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.

^{ 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 }

*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 }

*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.

*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:

- 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 } - 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 } - 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 } - 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 } - 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 }

- 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.

^{ 18 }

^{ 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.

^{ 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.

*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.

*X*

_{ i }is normally distributed about a constant

*c*:

*X*

_{ i }∼

*N*(

*c*, ς

^{2}). 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, ς

^{2}) 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.

_{ 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*(

*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

*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.

^{ 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.

*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 }

^{ 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.

^{ 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.

^{ 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.

^{ 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.

*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

*t*

_{ i }=

*i*in this case, and using the substitution

*Y*

_{ i }=

*X*

_{ i }−

*c*, we can simplify this to

_{ n+1}from the equation

*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,

*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 \({{\sum}_{i{=}1}^{n{-}1}}\ iY_{i}\mathrm{{\geq}}\ 0\)

_{ 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,

*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

^{2}is the variance of each

*Y*

_{ i }(assumed earlier to be constant). And so for testing the significance of the PLR slopes

_{ 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.

**Figure 1.**

**Figure 1.**

**Figure 2.**

**Figure 2.**

**Figure 3.**

**Figure 3.**

**Figure 4.**

**Figure 4.**

**Figure 5.**

**Figure 5.**

**Figure 6.**

**Figure 6.**

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 |

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