The Gaussian mixture-model with expectation maximization (GEM) produces a similar output, but learns 50 times faster and is a fully automated unsupervised learning approach.

^{28,31}The application of progression of patterns (POP) to VIM and GEM make the progression detectors VIM-POP and GEM-POP.

^{29–31}

^{33}that evaluates change at each individual test location over the entire follow-up duration based on a fixed number of changing test locations; permutation analysis of PLR (PoPLR),

^{34}an individualized analysis that uses a

*P*value combination function and permutation analysis to detect glaucomatous change; combined binomial tests with PLR,

^{35}and methods based on variational Bayesian analysis.

^{36}

^{28,31}We then compare the clustering performance of GEM with VIM, based on specificity and sensitivity for clustering VFs as healthy and glaucomatous. Next, we detect glaucomatous progression in study eyes based on significant change of longitudinal VF measurements (exams) along the previously generated GEM and VIM defect patterns, using POP. Finally, we compare the accuracy of GEM-POP and VIM-POP, to PLR,

^{33}PoPLR,

^{34}and linear regression of MD and VFI, with detect progression in VFs from known progressing eyes.

^{37}The institutional review boards of UC San Diego, NYEEI, and UAB approved all DIGS and ADAGES methods. All methods adhered to the tenets of the Declaration of Helsinki and to the Health Insurance Portability and Accountability Act. DIGS and ADAGES are registered as cohort clinical trials with www.clinicaltrial.gov (NCT00221897 and NCT00221923, respectively; September 14, 2005).

^{38}In this study, participants with normal VFs were defined as those with no evidence of repeatable abnormal VFs (as defined above) in each eye.

**Table 1**

**Table 2**

^{39}Independence of axes was forced within each cluster, not between different clusters. The generated visual field at ±2 SD from the cluster mean on each axis, and the VFs associated with each cluster, characterized the patterns of visual defect. To avoid working with a large number of axes, only axes with significant contributions (described later) were retained in each cluster. VIM-POP was equipped with a sliding window because it is expected that glaucoma-associated change in visual function and structural imaging is nonlinear in some eyes. For example, a 5-visit spurt of progression in a trend of 10 visits might be missed by a 10-visit linear regression while regression in a sliding 5-visit window along the 10-visit range would be more likely to capture the progression spurt.

^{29,32}Briefly, VIM is a combination of multiple ICA models weighted in a probabilistic manner. This combination allows the unsupervised identification of independent clusters of data, each containing statistically independent axes of information. Clustering and axis development are done simultaneously in VIM. VIM is a semiautomatic clustering method because the user selects the model with the highest average of sensitivity and specificity among a very large number of VIM models and the optimal model is retrained to further improve its diagnostic accuracy. In the current study, the VIM training feature set had 53 features; the absolute sensitivity values from 52 of the 54 VF test points (2 blind spot points excluded) and participant age, for each of the 1976 SAP tests. VIM varied the maximum number of clusters, the maximum number of axes within each cluster, and the number of Gaussians to create 720 models. Each model was iterated 500 times, employing a different number of axes and several random seeds at initialization. It was assumed that the single best model (highest average of sensitivity and specificity for identifying abnormal and normal VFs) would provide the best environment for finding glaucomatous patterns and for detecting progression. The optimal number of axes within each cluster was chosen based on the contribution of each axis in cluster decomposition. This number was called a “knee point” and was chosen by ranking the axes in each cluster based on their lengths or magnitudes and including the number of axes with the largest magnitudes and excluding axes with smaller magnitudes.

^{32}On a graph of axis contribution versus number of axes, the chosen number of axes occurred at the point where the slope of the graph changed from steep to nearly horizontal; hence, the term knee point. The optimal model was retrained 500 times to determine the final best specificity and sensitivity.

^{28}Briefly, GEM combines multivariate Gaussian components to model the VF data points and uses the expectation maximization (EM) procedure to estimate the parameters of the model, iteratively. Similar to VIM, we used absolute sensitivity at 52 SAP locations and participant age as inputs to GEM. Clusters were created by selecting the component that maximized the maximum a posteriori probability, based on the EM-estimated parameters. GEM is a probabilistic approach with a hierarchical modular framework that allows identification of clusters first, followed by identification of axes within each cluster using ICA. Unlike VIM, where clustering and axis development are done simultaneously, GEM is sequential. To select an optimal GEM model that represents glaucoma categories and VF defect patterns, we generated 600 GEM models (to be roughly comparable to VIM for assessing the computational complexity, however we have observed that 50 models generally are sufficient for GEM), and selected the model that provided the highest average of sensitivity and specificity for discriminating between abnormal and normal VFs. The optimal number of axes within each cluster was chosen based on the previously described knee point.

^{28,31}

^{33}To detect progression for PLR, for each of 52 VF points, two different parameters; the slope and the significance of the slope (

*P*value) were considered. Progression of the VF was defined based on one to three deteriorating points with significant

*P*value (smaller than 0.01) of the slope exceeding the specified threshold, allowing six different criteria for VF progression (two different slopes and three possible locations). We applied these six criteria to our stability definition group to compute specificity and to our progression study group to compute sensitivity to identify sensitivity/specificity trade-offs at several discrete points along the receiver operating characteristic (ROC) curve similar to O'Leary et al.

^{34}We also implemented PoPLR according to method described by O'Leary et al.

^{34}We combined the significance values across the VF and compared it with the null distribution of

*P*values of all combinations of the VF sequences obtained by permutation of visits. We considered progression if the combined significance value of actual visits was greater than 95th percentile of the null distribution of all permuted visits and computed the full ROC curve.

_{1}(representing early to moderate glaucoma, based on post hoc assessment of MD), and G

_{2}(moderate to advanced glaucoma, based on post hoc assessment of MD). The best VIM model was composed of nine axes: two axes for each of the first two clusters (N and G

_{1}) and five axes for the third cluster (G

_{2}). Similarly, the best GEM model was composed of nine axes: two axes for each of the first two clusters and five axes for the third cluster.

_{1}and G

_{2}combined contained primarily VFs from participants with glaucoma (abnormal VFs, 799 of 859, sensitivity = 93.0%). Cluster G

_{1}contained 503 eyes with abnormal VFs and 32 eyes with normal VFs. Cluster G

_{2}contained 296 eyes with abnormal VFs and one eye with a normal VF.

_{1}, and G

_{2}combined contained primarily VFs from participants with glaucoma (772 out of 859, sensitivity = 89.9%). Cluster G

_{1}contained 478 eyes with abnormal VFs and 69 eyes with normal VFs. Cluster G

_{2}contained 294 eyes with abnormal VFs and no eyes with normal VFs. Figure 1 shows scatter plots of the mean threshold values in the superior hemifield plot versus the inferior hemifield plot, to demonstrate cluster variability and Figure 2 shows MD versus PSD of all eyes assigned to each of the VIM and GEM clusters, to demonstrate clustering by VF defect severity.

**Figure 1**

**Figure 1**

**Figure 2**

**Figure 2**

_{2}was farther from the mean of cluster N than the mean of cluster G

_{1}, indicating that the overall defect was more severe in G

_{2}. Within each cluster, axes represented the pattern (shape) of each VF defect. VIM efficiently identified the magnitude of the various patterns of defect present in each study cluster by projecting each VF along each axis. If a point moved along any axis away from the cluster mean, the direction of motion would be positive if the distance (vector) of the point from the normal mean increased with movement, and the direction would be negative if the vector from the normal mean to the point decreased with movement.

_{1}, and G

_{2}. Similar to the total deviation plot in SAP, these patterns represent the degree of defect severity and deviation from normal. The two defect patterns of cluster N shown are normal, without glaucomatous damage. Cluster G

_{1}includes eyes with early to moderate glaucomatous damage (average MD = −2.15, SD = 1.52 dB); consequently, the patterns shown illustrate primarily mild defects. Axis 1 of G

_{1}(at +2 SD) displays early superior and inferior nasal steps and arcuate reductions of sensitivity. Axis 2 of G

_{1}(at +2 SD) suggests an early superior arcuate defect. Cluster G

_{2}includes eyes with more moderate to advanced glaucomatous damage (average MD = −7.98, SD = 6.27 dB); hence, the axis patterns shown in Figure 3c are more advanced. Axis 1 of G

_{2}(at +2 SD) represents a diffuse depression with increased depression in the superior hemifield. Axis 2 of G

_{2}(at +2 SD) shows early diffuse reduction of sensitivity with an inferior hemifield defect. Axis 3 of G

_{2}(at +2 SD) represents a peripheral defect with increased defect at the nasal steps and the superior arcuate zone. Axis 4 of G

_{2}(at +2 SD) presents a nasal step defect with inferior nasal exaggeration and axis 5 of G

_{2}(at +2 SD) suggests a central-to-nasal defect, with inferior weighting.

**Figure 3**

**Figure 3**

_{1}, and G

_{2}were displayed as TD plots. These generated TD plots at points on the axes within each cluster represented distinct VF defect patterns.

_{1}, and G

_{2}. The defect patterns of cluster N are normal, without glaucomatous damage. Cluster G

_{1}includes eyes with early to moderate glaucomatous damage (average MD = −2.19, SD = 1.55 dB). Axis 1 of G

_{1}(at +2 SD) represents a mild peripheral superior arcuate depression, and axis 2 of G

_{1}(at +2 SD) represents early superior arcuate and nasal step defects. Cluster G

_{2}includes eyes with moderate to advanced glaucomatous damage (average MD = −8.07, SD = 6.26 dB). Axis 1 of G

_{2}(at +2 SD) represents moderate diffuse reduction of sensitivity with a more pronounced superior hemifield defect and axis 2 of G

_{2}(at +2 SD) represents a diffuse reduction of sensitivity with increased defect in the nasal step positions. Axes 3, 4, and 5 (at +2 SD) primarily represent nasal step defects, with some variation in weighting among these three axes.

**Figure 4**

**Figure 4**

_{2}cluster of VIM. The left tail of the 95th percentile limit is shown as a blue line that indicates the stability limit for this axis toward progression. The right panel of Figure 5 shows the distribution of the slopes of projected points on each axis from all eyes from the Progression Study Group on the first axis of the G

_{2}cluster of VIM. The red circle shows the stability limit computed from the distribution of the slopes of the Stability Definition Group on the first axis of cluster G

_{2}(Figure 5, left, blue line). For a test eye, if its slope (estimated by least square regression of projected VFs) exceeded this limit, the eye was classified as a progressed along this axis.

**Figure 5**

**Figure 5**

_{2}cluster of GEM. The right panel of Figure 6 shows the distribution of the slopes of all Progression Study Group eyes on the first axis of the G

_{2}cluster of GEM. The red circle shows the stability limit that was computed from the distribution of the slopes of the Stability Definition Group on the first axis of cluster G

_{2}(Figure 6, left, blue line). As above, if the slope of a test eye exceeded this limit of stability, the eye was classified as a progressing eye along this axis.

**Figure 6**

**Figure 6**

_{2}present in the 52-dimensional VF space. The blue line indicates the slope (linear regression of the orange circles). The gray line indicates the 95% progression limit for the slopes of the first axis of cluster G

_{2}. If the linear model approximating the slope falls below the gray line (progression zone), then the eye is classified as progressed; otherwise, the eye is classified as nonprogressed. Therefore, GEM-POP is detecting progression in the study eye in Figure 7 (left) and is detecting no evidence of progression in the study eye in Figure 7 (right) along the first axis of cluster G

_{2}. In addition to event-related information, Figure 7 (left) provides information about the rate of progression in the example eye (blue circles).

**Figure 7**

**Figure 7**

**Figure 8**

**Figure 8**

**Figure 9**

**Figure 9**

**Table 3**

**Table 4**

^{24}for a similar VIM result using frequency doubling technology perimetry VFs). With VIM, the clusters and axes that constituted the VIM environment were determined simultaneously. In contrast, GEM used a modular approach to first generate the clusters of disease severity followed by identification of axes or defect patterns.

*P*value = 0.03), linear regression of MD (

*P*value <0.001), and VFI (

*P*value <0.001). The area under the ROC curve of GEM-POP was similar to the area under the curve of VIM-POP (

*P*value = 0.12). Progression of patterns by VIM and GEM is based on progression along any one of seven (in the current environment) axes, whereas progression by linear regression of MD and VFI is based on a single metric, indicating that detecting localized change in defect patterns likely is a more sensitive technique than detecting global change. This superiority is expected, because in VIM-POP and GEM-POP, uncontributing VF locations are ignored; whereas for change in global indices the noncontributing VF locations are included. In fact, all progression detection methods that relied on local analysis (VIM-POP, GEM-POP, PLR [based on best performing parameter sets], and PoPLR) outperformed methods that relied on global analysis. It should be noted that PLR sensitivities can be significantly changed based on the selection of particular parameters (e.g., varying the required number of deteriorating locations or varying the slope).

^{40}In GEM, one set of axes that are discovered a priori describe the general patterns of glaucoma defects. Progression in a study eye is determined based on progression along these predetermined defect patterns. Whereas in POD, a set of axes is identified for each eye that describes the baseline conditions of the eye (known as the baseline subspace of the eye). Progression is determined based on the deviation of follow-up measurements from the baseline subspace of the eye.

^{34}and Analysis with Non-Stationary Weibull Error Regression and Spatial Enhancement (ANSWERS).

^{41}We compared GEM- and VIM-based progression detection methods with PoPLR directly. However, PoPLR

^{34}requires a minimum of one baseline and six follow-up VF exams (to provide at least 5000 unique permutations of the VF series for building null distributions for hypothesis testing) to generate a reliable and robust outcome.

^{34}Because the Stability Definition Group in the current study included an average of five VFs per eye, allowing 120 permutations, we generated sequences of seven visits for each eye to fulfill the PoPLR requirements and used the newly generated simulated test–retest dataset to compute the specificity of all methods. Therefore, the comparison is valid because all methods used the same test–retest dataset. ANSWERS relies on a mixture of Weibull distributions to model variability and a Bayesian method to aggregate spatial correlation of local measurements to confirm repeatable defects in the same or adjacent locations in follow-up examinations. The addition of spatial correlations of measurements improves this method, compared with ANSWERS' precursor, without spatial enhancement. We did not compare GEM- and VIM-based progression detection methods with ANSWERS because such a comparison is beyond the scope of the current manuscript.

^{24,43}(Bowd et al.,

*IOVS*2014;55 ARVO Abstract 3008; Yousefi et al.,

*IOVS*2015;56 ARVO Abstract 4564).

^{43}One of the limitations of this approach is that the effects of aging, glaucoma management, and long-term measurement variability cannot be modelled in a longer pseudoseries using only five exams. Nevertheless, this limitation does not affect the comparison of progression detection performance of VIM-POP and GEM-POP, PoPLR, PLR, MD, and VFI because all of the progression detection methods used the same simulated test–retest dataset.

**S. Yousefi**, None;

**M.H. Goldbaum**, None;

**M. Balasubramanian**, None;

**F.A. Medeiros**, Ametek (C), Alcon Laboratories Inc, (C), Allergan Inc. (F, C), Bausch+Lomb (F), Carl Zeiss Meditec (F, C), Heideleberg Engineering GmbH (F, C), Sensimed (F);

**L.M. Zangwill**, Carl Zeiss Meditec Inc. (F), Heidelberg Engineering GmbH (F, R), Optovue Inc. (F), Topcon Medical Systems Inc. (F);

**R.N. Weinreb**, Alcon Laboratories Inc. (C), Allergan Inc. (C), Carl Zeiss Meditec Inc. (F, C), Heidelberg Engineering GmbH (F), Optovue Inc. (F), Topcon Medical Systems Inc. (F, C);

**C.A. Girkin**, None;

**J.M. Liebmann**, Alcon Laboratories Inc. (C), Allergan Inc. (C), Carl Zeiss Meditec Inc. (F), Diopsys Corp. (F, C), Heidelberg Engineering GmbH (F), Optovue Inc. (F, C), Pfizer Inc. (C), Topcon Medical Systems Inc. (F, C);

**C. Bowd**, None

*. 2014; 311: 1901–1911.*

*JAMA**. 2004; 363: 1711–1720.*

*Lancet**. 2006; 90: 262–267.*

*Br J Ophthalmol**. 2004; 82: 887–888.*

*Bull World Health Organ**. 2003; 110: 1890–1894.*

*Ophthalmology**. 2008; 145: 343–353.*

*AM J Ophthalmol**. 2005; 46: 3730–3736.*

*Invest Ophthalmol Vis Sci**. 2007; 16: 20–28.*

*J Glaucoma**. 2002; 43: 3444–3454.*

*Invest Ophthalmol Vis Sci**. 2012; 53: 2382–2389.*

*Invest Ophthalmol Vis Sci**. 2005; 46: 1322–1329.*

*Invest Ophthalmol Vis Sci**. 2004; 45: 2255–2262.*

*Invest Ophthalmol Vis Sci**. 2005; 46: 4147–4152.*

*Invest Ophthalmol Vis Sci**. 2002; 49: 963–974.*

*IEEE Trans Biomed Eng**. 2009; 50: 674–680.*

*Invest Ophthalmol Vis Sci**. 2002; 43: 162–169.*

*Invest Ophthalmol Vis Sci**. 1994; 35: 3362–3373.*

*Invest Ophthalmol Vis Sci**Perimetry Update 1990/1991*.

*New York: Kugler & Ghedini Publications;*1991: 287–290.

*. 1999; 8: 77–80.*

*J Glaucoma**. 1994; 159: 553–557.*

*Mil Med**Perimetry Update 1990/1991*.

*New York: Kugler & Ghedini Publications;*1991: 291–295.

*. 1994; 14: 239–248.*

*Ophthalmic Physiol Opt**. 2009; 247: 1517–1530.*

*Graefe's Arch Clin Exp Ophthalmol**. 2014; 9: e85941.*

*PLoS One**. 2005; 103: 270–280.*

*Trans Am Ophthalmol Soc**. 2009; 107: 136–144.*

*Trans Am Ophthalmol Soc**. 2004; 45: 2596–2605.*

*Invest Ophthalmol Vis Sci**. 2014; 2104:pii:90342M.*

*Proc SPIE Int Soc Opt Eng**. 2005; 46: 3684–3692.*

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

*Invest Ophthalmol Vis Sci**. 2014; 61: 2112–2124.*

*IEEE Trans Biomed Engin**. 2009; 107: 136–144.*

*Trans Am Ophthalmol Soc**. 1996; 80: 40–48.*

*Br J Ophthalmol**. 2012; 53: 6776–6784.*

*Invest Ophthalmol Vis Sci**. 2013; 8: e78630.*

*PLoS One**. 2014; 55: 8386–8392.*

*Invest Ophthalmol Vis Sci**. 2009; 127: 1136–1145.*

*Arch Ophthalmol**. 1992; 110: 812–819.*

*Arch Ophthalmol**. 2000; 22: 1078–1089.*

*IEEE Trans Pattern Anal Mach Intell**. 2012; 53: 3615–3628.*

*Invest Ophthalmol Vis Sci**. 2014; 9.*

*PLoS One**. New York, NY; 2014.*

*21st International Visual Field and Imaging Symposium**. New York: Chapman & Hall; 1993.*

*An Introduction to the Bootstrap*