July 2011
Volume 52, Issue 8
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Eye Movements, Strabismus, Amblyopia and Neuro-ophthalmology  |   July 2011
Automated Perimetry: Using Gaze-Direction Data to Improve the Estimate of Scotoma Edges
Author Affiliations & Notes
  • Harry J. Wyatt
    From Biological Sciences, SUNY State College of Optometry, New York, New York.
  • Corresponding author: Harry J. Wyatt, Biological Sciences, SUNY State College of Optometry, New York, NY 10036; wyatt@sunyopt.edu
Investigative Ophthalmology & Visual Science July 2011, Vol.52, 5818-5823. doi:10.1167/iovs.10-6398
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      Harry J. Wyatt; Automated Perimetry: Using Gaze-Direction Data to Improve the Estimate of Scotoma Edges. Invest. Ophthalmol. Vis. Sci. 2011;52(8):5818-5823. doi: 10.1167/iovs.10-6398.

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Abstract

Purpose.: To make an initial assessment of the feasibility of using records of eye movements during perimetry to improve the estimate of scotoma edge location.

Methods.: The nasal edge of the blind spot was mapped in seven normal subjects with a 2° grid of test locations, using a custom test station, while gaze direction was monitored with an eye tracker. Records were analyzed to determine whether the combined sensitivity and eye movement data could be used to estimate the nature of the blind spot edge.

Results.: Analysis was conducted for 15 high-variability test locations. For 11 locations the blind spot edge estimates fit plausibly with the general form of the blind spot (edge orientation within 90° of expected); for four locations the agreement was poor. One consequence of interpreting the test results using the edge estimates was an average reduction of test–retest variability by 58%.

Conclusions.: Recordings of eye movements during perimetry can be used to generate an improved estimate of scotoma boundaries. A byproduct of the new estimate is a substantial reduction of test–retest variability.

Visual fields of patients are most frequently assessed by means of computerized perimetry or standard automated perimetry (SAP). Although in the normal visual field, test–retest variability is relatively low, on the order of 1 to 2 dB, 1,2 in damaged visual fields it can become much higher. 3 8 This makes it difficult to determine whether a patient's condition is stable or progressing. 
Small fixational eye movements during testing may contribute to increased test–retest variability in damaged fields. 2,9 12  
Assessing damaged visual fields often involves assessing size and location of scotomas, and one difficulty in static perimetry is determining where scotoma edges are located. To date, in experimental work studying sensitivity near edges of scotomas or the edge of the blind spot (using it as a physiological scotoma), the location of the edge relative to the test locations has generally been unknown. To clarify this, in a commonly encountered situation, a row of three test locations gives the after results: At the first location, sensitivity is repeatedly found to be near normal, while at the third location sensitivity is repeatedly found to be essentially absent. At the second (middle) test location, results may be like those of the first location, or like those of the third location, or highly variable from test to test. In the case of high variability at the middle location, all that can be said about the boundary between healthy and damaged fields is that it lies between the first and third locations, which—in the case of the usual 6° or 2° grids—locates the boundary to within 12° or 4°, respectively. Examples of such situations are commonplace findings in repeated visual fields near edges (e.g., Fig. 2 of Ref. 2); upper subject: x = 5°, y = 1°, 3°, 5° deg; lower subject: x = –1°, 1°, 3°, y = 3°). 
One reason why scotoma edge location is an important issue is that sensitivity can change rapidly as one crosses such an edge. If a visual field test location lies near an edge, a small eye movement can potentially cause a large difference in sensitivity at the location to which a test stimulus is delivered. A number of researchers have considered the possibility that fixational eye movements could be involved in test–retest variability. There have been reports that variability increases near the edges of scotomas 9,13,14 and that the number of steep scotoma boundaries in a field is correlated with the variability. 10 Pan et al., 15 modeling the retina-plus-cortex, looked at the effect of randomly damaging the retinal ganglion cell array together with random fixation shifts. As might be expected, in the presence of damage this created variability in sensitivity in an otherwise noise-free model, at least for smaller stimuli (Goldmann size III). In visual fields of patients with glaucoma, Wyatt et al. 2 found a strong spatial correlation between the gradient of sensitivity in the visual field and test–retest variability, and argued that it was most easily explained by small eye movements causing variability where the gradient was steep. However, findings in this area have been mixed: Haefliger and Flammer 14 concluded that the gradient of sensitivity has less effect on variability for glaucomatous defects than for the physiological blind spot, and Henson et al. 12 concluded that fixation errors may have only a minor influence on perimetric variability. 
If fixational eye movements contribute to test–retest variability, studying eye movements during visual field testing might illuminate the issue. (This need not necessarily be the case; random or patchy damage on a fine scale could make it difficult to see any relationship.) However, there have been no studies of sensitivity near scotoma edges in which the time course of eye movements has been related to the time course of the test results—that is, the time course of the staircase of test presentations used to arrive at the test results. 
If the gaze direction (fixation error) were known for each test flash in a perimetric determination, it would mean that the actual retinal location of each test flash would be known. It seemed possible that in such a situation the spatial distribution of test flash contrasts and subject responses could be used to determine the most likely boundary between healthy and damaged fields. If the spatial pattern of damage is relatively simple, with substantial areas of healthy and damaged fields separated by simple boundaries, then a manageable number of repeat tests might provide enough data to estimate the retinal location of the boundary. In the present work, visual sensitivity near the blind spot was measured in normal subjects and gaze direction was measured concurrently. A post hoc analysis indicated that a substantial part of the test–retest variability could be accounted for by the eye movements measured for each subject. 
Methods
General
Apparatus.
The apparatus used in the present work has been previously described in detail. 2 Briefly stated, visual stimuli were presented on a CRT display monitor (Radius Press View 21SR; Miro Display GmbH, Dietzenbach, Germany) driven by a computer (Power Macintosh G3; Apple, Cupertino, CA). The display monitor had a 38.0 × 27.8 cm active area, resolution 832 × 624 pixels, and frame rate 75 Hz. The monitor was calibrated with a luminance meter (LS-100, Minolta, Tokyo, Japan). The monitor was 75 cm from the recorded eye, so it subtended 29.1° horizontally and 21.8° vertically at the eye. 
Eye movements (gaze direction) and pupil diameter were measured using a PC-based infrared eye tracker (ISCAN EC-101, ISCAN, Inc., Burlington, MA) at a sample rate of 60/s. Experiments were controlled by the computer (Macintosh; digital I/O lines allowed the computer to turn the eye tracker recording on and off. 
Visual Stimuli.
The visual stimuli were Goldmann size III (circular luminance increments 26 minutes of arc in diameter), 100 ms duration, presented on a background luminance of 5 cd/m2 at 1-second intervals. The array of test locations is shown in Figure 1; a rectangular test array (4 wide × 7 high, 2° spacing) extended from 11° to 17° in the temporal visual field, encroaching on the blind spot. Twelve additional test locations were placed in nasal, superior, and inferior fields, to distribute subjects' attention broadly in the visual field. Because the right eye was tested, the fixation point was placed 6° left of the monitor center, allowing stimuli out to 20° in the temporal visual field. The complete sequence of visual stimuli presented, including location, time, and luminance of each presentation, and subject response (seen/not seen) was recorded by a technical computer language (MATLAB; MathWorks, Natick, MA) controlling the stimuli. 
Figure 1.
 
The test array used for the right eyes studied. The diamond represents the fixation target; the dark oval is an average blind spot in position and size.
Figure 1.
 
The test array used for the right eyes studied. The diamond represents the fixation target; the dark oval is an average blind spot in position and size.
The stimulus system provided a maximum test luminance of 54 cd/m2. With the apparatus and parameters used, this was approximately 16 dB (1.6 log units) brighter than threshold for locations away from the blind spot at the same eccentricity. 
Subjects.
Seven normal subjects participated in these experiments. All subjects had undergone a complete ocular examination within 1 year of the experiments and had been found to be free of ocular disease. Average age was 30.2 years (range 22–64). Data were collected from the right eyes of all subjects. The study was approved by the SUNY College of Optometry Institutional Review Board (IRB) and adhered to the principles of the Declaration of Helsinki. Written informed consent was obtained from each subject after the nature of the experiment was explained in detail. Subjects wore appropriate refractive correction and their left eyes were occluded with an eye patch. Each subject participated in at least three sessions. 
Experimental Protocol.
After subjects were positioned in the apparatus, the eye tracker was calibrated by having subjects sequentially fixate steady stimuli at an array of five locations: the central fixation target, ±3° horizontal, and ±3° vertical, for 1.5 seconds each while the eye tracker recorded “raw” gaze direction data. (Gaze direction data consisted of horizontal and vertical coordinates of the location of the pupil center and of the reflection of the eye tracker infrared source in the first corneal surface.) 
In addition to the x–y calibrations, subjects also participated in a “light–dark” trial, in which they fixated the central calibration target while a large, bright (54 cd/m2) stimulus was turned on and off with a 4-second period (2 seconds on, 2 seconds off, etc.), and pupil and gaze direction data were recorded for 16 seconds. This produced substantial pupil responses, and the data were used in analysis of experimental eye tracker data (see below). 
The visual field testing, using the test array of Figure 1, used a 2 dB/1 dB, two-reversal staircase. Initial stimulus luminance was randomly set at either 8.09 or 6.23 cd/m2. For test locations presented within the average location of the blind spot, initial stimulus luminance was set at 20.50 cd/m2. Subjects pressed a button connected to the Macintosh computer to indicate that they had seen a given test flash. Blank trials were presented at a rate of 1 in 6. For each test flash (including blanks), the computer recorded the time, location, and luminance of the flash and the subject's response. 
Gaze Direction Data Analysis.
Gaze direction data were converted from raw data into an estimate of gaze direction by (1) removing blinks using a blink detection algorithm based on rate of change of pupil diameter, (2) calculating the horizontal and vertical distances between pupil center and corneal reflex, (3) correcting the calculated distances according to pupil diameter (see below), and (4) using the calibration values to calculate gaze direction relative to the fixation target. The resulting records of gaze direction as a function of time during the trial were smoothed using 7-bin (7/60 seconds = 0.117 second) “boxcar” smoothing (running average of 7 bins). Under the conditions of these experiments, five of the seven subjects had pupils large enough and palpebral fissures small enough so that vertical pupil diameter measures were contaminated. Therefore, vertical eye position measurements were unreliable for these subjects and only horizontal eye position data were used in subsequent analyses. 
Correction of Gaze Direction for Pupil Diameter.
In previous work, it was shown that pupil centration in the eye is an idiosyncratic function of pupil diameter, but that the behavior is reasonably fixed for each subject. The “light–dark” trials described above were used to construct functions of pupil center versus pupil diameter for each subject. Data of the level of step 2 above were then corrected to a standard pupil size, which was taken to be the pupil diameter during gaze direction calibration. This step can be particularly significant in eyes of younger subjects whose pupils can change diameter substantially during field testing. The basis for such pupil changes, which are not a result of changes in environmental illumination, are likely to be responses to changes in balance of parasympathetic/sympathetic activity due to various internal variables—for example, concern about not doing well on the test. The effect of pupil changes on video eye tracker data, and a method for compensating for the changes, are described in Wyatt. 16  
Post Hoc Analysis of Gaze-Direction and Sensitivity Data
For the post hoc analysis, test locations were selected where test–retest variability was high. A “look-up” of data for the staircase information for that location was conducted: For each session, the staircase for the selected test location was extracted from the complete record of the session, and the time, luminance, and subject response were noted for each test flash presented at the location. The time was then used to look up gaze direction at the time of each flash. Pooling data for all sessions for the particular subject created a data set of actual retinal test locations, test luminances, and subject responses for the selected test location. 
The data were then fitted with a spatial function, consisting of two spatial regions of differing sensitivities, separated by a sharp edge. For both regions comprising the spatial function, the probability distribution for seeing a stimulus of contrast z was taken to be Quick's version of a Weibull function 17 :   where α is threshold contrast (P = 0.5) and β determines the steepness, with larger β giving a steeper curve corresponding to less variability. For two-dimensional (2-D) gaze-direction data, there were five parameters: angle and placement of the edge, two values of sensitivity α (one value on each side of the edge), and the steepness parameter β. For one-dimensional (1-D) gaze-direction data, the edge was assumed to be vertical, leaving four parameters. 
The functions were fitted using maximum likelihood estimation (MLE), in which each test delivered is assigned probability P(z) if seen and (1 – P(z)) if not seen, the probabilities being evaluated for the current set of parameters. The MLE approach maximizes the product of these probabilities for the entire set of test presentations. The fitting was carried out using a Monte Carlo technique in a data analysis program written (IGOR; WaveMetrics, Inc., Lake Oswego, OR). Some constraints were placed on parameters; in particular, β was allowed to vary from 0.5 to 6, covering a reasonably broad range of steepness. 106 trials were performed for an initial fit and 105 trials were performed to refine the parameters in smaller ranges near the best values from the initial fit. 
For each subject, the blind spot contour map of sensitivity generated from the basic test data were fitted by eye with an ellipse, and tangents to that ellipse were used to estimate the orientation (and polarity) of the blind spot edge for each test location studied in the post hoc analysis. 
To permit a comparison between test–retest variability with and without consideration of gaze direction, a fit was performed as above, but with the assumption that the eye did not move; the single “fit” value was determined by fitting all the test contrasts and responses with a single probability function of the type above. 
Results
The average sensitivity and test–retest variability for one subject are shown in Figure 2. (Variability in these plots was taken to be the SD of sensitivity estimates for each test location.) The star indicates the test location selected for post hoc analysis. For the subject of Figures 2 and 3, only horizontal gaze-direction data were available; therefore, the test location selected for subsequent analysis was close to a near-vertical blind spot boundary. 
Figure 2.
 
Average sensitivity and variability for one subject (S6) shown as grayscale contour plots. Star indicates the location used in subsequent analysis. Sensitivity and variability varied from 0.0 to 16.3 dB and 0.0 to 6.9 dB, respectively.
Figure 2.
 
Average sensitivity and variability for one subject (S6) shown as grayscale contour plots. Star indicates the location used in subsequent analysis. Sensitivity and variability varied from 0.0 to 16.3 dB and 0.0 to 6.9 dB, respectively.
Figure 3.
 
Post hoc data analysis for the subject in Figure 2 and the test location indicated by the star in Figure 2: 13°, –3° relative to fixation. Open symbols: stimuli that were seen; filled symbols: stimuli that were not seen. Left: all data are shown at one location; right: data are plotted at the retinal location determined by looking up the gaze direction for each flash. The fitted functions (gray lines) that maximized likelihood were sensitivity = –12.6 dB (ignoring eye position) and a step function from –3.0 dB relative to normal to approximately –19dB, positioned near to the nominal test location. The steepness of the probability distribution was β = 0.50 (left) and β = 2.64 (right), amounting to an 81% reduction in variability when gaze direction was considered.
Figure 3.
 
Post hoc data analysis for the subject in Figure 2 and the test location indicated by the star in Figure 2: 13°, –3° relative to fixation. Open symbols: stimuli that were seen; filled symbols: stimuli that were not seen. Left: all data are shown at one location; right: data are plotted at the retinal location determined by looking up the gaze direction for each flash. The fitted functions (gray lines) that maximized likelihood were sensitivity = –12.6 dB (ignoring eye position) and a step function from –3.0 dB relative to normal to approximately –19dB, positioned near to the nominal test location. The steepness of the probability distribution was β = 0.50 (left) and β = 2.64 (right), amounting to an 81% reduction in variability when gaze direction was considered.
In Figure 3, the post hoc analysis results for the starred test location of Figure 2 are shown. Ignoring gaze direction, the data were fitted by a single sensitivity of –12.6 dB relative to normal, with a slope parameter β = 0.50 (the minimum value allowed). Using gaze-direction information, the data were fitted by a step change from –3 dB to –19 dB relative to normal, located near the nominal test location. The use of gaze information increased β to 2.64, amounting to a reduction of variability by 81%. There was some “play” in the optimum placement of the sensitivity step; varying the step location from –0.2 to +0.1° did not change the value of the likelihood. 
Results for one of the subjects for whom 2-D gaze-direction data were available are shown in Figure 4. Four test locations were analyzed; the edge estimated for each location is shown by the rectangles. The fits for three locations appear plausible, while the fit at (11, 1), although it reduced the variability, was contrary to expectations in terms of the general form of the blind spot; that is, the fit had greater sensitivity on the side of the edge closer to the blind spot center. 
Figure 4.
 
Fits for a subject (S7) with 2-D gaze-direction data. Four high-variability locations were analyzed. Sensitivity varied from 0.0 to 15.9 dB. The rectangles indicate estimated edges using the same grayscale as sensitivity.
Figure 4.
 
Fits for a subject (S7) with 2-D gaze-direction data. Four high-variability locations were analyzed. Sensitivity varied from 0.0 to 15.9 dB. The rectangles indicate estimated edges using the same grayscale as sensitivity.
The results for all subjects are summarized in Table 1. The data from S5 proved inadequate to obtain a meaningful fit. 
Table 1.
 
Summary of Fitting Results
Table 1.
 
Summary of Fitting Results
Subject No. of Locations Plausible Fit Contrary Fit
S1 5 2 3
S7 4 3 1
S2 1 1
S6 1 1
S3 2 2
S4 2 2
S5 1 Inadequate data
The parameter β in the Weibull function is related to steepness of the psychometric function and inversely related to variability. Across all test locations, without considering gaze information β was found to be 1.1 ± 1.4 (mean ± SD; median, 0.8). Taking gaze information into consideration, β was found to be 2.6 ± 2.0 (mean ± SD; median, 1.9). This amounts to an average reduction of variability of approximately (1.1−1–2.6−1)/1.1–1 = 58%. Taking gaze information into consideration also increased the log likelihood (in the MLE technique) by 1.6 ± 1.4 dB (mean ± SD; median, 0.9). 
By independently varying the parameters for each fit, it was possible to obtain estimates of the confidence intervals for the key parameters provided by the MLE technique. The 95% confidence interval can be estimated as the width of the parameter range leading to a 1.09 log unit falloff in log likelihood on either side of the maximum. 18 This interval was 0.3° ± 0.3° (mean ± SD; median, 0.3) of visual angle for placement of the edge;13° ± 25 (mean ± SD; median, 4) of orientation for orientation of the edge in cases of 2-D data; and 1.1 ± 1.4 (mean ± SD; median, 0.8) for the steepness parameter β. 
Discussion
The results presented here are consistent with the suggestion that small eye movements underlie a significant portion of test–retest variability in damaged visual fields. It is possible that more precise measurement of gaze direction might increase the proportion further, though increased intrinsic variability in damaged areas could account for some of the variability. 
In trying to reconcile these findings with earlier work, it is important to keep in mind that the interaction between eye movements and sensitivity variations depends on the spatial nature of the latter; if the boundaries between normal and damaged retina or at the edge of the blind spot are gradual, extending over a number of degrees, extent of eye movements and amount of test–retest variability should correlate. However, there is considerable evidence that such boundaries can be steep. Israel 19 measured sensitivity across the blind spot along the horizontal meridian in 1° intervals, and found changes as rapid as 14 dB/deg. Haefliger and Flammer 13,14 found steep edges (approximately 15 dB/deg) at both blind spot and scotoma edges. Recent work using 1° intervals has found blind spot and scotoma edges as sharp as 30 dB/deg (Wyatt HJ, et al. IOVS 2008;49:ARVO E-Abstract 1089). 20 If at least some boundaries are sharp, what takes place during assessment of sensitivity at a test location near such a boundary may differ markedly from the usual view of testing. Even if zero intrinsic variability is assumed, each test flash has a certain probability of crossing the boundary and being seen or not seen accordingly, so one test with a staircase as used in perimetry will amount to a series of tosses of a coin that is weighted toward the more likely side of the boundary. 
The way in which extent of eye movement and distance of a test location from an edge affect the probability of crossing the edge is shown in Figure 5. The key variable is separation of test and edge measured in SDs of eye position, which explains the linear contours; the two parameters are separated here to simplify visualization of the behavior. As an example, for a typical SD (eye position) of 0.4° (Wyatt HJ, et al. IOVS 2007;48:ARVO E-Abstract 1621), variations of 0.1° or 0.2° in separation can make large differences in probability of crossing the edge, and therefore in test–retest variability. In addition, for locations on the other side of the edge (negative x-axis values), the graph is symmetric about the y-axis. For a location on the y-axis, there is a 50% chance of falling on either side. 
Figure 5.
 
Probability of an actual test location crossing an edge as function of distance to edge and SD (eye position). At the origin, the value is not defined.
Figure 5.
 
Probability of an actual test location crossing an edge as function of distance to edge and SD (eye position). At the origin, the value is not defined.
In the present results, many of the fits to the data sets appeared to be “plausible” in terms of the subjects' blind spots (Table 1). It is worth noting that two of the three test locations with 2-D data and “contrary” fits were the locations with estimated blind spot edge-orientation nearest horizontal. (Video-based eye trackers typically give somewhat less reliable data for the vertical dimension.) Although contour plots shown here (Figs. 2 and 4) and previously 2 appear to represent details of blind spot shape accurately, the underlying data are a set of values estimated at grid intersections, with each estimate made without considering fixational eye movements. Those data are then interpolated to estimate values for intermediate locations. For the most part, the details of the boundary shape are not known. Modest further support for the analysis comes from the values of parameter β for plausible versus contrary fits: pooling 2-D and 1-D data, β for plausible fits = 3.0 ± 2.1 (mean ± SD; median, 2.5), whereas β for contrary fits = 1.6 ± 1.2 (mean ± SD; median, 1.0). 
In the case of scotomas due to retinal damage there is even less certainty about boundaries. For example, if no test flashes are seen at a location, one knows that the location lies on the damaged side of the boundary, but not its distance from the boundary. This could account for results such as those of Henson et al., 12 who did not find a correlation between amount of eye movement and amount of test–retest variability; the values of a critical parameter (distance of test location from boundary) were not known. As an example, a patient with SD (eye position) = 1.2° and a nominal test location 2.1° from a scotoma edge would have a 4% chance of the test location crossing the edge. Another patient with SD (eye position) = 0.6° and nominal test location 0.4° from the edge would have a 25% chance of crossing the edge; thus, the patient with twice the extent of eye movement would have approximately 1/6 the likelihood of crossing the edge. Henson et al. also found that excluding test responses associated with fixation errors >1° did not systematically reduce variability (i.e., steepen the frequency of seeing curves), as might be expected. In that work, the abnormal test locations were selected from among test locations used in full-threshold testing with a 6° square grid; they also lay near the edge of a scotoma and showed “some sensitivity deficit” (but not a severe deficit) in full-threshold testing. In such a situation, neither the distance from test location to edge, nor the side of the edge on which the location lay, would be known. Although some variability decrease might be expected to result from excluding the points with larger fixation errors, the results on particular datasets could vary depending on details of eye movements and edge location. In the present work, the fitting process included all available data and “freed” the details of the boundary, to see what boundary would best account for the data set. 
Edges of Pathologic Scotomas versus the Blind Spot
Can results from the edge of the blind spot in normal subjects be expected to apply to scotoma edges in patients? A full assessment will require testing patients, but there are reasons for optimism: (1) There is evidence that test–retest variability in patients with glaucoma is greatest near boundaries. 2 (2) In the same work on glaucomatous fields, locations with intermediate sensitivities were typically associated with large—not moderate—gradients of sensitivity, supporting the view that scotoma edges are often steep. (3) Chan 20 found that scotoma edges could be strikingly steep. Given the presence of edges, some very steep, and the finding that patients undergoing perimetry do make fixational eye movements, 9,12 it could be argued that some variability driven by eye movements should be expected. 
Implications and Directions for Visual Field Assessment
In a clinical setting, stabilizing test targets on the retina, as proposed in a 1975 patent by Lynn and Tate, might seem preferable to post hoc correction. However, the technology for stabilization is still somewhat limited; microperimeters stabilize by registering selected retinal landmarks; they currently operate at approximately 20 updates per second and are not invulnerable to eye movements. Heidelberg Engineering (FRG) employs a similar registration system to provide some degree of stabilization for their ophthalmic scanning systems and their edge perimeter. All video eye trackers, even the fastest models available, are subject to imprecision in using pupil center and corneal reflex to estimate gaze direction, 21 though some improvement through corrections such as those used here is possible. Microperimeters are also expensive at present, compared to standard automated perimeters. Other high-precision devices for evaluating gaze direction, such as Purkinje image-based eye trackers, are even less practicable. Less elaborate video-based eye trackers could be used in stabilization, but there are still issues such as lid intrusion as well as the shift with pupil size change. In many less industrialized parts of the world, stabilized perimetry is probably not practicable at present. 
From a different perspective, if perfect stabilization were possible, would standard test arrays such as 24 to 2 or 10 to 2 provide the greatest amount of information? The chance of test locations falling on very steep scotoma boundaries may be fairly small because steep boundaries have little spatial extent. Thus, with a perfectly stabilized array, especially a 24 to 2 array with 6° spacing, boundaries would probably fall between test locations and real but modest changes in those boundaries might escape detection. Manual selection of test locations would be a possible approach, but that is demanding in a clinical setting. The presence of small eye movements, which vary the actual test location, might arguably be considered an advantage, if devices were capable of acquiring and using information about those eye movements. 
The post hoc correction approach does have drawbacks; a substantial amount of data (and therefore time) is required to perform successful fits. In the present study, subjects participated in at least three sessions. On the other hand, because the uncorrected data comprise standard perimetric findings, or would do so if a standard test array were used, conventional perimetric results would be available until there were enough repeats for a post hoc analysis. In current clinical practice, repeat fields are the basis for determining whether a patient with glaucoma is progressing; thus, acquiring the data to perform later post hoc corrections would take no extra time. The addition of post hoc corrections might then clarify the question of progression. In fact, if there were enough repeats for, say, the first and second half-sets to provide independent edge estimates, it is possible that movement of the edge might be observable for a case of progression. 
Footnotes
 Supported by NEI Grant R03EY014549.
Footnotes
 Disclosure: H.J. Wyatt (P)
The author thanks William H. Swanson for many helpful comments and much discussion. 
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Figure 1.
 
The test array used for the right eyes studied. The diamond represents the fixation target; the dark oval is an average blind spot in position and size.
Figure 1.
 
The test array used for the right eyes studied. The diamond represents the fixation target; the dark oval is an average blind spot in position and size.
Figure 2.
 
Average sensitivity and variability for one subject (S6) shown as grayscale contour plots. Star indicates the location used in subsequent analysis. Sensitivity and variability varied from 0.0 to 16.3 dB and 0.0 to 6.9 dB, respectively.
Figure 2.
 
Average sensitivity and variability for one subject (S6) shown as grayscale contour plots. Star indicates the location used in subsequent analysis. Sensitivity and variability varied from 0.0 to 16.3 dB and 0.0 to 6.9 dB, respectively.
Figure 3.
 
Post hoc data analysis for the subject in Figure 2 and the test location indicated by the star in Figure 2: 13°, –3° relative to fixation. Open symbols: stimuli that were seen; filled symbols: stimuli that were not seen. Left: all data are shown at one location; right: data are plotted at the retinal location determined by looking up the gaze direction for each flash. The fitted functions (gray lines) that maximized likelihood were sensitivity = –12.6 dB (ignoring eye position) and a step function from –3.0 dB relative to normal to approximately –19dB, positioned near to the nominal test location. The steepness of the probability distribution was β = 0.50 (left) and β = 2.64 (right), amounting to an 81% reduction in variability when gaze direction was considered.
Figure 3.
 
Post hoc data analysis for the subject in Figure 2 and the test location indicated by the star in Figure 2: 13°, –3° relative to fixation. Open symbols: stimuli that were seen; filled symbols: stimuli that were not seen. Left: all data are shown at one location; right: data are plotted at the retinal location determined by looking up the gaze direction for each flash. The fitted functions (gray lines) that maximized likelihood were sensitivity = –12.6 dB (ignoring eye position) and a step function from –3.0 dB relative to normal to approximately –19dB, positioned near to the nominal test location. The steepness of the probability distribution was β = 0.50 (left) and β = 2.64 (right), amounting to an 81% reduction in variability when gaze direction was considered.
Figure 4.
 
Fits for a subject (S7) with 2-D gaze-direction data. Four high-variability locations were analyzed. Sensitivity varied from 0.0 to 15.9 dB. The rectangles indicate estimated edges using the same grayscale as sensitivity.
Figure 4.
 
Fits for a subject (S7) with 2-D gaze-direction data. Four high-variability locations were analyzed. Sensitivity varied from 0.0 to 15.9 dB. The rectangles indicate estimated edges using the same grayscale as sensitivity.
Figure 5.
 
Probability of an actual test location crossing an edge as function of distance to edge and SD (eye position). At the origin, the value is not defined.
Figure 5.
 
Probability of an actual test location crossing an edge as function of distance to edge and SD (eye position). At the origin, the value is not defined.
Table 1.
 
Summary of Fitting Results
Table 1.
 
Summary of Fitting Results
Subject No. of Locations Plausible Fit Contrary Fit
S1 5 2 3
S7 4 3 1
S2 1 1
S6 1 1
S3 2 2
S4 2 2
S5 1 Inadequate data
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