Examination of Different Pointwise Linear Regression Methods for Determining Visual Field Progression

Abstract

purpose. To compare the specificity and sensitivity of several different methods for using pointwise linear regression (PLR) to detect progression (deterioration) in visual fields.

methods. First, theoretical results were derived to predict which of the considered PLR methods would be the most specific and hence the least sensitive. Then, a “Virtual Eye” simulation model was developed that simulates series of sensitivity readings for a point over time. The model adds normally distributed noise (estimated from published results) to the sensitivity at each point to produce a series of fields to be analyzed using each method. Stable and deteriorating eyes were simulated, with the latter defined to have a noise-free loss of 2 dB/y at a significant cluster of points over the series.

results. The most sensitive method tested was to flag a visual field as progressing if it had a point that exhibited a statistically significant slope (at the 1% level) of at least −1 dB/y in the sensitivity. The most specific was a new “Three-Omitting” method that is being proposed, using two confirmation fields in a novel way. Current methods of using confirmation fields to verify a significant slope incorrectly flagged up to twice as many stable eyes as having progressing fields as did our new method.

conclusions. Using the new proposed PLR method is recommended in preference to current PLR methods in any applications when a high degree of specificity is the main priority.

It has been well documented that one of the biggest clinical challenges when managing glaucoma is identifying whether the visual field is actually deteriorating over time, or remaining stable, with the variation between fields in a series being accounted for by the variability (noise) present in the measurement methods. Naturally, results from a set rule or automated computer program for detecting change will only partially contribute to a clinical management decision for a patient. However, when it comes to comparing, say, two different groups of patients who are undergoing, for example, different treatment regimens, then a benchmark for change is desirable to identify any difference in the amount of progression in the two groups. 

Several different methods of determining visual field progression have been proposed, and there is no agreement about which is the best. Indeed, the method that is most suitable for use depends on the purpose to which the results are to be put. When the outcome of an eye’s being labeled as progressing would be the patient’s undergoing a risky or costly change in clinical management, a method should be used that is known to falsely label very few stable eyes as progressing (i.e., a high degree of specificity). Conversely, when failing to treat a progressing eye would be much more damaging than treating a stable eye, it is better to use a test that will correctly flag progressing eyes quicker (i.e., a more sensitive test). Clearly, the key is which way the method flags eyes in the middle ground that could be judged to be either progressing or stable; normally, tests that are more sensitive are less specific, and vice versa. 

Automated visual fields are essentially made up of a grid of numerical sensitivity values, making them amenable to an assortment of statistical analyses. One set of methods for detecting visual field progression relies on estimates of change in summary measures of the field (so-called global indices) such as mean deviation (MD) ¹ or visual field defect scores. ² The advantageous simplicity of these methods is outweighed by the fact that they largely or completely ignore the detailed spatial information contained within computerized field tests, and they are reported to be insensitive to glaucomatous change. ^{3 4 5} Methods considering change in parts of the fields or at individual locations are more sensitive to change. ^{6 7} An example of this is known as glaucoma change probability (GCP), which examines the difference in threshold deviation at individual locations between a given field and baseline test results. ⁸  

Alternatively, pointwise linear regression (PLR) examines the sensitivity of each test location in the field against time over a patient’s series. This provides a measure of the rate of loss (decibels per year) at each test location and a measure of the error associated with this change, summarized by the statistical significance or P-value. PLR, fully described elsewhere, ⁹ has been shown to be clearly more sensitive at identifying visual field loss than monitoring summary measures of the field ^{5 6 7} and compares favorably to GCP analysis at detecting and predicting progression. ^{10 11} PLR has also been found to agree more closely with expert clinical judgment about the status of progression than GCP analysis. ¹² Moreover, in a study of untreated glaucomatous eyes, simple linear regression has been shown to perform better than polynomial regression in predicting deterioration of visual fields. The latter merely imitates the noise in the series of readings. ¹³ Furthermore, PLR has been used to demonstrate the benefits of treatment changes in normal-tension glaucoma, ^{14 15 16} and several different research groups have reported on the usefulness of the technique. ^{5 6 7 12 17 18 19}  

In spite of the obvious claims of PLR’s being a clinically useful tool for examining longitudinal visual field data, there is no consensus on what value of regression slope and P-value constitutes progression and whether it should be maintained in subsequent fields. Indeed, the latter idea of “confirmation fields” or “confirmation criteria” has been shown to improve the specificity of other methods for detecting progression in visual fields, ^{20 21} but has yet to be formally examined for PLR. Yet, such criteria are still used ad hoc to demonstrate the “benefits” of treatment changes in glaucoma. ¹⁶ In this study, we examined both the sensitivity and specificity of a selection of the different PLR criteria for confirming progression, and we propose an improved method. Studies of deterioration of visual fields are hampered by the lack of a gold standard for progression and complications inherent in using patients’ data; and so, in this article we offer a novel approach to this difficulty by comparing the PLR methods theoretically and follow this by using a purpose-written “Virtual Eye” simulation program exploiting newly published estimates of the variability (noise) inherent in visual field data. ^{22 23} Whether or not PLR is the best way of detecting progression (which is a widely debated question and one that has no firm answer at present), it is a widely used method, and as such any refinements to its methodology are to be welcomed. 

Methods

Seven different PLR methods of determining whether a point of an eye is progressing or not, given the sensitivities at n points (equally spaced visual field tests in time) were compared—the purpose being to rank them according to their specificity and sensitivity. We define these as:

1.  
   Standard Criteria: a point is flagged as progressing if it shows a significantly negative slope at the 1% level, together with an observed slope of at least −1 dB/y in sensitivity; this is written as Z _n = 1. The significance level is calculated by comparing the slope with the t-distribution, with (n − 1) degrees of freedom. This is the simplest commonly used PLR method in published studies, ^{5 6 7 10 11 12 17 18 19 24 25} and it is used clinically as an indicator for change in some centers. ²⁶
2.  
   Two of Two: a point is flagged as progressing if it satisfies the standard criteria and continues to satisfy them after addition of a further observed (confirmation) point: Z _n = 1 and Z _n+1 = 1. This has been used by Hitchings et al. ²⁷
3.  
   Three of Three: a point is flagged if it satisfies the standard criteria and continues to satisfy them after the addition of each of a further two observed (confirmation) points in turn: Z _n = 1, Z _n+1 = 1, and Z _n+2 = 1. This approach, applied to a form of GCP analysis, is being used in a clinical trial to evaluate the role of immediate intraocular pressure reduction in glaucoma. ²⁸
4.  
   Two of Three: a point is flagged if it satisfies the standard criteria and continues to do so after the addition of either the following one or two points: Z _n = 1 and either Z _n+1 = 1 or Z _n+2 = 1 (or all three). This has recently been used in a study comparing PLR and the Advanced Glaucoma Intervention Study (AGIS) visual field score. ²⁹
5.  
   Three of Four: a point is flagged if it satisfies the standard criteria and continues to do so with the addition of two of three successive points to the series: Z _n = 1 and at least two of Z _n+1 = 1, Z _n+2 = 1, and Z _n+3 = 1. This method has been used in studies examining the treatment of normal-tension glaucoma. ^{15 16}

And our two new proposed methods:

•  
  Two-Omitting: a point is flagged if it satisfies the standard criteria and the slope obtained by adding one confirmation point, but excluding point n also satisfies the criteria. This is written as Z _n = 1 and Z^′ _n+1 = 1.
•  
  Three-Omitting: a point is flagged if it satisfies the standard criteria, and the two slopes obtained by using points 1 to (n − 1) and either point (n + 1) or (n + 2) both satisfy the criteria: Z _n = 1, Z^′ _n+1 = 1, and Z^′ _n+2 = 1.

The logic behind these new methods is that if the latest sensitivity reading in the series is significantly worse than preceding readings purely by chance because of the noise, then a confirmatory regression, performed once the following point has been added to the series, will be biased by this low-sensitivity reading, and so omitting the point will give a more conservative estimate of the true rate of progression. As seen in Figure 1 , when the point is actually stable, there is less chance of its incorrectly being flagged as progressing—that is, fewer false-positive results are obtained. 

Assumptions for the Theoretical Approach

Several simplifying assumptions must be made to make the theoretical problems tractable. First, the noise is assumed to be normal: This is a reasonably accurate assumption but may be less true when the sensitivity is at the extremes of the range of measurements because of the censoring nature of the instrument. In other words, a point with a sensitivity of say 5 dB has positively skewed noise, because no readings below 0 dB are possible, whereas a stimulus seen at low levels, such as 36 dB, has a negatively skewed noise. The mean for this normal distribution is estimated from the readings taken so far. Second, it is assumed that the visual fields for each patient are taken at regularly spaced intervals. Both of these assumptions are in keeping with published work. ¹⁸  

Virtual Eye Simulation Model

We simulate sets of readings for a whole visual field in two circumstances: 

Stable Eye.

A Humphrey 30-2 visual field (Humphrey Instruments, San Leandro, CA) for a Virtual Eye was constructed and is shown in gray-scale form at the top of Figure 2 . This left eye has an early-to-moderate inferior arcuate scotoma and some general depression of sensitivity in the superior field. This field has an abnormal Humphrey MD of approximately −4 dB. The individual pointwise sensitivities that make up this field are assumed to be the true physiological data for this eye. These are generally not identical with the measured or estimated data derived in a visual field test. Three typical examples of the latter, where noise has been added to the true visual field, are shown as grayscales at the bottom of Figure 2 . 

The key to any accurate visual field simulation is how well this noise or variability is estimated. The variability of standard threshold automated perimetry is complex, with it being suggested that it is made up of at least two components: intra- and intertest variability, sometimes referred to as short-term fluctuation (SF) and long-term fluctuation (LF), respectively. Our approach uses one estimate for the noise, dependent on the level of the true thresholded sensitivity according to Henson et al., ²² and is based on frequency-of-seeing (FOS) data collected in patients and normal subjects at different visual field locations. This study demonstrated that the response variability was well represented by the function: log_e (SD) = A × sensitivity(dB) + B, where the constants A and B are −0.081 and 3.27, respectively. Hence, at each test location of the Virtual Eye, the noise is determined by independent random samples from the normal distribution, with the mean set at the true value for that location, and SD derived from the preceding function. For example, if the true sensitivity was 28 dB, then a measured sensitivity at that point is found by the Monte-Carlo simulated value drawn randomly from a normal distribution with mean 28 ± 2.72 dB (SD). This is repeated across the field to give a simulated noisy field. An entire visual field series for this eye, mimicking n tests per year over t years, (nt + 1) fields in total, can be generated in this manner. Note that, even when the eye is stable, the visual fields at the beginning and end of the series are not identical, because each has noise. 

One extra feature of this stable Virtual Eye is that a normal age-related decline of 0.1 dB per year was subtracted from the true value in all cases. Returning to the numerical example, in which the true sensitivity is 28 dB, this means that a measured sensitivity at that point recorded 5 years after the first test is determined by the simulated value drawn randomly from a normal distribution with mean 27.5 ± 2.84 dB (SD). Note the increase in SD, imitating the established fact that variability is dynamic and increases as sensitivity declines. 

Deteriorating Eye.

The stable Virtual Eye forms the basis and starting point for the deteriorating eye. A progressive visual field defect is then added to sites in and around the initial inferior arcuate defect. More precisely, this means that six points (with starting sensitivities of 32, 28, 24, 20, 16, and 12 dB) in the visual field are given a rate of loss of 2 dB per year. The consequence of this magnitude of progression on the noise-free eye over a 6-year period is illustrated in Figure 3 (naturally, the simulation subsequently adds noise to each point to generate visual field series). 

The visual field at 6 years would have a Humphrey MD of approximately −5 dB. This level of deterioration could be considered clinically moderate, and, with noise added in, it is doubtful that progression would be diagnosed by visual inspection or monitoring changes in global indices. Nevertheless, the cluster of points are decaying at more than 10 to 20 times age-related data, ^{30 31} and the threat of the progressing defect to the functionally important area of fixation means that it is precisely the type of progression that we suggest any decent pointwise method should detect with reasonable sensitivity and specificity. As with the stable eye, the important feature of this type of simulation is the noise, and this is added to deteriorating points in the visual field as before. For example, the progressing point at −3°, + 3° on the Humphrey visual field starts with a true value of 32 dB. The measurement at this point is imitated by the simulated value drawn randomly from a normal distribution with a mean 32 ± 1.97 dB (SD). After a period of follow-up, say 3 years, the true value at this point is 26 dB and the simulated measurement is arrived at by the value drawn randomly from a Normal distribution with mean 26 ± 3.20 dB. Visual field series for this deteriorating eye can be generated in a manner similar to those for the stable eye. 

Another feature of note for both the stable and deteriorating eye is a simple correction for the censored nature of the dB values. That is, if a generated value is below 0 dB or above 36 dB, then it is assigned to be 0 or 36 dB, respectively. Also, the dB values are all rounded to the nearest integer (as they would appear from a Humphrey perimeter). The Virtual Eye simulation program was purpose written in object-orientated statistical software (S-PLUS 2000 for Windows; StatSci Europe, MathSoft Inc., Oxford, UK). The visual fields are represented, stored, and output as numerical arrays on which further analysis can be performed (including the production of Humphrey-type grayscales as shown in Figs. 2 and 3 ). 

Simulation Experiment

One thousand series from the stable eye and 1000 series from the progressing eye, all with two tests per year over a 6-year follow-up (13 fields in total), were generated. Then PLR was applied to each of the 74 non–blind-spot locations within each 30-2 visual field series sequentially for each test starting from the 4th test (i.e., using the first 1.5 years of readings from the series) and finishing at the 13th test (using all 6 years of readings). Then, each criterion for determining progression was used in turn, and the eye labeled as progressing if one point in the field satisfied that particular criterion. The test at which the eye was first detected as having progressive disease was recorded. For those methods using at least two confirmation fields, it means that the fifth field tested was the earliest that progression could be flagged, and, for those using three confirmation fields, it means the sixth test was the earliest possible. Also note that if, for example, the Two-of-Two criterion is satisfied by using the 9th and 10th fields, then the visual field series was only labeled as progressing at the 10th field tested. 

Results

Theoretical Derivation

Because it is known that a more specific method of determining progression is always less sensitive if the level of noise is unchanged (methods of reducing the amount of noise are a separate issue entirely), it is only necessary to look at the specificity of each method—that is, how many stable eyes are incorrectly flagged as deteriorating. Suppose we have a series of visual fields, taken at times t ₁, t ₂, … t _n . At a given point in the field, the deviations from the age-corrected normal sensitivities are X ₁, X ₂, … X _n . Rather than the raw values, it is chosen to work with the deviations X _i below the age-corrected normal sensitivities. If the expected sensitivity for a patient of this age is S _E, then X _i = S _E − S _i . This means that at a stable point, the true (noise-free) deviation remains constant, and the X _i therefore all have the same distribution. If the point is progressing, the X _i increases over time. 

According to our first assumption described earlier, we can say that for a stable eye, the X _i is normally distributed about a constant c: X _i ∼ N(c, ς²). Note that the assumption is that the actual deviation of the point is constant, but we do not assume that the first or last readings are exactly equal (unlike Spry et al. ¹⁸ ), because those two readings also have noise, and so the readings are not the same as the actual deviation in the eye. The constant c is estimated by ĉ—the mean of the readings taken so far—so that the sum of the first (n − 1) readings is nĉ. But the PLR slope is unaffected by the addition of a constant to each reading X _i ; and so if we define Y _i = X _i − ĉ, we can perform PLR on the Y _i to produce the same results, where Y _i ∼ N(0, ς²) and the sum Y ₁ + Y ₂ … + Y _n−1 = 0. Thus, the Y _i values are pure noise, because if there were no noise, Y _i would equal 0 for each i = 1 … n − 1. We are also assuming that t _i = i —that is, the readings have been taken at equally spaced time intervals. 

Now, let β _n be the PLR slope based on the first n readings in the series, and with our new Two-Omitting method, the new PLR slope is θ _n+1. We are interested in the case in which X _n has been added to the series and has made the PLR slope significant, and the clinician wants to perform a confirmation test; therefore, we know that β _n > β _n−1. But then, according to the work fully described in the Appendix  

\[E({\beta}_{n{+}1}\ \mathrm{given}\ Y_{1},{\ldots}Y_{n}){>}E({\theta}_{n{+}1}\ \mathrm{given}\ Y_{1},{\ldots}Y_{n})\]

Where E(Z) is the expected (average) value of Z. Therefore in either case, the Two-Omitting method for PLR would be expected to produce a slope closer to the actual slope for a stable eye than the more standard Two-of-Two method. But also  

\[E\left(\left|\frac{{\theta}_{n{+}1}}{se({\theta}_{n{+}1})}\right|\ Y_{1},{\ldots}Y_{n}\right)\ {<}E\left(\left|\frac{{\beta}_{n{+}1}}{se({\beta}_{n{+}1})}\right|\ Y_{1},{\ldots}Y_{n}\right)\]

Where se(Z) is the standard error of Z. (Again, see Appendix for details). Thus, the slope θ _n+1 would be expected to be less steep and also less significant than β _n+1, proving that our new Two-Omitting method is more specific than the current Two-of-Two method. 

Naturally, it is straightforward to extend these two results to prove that the Three-Omitting PLR method is more specific than the Three-of-Three method. 

It is obvious (and easily provable) that the Three-of-Four methods flags more points as progressing than Three-of-Three, because any series that satisfies the criteria for the latter method necessarily satisfies the criteria for the former method. We can therefore sum up all the information in Figure 4 , where when a line connects two methods, it means that the upper method is known from theoretical results to be more specific than the lower method. 

Simulation Model

The graphs in Figures 5 and 6 show the cumulative percentage of visual field series in the Virtual Eye (of 1000) that were flagged as progressing by each time point, using the specified methods. In each graph, the uppermost line flags most eyes as progressing, whether this is incorrect (for a stable eye; this method is then the least specific) or correct (for a deteriorating eye; this method is the most sensitive). For example, Figures 5 and 6 show that, after 3 years of follow-up, the Standard Criteria had already falsely labeled 74.6% of the stable visual field series as progressing. By 6 years, virtually all the stable eyes had been falsely labeled as progressing. In contrast, by 6 years, the Three-Omitting method had falsely labeled just 11.6% of the stable eyes as having progressive disease. The sensitivity (proportion of correctly labeled progressing eyes) of the Standard Criteria and the Three-Omitting methods after 6 years were 97.1% and 65.7%, respectively (a significant but much smaller difference). 

Of those methods that use at least two confirmation fields (i.e., the three lower lines in Fig. 6 ), the Three-Omitting method affords a significant improvement in specificity, whereas the accompanying loss in sensitivity is much less marked. 

Thus, it is clear that, as predicted theoretically, our new Two-Omitting and Three-Omitting methods are the most specific of those using one and two confirmation fields. The currently used methods can incorrectly flag twice as many stable eyes as having progressive fields. These two new methods improve the specificity of PLR, without having such a severe negative effect on the sensitivity. 

Discussion

When choosing a method for determining whether a visual field is progressing or stable, the specificity and sensitivity both have to be taken into account. PLR has previously been shown to be a sensitive technique for detecting progression of deterioration in visual fields in patients with glaucoma. ^{5 6 7 11 12} However, invoking so many hypothesis tests of the individual rates of loss (slopes) several times in a follow-up period using standard interpretation of the statistical type 1 errors (P-values) means that specificity is difficult to estimate and control. ³² Confirmation tests or fields require more examinations of patients but afford better specificity. Of the seven PLR methods we have examined in this article, the Standard Criteria method was the most sensitive (it successfully detected progressing eyes quicker), whereas the Three-Omitting method was the most specific (it wrongly identified the fewest stable eyes as progressing). This work has also highlighted that fulfilling the Standard Criteria for PLR at just one point in the visual field without any confirmation criteria is clinically unreliable as a means of diagnosing progression, because of its absurd levels of specificity. 

Attempting to develop specific methods for detecting visual field progression has several benefits. There are 74 non–blind-spot points in the 30-2 Humphrey field, and therefore if, for example, the probability of each point’s being flagged as progressing is 3%, then the probability of at least one point in the eye’s being incorrectly flagged by the Standard Criteria method is  

\[\begin{array}{c}P\ \mathrm{(at\ least\ 1\ false\ positive)}{=}1-P\ \mathrm{(no\ false\ positive\ points)}\\{=}1-P\ \mathrm{(given\ point\ is\ not\ false\ positive)^{74}}\\{=}1-0.97^{74}{=}89.5\%\end{array}\]

This alarmingly poor performance is improved dramatically by a seemingly small improvement in the specificity at one point, as seen in Table 1 . 

Thus, there are substantial gains in specificity to be had by using our new proposed Omitting PLR methods. The other priority is to improve the sensitivity of the test. All the methods using more than one confirmation field exhibit only modest levels of sensitivity. Of course, this would improve if the rate of loss were greater or more widespread than that described by the progressing defect ascribed to our deteriorating Virtual Eye. It would also improve if the amount of noise in the measurements decreased. New perimetric testing strategies such as the Swedish interactive test algorithm (SITA; Humphrey) ³³ are primarily designed to offer reduced testing time, rather than directly reducing this noise. Alternative types of perimetry, exhibiting less noise, may be better equipped for detecting visual field progression. ^{23 34}  

One of the reasons that there is little consensus about which method of visual field detection offers the best specificity and sensitivity in diagnosing true deterioration is the lack of an independent gold standard. Agreement on progression between expert clinicians is poor ^{35 36} or moderate ^{12 37} at best, and this precludes it from being a good arbiter of different methods. In this article we offer an alternative approach. We have separated the diagnostic precision of each PLR criterion under examination by using statistical theory. This demonstrated that our newly proposed techniques offer higher levels of specificity, which may appear surprising because they use less of the data (PLR is applied to one field less than is available). A computer simulation of a stable and deteriorating Virtual Eye, which shows the improved specificity of the new techniques, confirms this result. 

Computer simulations are used as an alternative to analyzing the actual data of patients to examine data acquisition in automated perimetry. ^{33 38 39 40 41} It seems natural, therefore, to use them on the difficult problem of assessing visual field series. Spry et al. ¹⁸ designed a simulation of progressive and stable sequences of data by interpolating between two “real” measured fields. Their model had several parameters controlling the noise in the data, including values for SF, LF, and eccentricity-related fluctuation. Only one measure of noise, directly related to the pointwise sensitivity, was used in our Virtual Eye, based on that reported by Henson et al. ²² The latter reported the noise (measured by accurate FOS techniques) to be independent of stimulus eccentricity, and therefore we have not included a parameter for this in our model. It must be stipulated, however, that the estimates of noise used in our Virtual Eye are unlikely to be perfect. For example, the noise is expected to have a component based on the spatial configuration of a defect within a visual field, and work is under way to estimate this. 

Statistically speaking, there is very little justification for classifying a point within a visual field as either stable or deteriorating. The rationale behind such a distinction is that if a defect containing one or more points is progressing, then treatment is needed. A clinical decision of whether to treat is based on much more complex factors than whether the slope satisfies one strict criterion. Although we have used a significant slope of −1 dB/y to indicate progression, in accordance with published studies, this figure is, in essence, arbitrary, as is describing a typical progressing point as being one with a deterioration of −2 dB/y. Any software used clinically for the analysis of visual field data should allow users to alter the level of slope required for a point to be flagged, according to their clinical judgment based on other factors. Some software has this feature. ⁹ Also, when comparing two treatments, for example, it may be better to compare the distribution of slopes of points, rather than to compare perfunctorily the proportion of slopes that satisfy such criteria. 

This study has been confined to a consideration of different PLR methods. No comparison is made against nonlinear or non–pointwise methods. There is disagreement about whether PLR really is the best method of determining progression, but it is a commonly examined method, ^{6 7 9 10 11 12 13 14 15 16 17 18 19 24 25 26 27} and, as such, any potential refinements to the method should be considered. Moreover, PLR may be suitable for examining any deepening of an existing defect, but its effectiveness in examining enlargement of defects is more open to question. Potential disadvantages of PLR are not addressed by the present study. However, the conclusion that omitting techniques may be of benefit to regression methods has wider applications than the limited set of conditions used herein. For example, different levels of noise, different defect sizes, and different rates of progression would all affect the quantitative results in Figures 5 and 6 , but they would not affect the qualitative comparisons between the methods. The simulation should be viewed as an example, which supports the theoretical comparisons between the methods. 

Another way of evaluating the new method would be to use real data from patients. However, when the actual state of the eye is unknown (as in the case of real data), there is currently no gold standard against which to judge different techniques. A further advantage of the techniques used herein is that they are not limited to glaucomatous eyes. Thus, the methods developed and the principles behind confirmation techniques could be applied in other situations in related fields. Nevertheless, before any methods are recommended for widespread use, their usefulness and practicality would have to be tested in a clinical situation. 

In the clinical setting, the interpretation of any method remains a subjective matter, but using the approach described in this article provides decent estimates of sensitivity and specificity for particular methods, arming the clinician to make more informed management decisions for patients than with other PLR methods. It is very clear from these results how much specificity is gained by adopting a confirmation approach. We have demonstrated that (when the criteria of −1 dB/y and a 1% significance level are fixed) any gain in specificity when the PLR method is changed will be accompanied by a loss of sensitivity. But because of the disproportionate effect of a small change in specificity at each point, as demonstrated in our results, and because progressing eyes may often form a small proportion of the study population, seeking out specific methods is generally more important than sensitivity. For this reason, we suggest considering the new Three-Omitting method as an alternative to current confirmation methods applied to PLR. 

Appendix 1

When the first (n − 1) readings have been taken, the PLR slope is β _n−1. We are interested in the effect that X _n (and hence, Y _n ) has on β _n . For a pointwise linear least-squares regression, the slope is calculated by the equation  

\[\left({{\sum}}\ t_{i}^{2}\ {-}\ \frac{({{\sum}}t_{i})^{2}}{n}\right)\ {\beta}_{n}{=}{{\sum}}\ t_{i}X_{i}\ {-}\ \frac{{{\sum}}t_{i}}{n}\ {{\sum}}\ X_{i}\]

Because t _i = i in this case, and using the substitution Y _i = X _i − c, we can simplify this to  

\[\frac{1}{12}\ n(n{+}1)(n{-}1){\beta}_{n}{=}\ {{\sum}}\ iY_{i}\ {-}\ \frac{n{+}1}{2}\ {{\sum}}\ Y_{i}\]

Next, using our new Two-Omitting method, we can calculate the new PLR slope θ _n+1 from the equation  

\[\frac{1}{12n}\ (n^{2}{-}1)(n^{2}{+}12){\theta}_{n{+}1}{=}\ {{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}{+}(n{+}1)Y_{n{+}1}{-}\left(\frac{n{+}1}{2}{+}\frac{1}{n}\right)\ ({{\sum}_{i{=}1}^{n{-}1}}\ Y_{i}{+}Y_{n{+}1})\]

We are interested in the case in which X _n has been added to the series, and has made the PLR slope significant, and the clinician wants to perform a confirmation test. Therefore, we know that β _n > β _n−1 or, equivalently,  

\[\frac{(n{-}2)(n{-}1)}{2}\ Y_{n}{>}3\ {{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}\]

From our definition of Y _i , given the first (n − 1) readings, we know that the expected values E(Y _n ) = E(nY _n ) = E(Y _n+1) = E[(n + 1)Y _n+1] = 0, and remembering that \({{\sum}_{i{=}1}^{n{-}1}}\ Y_{i}{=}0\) , we see that when \({{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}\mathrm{{\geq}}\ 0\)  

\[E({\beta}_{n{+}1}\ \mathrm{given}\ Y_{1},{\ldots}Y_{n}){=}\ \frac{12}{n(n{+}1)(n{+}2)}\ \left({{\sum}_{i{=}1}^{n}}\ iY_{i}\ {-}\ \frac{n{+}2}{2}\ {{\sum}_{i{=}1}^{n}}\ Y_{i}\right)\]

 

\[{=}\frac{12}{n(n{+}1)(n{+}2)}\ \left({{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}\ {-}\ \frac{n{+}2}{2}\ {{\sum}_{i{=}1}^{n{-}1}}\ Y_{i}{+}\left(n{-}\frac{n{+}2}{2}\right)\ Y_{n}\right)\]

 

\[{=}\frac{12}{n(n^{2}{-}1)}\ \left(\frac{n{-}1}{n{+}2}\ {{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}{+}\ \frac{(n{-}2)(n{-}1)}{2(n{+}2)}\ Y_{n}\right)\]

 

\[{>}\frac{12}{n(n^{2}{-}1)}\ \left(\frac{n{-}1}{n{+}2}\ {{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}{+}\ \frac{3}{n{+}2}\ {{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}\right)\ {=}\frac{12n}{n^{2}(n^{2}{-}1)}\ {{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}\]

 

\[{\geq}\frac{12n}{(n^{2}{+}12)(n^{2}{-}1)}\ {{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}{=}E({\theta}_{n{+}1}{\vert}Y_{1},{\ldots}Y_{n})\]

Also, because β _n > 0 (because the PLR slope satisfies theprogression criteria), if \({{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}\) < 0 then [(n − 1)/2] Y _n > − \({{\sum}_{i{=}1}^{n{-}1}}\) iY _i and in this case,  

\[E({\beta}_{n{+}1}\ \mathrm{given}\ Y_{1},\ {\ldots}\ Y_{n}){=}\ \frac{12}{n(n{+}1)(n{+}2)}\ \left({{\sum}_{i{=}1}^{n}}\ iY_{i}{-}\ \frac{n{+}2}{2}\ {{\sum}_{i{=}1}^{n}}\ Y_{i}\right)\]

 

\[{=}\frac{12}{n(n{+}1)(n{+}2)}\ \left({{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}{-}\ \frac{n{+}2}{2}\ {{\sum}_{i{=}1}^{n{-}1}}\ Y_{i}{+}\left(n\ {-}\ \frac{n{+}2}{2}\right)\ Y_{n}\right)\]

 

\[{=}\frac{12}{n(n^{2}{-}1)(n{+}2)}\ \left((n{-}1)\ {{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}{+}\ \frac{(n{-}2)(n{-}1)}{2}\ Y_{n}\right)\]

 

\[{>}\frac{12}{n(n^{2}{-}1)(n{+}2)}\ \left((n{-}1)\ {{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}{-}(n{-}2)\ {{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}\right)\]

 

\[{=}\frac{12n}{n^{2}(n^{2}{-}1)(n{+}2)}\ {{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}{\geq}\ \frac{12n}{(n^{2}{+}12)(n^{2}{-}1)}\ {{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}\]

 

\[{=}E({\theta}_{n{+}1}{\vert}Y_{1},{\ldots}Y_{n})\]

Therefore, in either case, the Two-Omitting method for PLR would be expected to produce a slope closer to the actual slope for a stable eye than would the more standard Two-of-Two method. 

The advantage to assuming the Y _i values to be normally distributed is that any linear combination of them is also normal. In particular, β _n+1 and θ _n+1 are, with variance  

\[\mathrm{Var}\ ({\beta}_{n{+}1}){=}\ \frac{12{\varsigma}^{2}}{n(n{+}1)(n{+}2)}\ \mathrm{and\ Var}\ ({\theta}_{n{+}1}){=}\ \frac{12n{\varsigma}^{2}}{(n^{2}{+}12)(n^{2}{-}1)}\]

where ς² is the variance of each Y _i (assumed earlier to be constant). And so for testing the significance of the PLR slopes  

\[E\left(\left|\ \frac{{\theta}_{n{+}1}}{se({\theta}_{n{+}1})}\right|\ {\vert}Y_{1},\ {\ldots}\ Y_{n}\right)\ {=}\sqrt{\left(\frac{12n}{(n^{2}{+}12)(n^{2}{-}1){\varsigma}^{2}}\right)}\left|{{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}\right|\]

 

\[{<}\sqrt{\left(\frac{12(n{+}1)(n{+}2)}{n(n{-}1)^{2}(n{-}2)^{2}{\varsigma}^{2}}\right)}\left|{{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}\right|\]

 

\[{=}\sqrt{\left(\frac{12}{n(n{+}1)(n{+}2){\varsigma}^{2}}\right)}\left|{{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}{+}\ \frac{2}{(n{-}1)(n{-}2)}\ 3n\ {{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}\right|\]

 

\[{<}\sqrt{\left(\frac{12}{n(n{+}1)(n{+}2){\varsigma}^{2}}\right)}\left|{{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}{+}nY_{n}\right|\]

 

\[{=}E\left(\left|\frac{{\beta}_{n{+}1}}{se({\beta}_{n{+}1})}\right|{\vert}Y_{1},{\ldots}Y_{n}\right)\]

Thus, the slope θ _n+1 would be expected to be less steep and also less significant than β _n+1, which shows that our new Two-Omitting method is more specific than the current Two-of-Two method. 

 Submitted for publication September 7, 2001; revised December 27, 2001; accepted January 11, 2002.

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 The publication costs of this article were defrayed in part by page charge payment. This article must therefore be marked “advertisement” in accordance with 18 U.S.C. §1734 solely to indicate this fact.

 

Figure 1.

 

Graphs of sensitivities (y-axis) against time. After seeing the PLR slope based on the first five readings (the first graph), a confirmation field is sought to determine whether the series really is progressing (it clearly is not, given the next point). But the PLR slope including the sixth point (the second graph) is still flagged as progressing, because it is biased by the low fifth-point reading. Our Two-Omitting method (the third graph) solves this problem.

Figure 2.

 

The effect of adding noise to the actual input sensitivity values for each point in a Virtual Eye. Visual fields are represented as 30-2 grayscales (Humphrey, San Leandro, CA), with darker areas indicating lower sensitivity readings at that point. The three noisy eyes generated are notably different, both from each other and the true physiological values represented by the top grayscale.

Figure 3.

 

The progression in the noise-free eye, as used in the Virtual Eye simulation.

Figure 4.

 

Theoretical relative specificity and sensitivity of the different PLR methods.

Figure 5.

 

The relative specificity and sensitivity (respectively) of four PLR methods, as determined by the Virtual Eye simulation.

Figure 6.

 

The relative specificity and sensitivity (respectively) of the remaining PLR methods (plus the Standard Criteria method again, for reference), as determined by the Virtual Eye simulation.

Table 1.

 

The Dramatic Effect of a Relatively Small Increase in Specificity

Prob (Given Point FP)	Prob (≥1 FP in 24-2 Field)	Prob (≥1 FP in 30-2 Field)
3	79.5	89.5
2	65.0	77.6
1	40.7	52.5

 A small probability of an individual point’s being falsely labeled as progressing (FP) leads to a high probability of at least one point in the eye’s (52 points for a 24-2 Humphrey field, 74 points for a 30-2 Humphrey field) being FP. Data are percentages. Prob, probability.