purpose. To determine, in stable glaucoma, the characteristics of the between-examination variability of the visual field recorded with the Humphrey Field Analyser (HFA; Humphrey Systems, Dublin, CA) using the homogeneous, LF(Ho), and heterogeneous, LF(He), components of the long-term fluctuation (LF), thereby providing a technique for separating progressive loss from fluctuation in sensitivity.

methods. The LF components were calculated using a two-factor analysis of variance (ANOVA) with replications and were determined between each pair of three successive HFA program 30-2 fields for each patient from two groups, each containing 30 patients with primary open-angle glaucoma. The interval between examinations for the first group was 6 to 9 months and for the second group was 3 weeks.

results. The group mean values for LF(Ho) ranged from 1.50 to 2.19 dB and for LF(He) from 1.70 to 2.05 dB. The average difference between examinations was within ±0.35 dB for each component, and the 95% limits of agreement for the two groups, respectively, were ± 2.31 and ± 2.39 dB for the LF(Ho) and ± 2.36 and ± 2.09 dB for the LF(He). The estimate of the 90% confidence limit for the LF(Ho) was 3.30 dB and for the LF(He), 3.60 dB. Little relationship was present between the LF components and the modulus differences in mean deviation (MD), the corrected pattern SD (CPSD), or the mean MD, mean short-term fluctuation, and mean CPSD, of the two fields.

conclusions. Estimation of the LF components and of the corresponding confidence limits yields an expression of the normal between-examination variability of two consecutive fields that can be used as a reliability index. A value outside the confidence limits indicates the necessity for a confirmatory follow-up field.

^{ 1 }

^{ 2 }

^{ 3 }refractive error,

^{ 4 }

^{ 5 }

^{ 6 }media opacities,

^{ 7 }

^{ 8 }

^{ 9 }

^{ 10 }learning,

^{ 11 }

^{ 12 }fatigue,

^{ 13 }

^{ 14 }

^{ 15 }

^{ 16 }medical therapy,

^{ 17 }

^{ 18 }and random variations in the physiological status of the patient.

^{ 19 }These factors are such that the variability within and between any two successive examinations can mask or even mimic true deterioration in the visual field. The variation in the threshold estimate within a perimetric examination at a given stimulus location is known as the short-term fluctuation (SF), and the variation between examinations is known as the long-term fluctuation (LF).

^{ 20 }The SF is generally evaluated for a given number of stimulus locations but can also be considered for each individual stimulus location. It essentially represents the measurement error associated with the determination of threshold.

^{ 19 }The LF is the variance additional to that of the SF and is present between two or more examinations. It is divided into two components: the homogeneous component, LF(Ho), which affects all locations equally, and the heterogeneous component LF(He), which varies between locations.

^{ 21 }The two components each represent summary measures of the fluctuation occurring across all the specified locations and, as such, are global measures. Several alternative measures of between-examination variation have been proposed for the field as a whole

^{ 21 }

^{ 22 }

^{ 23 }

^{ 24 }

^{ 25 }or for a given stimulus location.

^{ 24 }

^{ 26 }

^{ 27 }

^{ 28 }

^{ 29 }

^{ 30 }These additional measures, which have also been labeled as the LF, describe the total variation and do not differentiate between the two classically defined components of between-examination fluctuation, LF(Ho) and LF(He).

^{ 23 }Nevertheless, the LF is not routinely specified, despite the overriding need to separate fluctuation from true progression.

^{ 20 }This omission is perhaps due to the computational complexity of the calculation. Progressive loss is frequently only identified retrospectively from a series of visual field examinations obtained over a period of several years or more, despite suggestions that confirmation of deterioration should be based on repeat examinations performed over a period of weeks.

^{ 31 }Additional confirmatory examinations add to the financial and resource burdens and may delay appropriate therapeutic intervention. Ideally, repeat examinations would be performed in patients only when the additional information would enhance the management of the glaucoma (i.e., only in those in whom the LF lies outside the normal limits encountered in stable glaucoma).

^{ 32 }The effect on the magnitude of the LF components when these additional double determinations of sensitivity are included is also unknown.

^{ 33 }For the statistical analysis, three separate measures of the MD were evaluated: the weighted MD of all 76 locations, the corresponding unweighted MD, and the unweighted MD derived from the 10 standard stimulus locations at which a double determination of threshold is undertaken. The weighted MD was available from the perimeter printout and is a summary measure of the differences between the measured values of sensitivity and the corresponding expected age-matched normal values. The unweighted indices were calculated because the details of the HFA weighting function have not been published. Two measures of the SF were analyzed: the weighted SF derived from the HFA printout and the unweighted SF. Three measures of the CPSD were analyzed: the weighted CPSD from the HFA printout, the unweighted 76 location CPSD, and the unweighted CPSD derived from the 10 standard locations. The unweighted mean deviation and unweighted corrected pattern SD require the age-matched normal sensitivity at each stimulus location for their calculation; the necessary normal values were obtained from the manufacturer. The values were identical with the normal data used in the software (Statpac; Humphrey).

^{ 32 }A separate repeated measures analysis of covariance (ANCOVA) was undertaken for each of the three indices, MD, SF, and CPSD. Age was considered as a covariate and subtype of each index (i.e., weighted 76 point, unweighted 76 point, or unweighted 10 point) and examination (i.e., first, second, and third) as within-subject factors. The three measures of each index were each stable over the three examinations (MD

*P*= 0.148; SF

*P*= 0.644; CPSD

*P*= 0.475).

^{ 21 }

^{ 22 }

^{ 23 }The replications comprised the double determinations of threshold at each of the 10 stimulus locations used by the field analyzer for calculation of the SF. The procedure assesses the difference in the average overall sensitivity between the two examinations, LF(Ho), and whether the difference varies between stimulus locations, LF(He). The two components are separately derived from the root of the corresponding mean square estimate of the ANOVA corrected for the aggregate measurement error across the two examinations.

^{ 21 }

^{ 22 }

^{ 23 }

^{ 34 }The LF(Ho) and LF(He) components for each patient were calculated between the first and second, and between the second and third, fields of the series (see Appendix). The within-subject factors were stimulus location and examination order. Custom software was used to convert and analyze the HFA data files (HFA Tools, 1989; Microsystems Technology

*,*Waterloo, Ontario, Canada).

^{ 23 }A negative value for the variance estimate,σ

_{ jk }

^{2}, corresponding to the LF(He) was obtained in cases in which the magnitude of the mean square estimate was considerably smaller than that of the variance estimate of the measurement error (σ

^{ 2 }; see Appendix). Similarly, a negative value for the variance estimateσ

_{ k }

^{2}was obtained when the mean square estimate corresponding to the LF(Ho) was small relative to that of LF(He). In such cases, the mean square estimates corresponding to the measurement error and to the particular negative variance estimate were combined to produce a pooled aggregate error, the magnitude of which better described the between-examination fluctuation. In all cases, the components of the LF were expressed in decibels as the square root of the given variance estimate.

*P*= 0.79), in terms of intraocular pressure (group mean 15.4 ± 2.21 mm Hg;

*P*= 0.110) and of age (group mean, 67.8 ± 7.14 years;

*P*= 0.532). The group mean difference between the long and short follow-ups for the MD was 1.32 ± 1.18 dB and for the CPSD was 2.85 ± 1.58 dB. All patients were experienced in automated perimetry having had numerous previous examinations. The exclusion criteria and optic nerve head evaluation were identical with that for the long follow-up.

*P*= 0.877; SF

*P*= 0.126; CPSD

*P*= 0.064), including all subtypes. The LF(Ho) and LF(He) components for each patient were then separately calculated between the first and second and the second and third fields of the series, using a procedure identical with that of the long follow-up.

*P*= 0.940; LF(He)

*P*= 0.426; error

*P*= 0.652.

*P*= 0.008), and this difference remained the same over both the first and second pairs of examinations (

*P*= 0.263). The discrepancy in the magnitude of the LF(Ho) between the two arms of the study can be attributed to the increased number of patients in the short follow-up exhibiting a measurement error that initially produced a negative value of the LF(Ho) component. To overcome this problem, the negative value was combined with the measurement error across the two examinations (see Appendix). The aggregate error was greater in the short-term follow-up (

*P*= 0.005).

*R*

^{2}) ranged from 1.0% to 32.5% with the larger values clearly attributable to the presence of obvious outliers. Little relationship was present for the corresponding functions between the LF(He) component and the modulus difference in the three types of CPSD. The

*R*

^{2}values ranged from 1.7% to 31.3%.

^{ 32 }However, this difference did not reach statistical significance for either component (LF[Ho]

*P*= 0.132; LF[He]

*P*= 0.082), regardless of examination pair (LF[Ho]

*P*= 0.671; LF[He]

*P*= 0.357) or length of follow-up (LF[Ho]

*P*= 0.622; LF[He]

*P*= 0.461).

^{ 23 }The disparity in the values between the Humphrey and Octopus perimeters is likely to be due in part to the differences in the thresholding algorithms,

^{ 35 }in the number of stimulus locations at which a double determination of sensitivity is undertaken, and in the stimulus parameters of the two instruments.

^{ 36 }

^{ 21 }

^{ 22 }

^{ 23 }that originally had ignored the presence of negative LF components. The treatment of negative values of the variance estimate is a notoriously challenging statistical issue. The procedure described here incorporated an approach for the handling of negative estimates of LF. This difference in methodologies accounts, to some extent, for the disparity between our results and those reported for the Octopus, because the effect of the modified technique is to skew the distribution toward a more positive value. Another approach would have been either to allocate a value of zero to the negative values or to omit such values from the summary distributions. The former was considered inappropriate, because it would have ignored the influence of the measurement error but would have resulted in smaller estimates, whereas the latter would not have been representative of the clinical reality. Alternative statistical techniques for approximating interval estimates have also been described.

^{ 37 }

^{ 38 }The probable effect of the method for dealing with negative estimates of variances used in the study is to overestimate the magnitude of those components of the LF that exhibit a negative variance. The consequence of this overestimation in relation to the identification of progressive visual field loss is the provision of a conservative standard for the definition of excessive long-term fluctuation.

^{ 29 }In any patient, the precision of the estimate of the LF for the field as a whole is affected by the degree of similarity in the position and depth of the field loss for the complete field compared with that manifested at the 10 locations. However, any sampling error derived from using the 10 locations is constant between any two tests of the same program for any patient. An alternative would be to increase the number of stimulus locations at which a double determination of threshold is routinely undertaken. The resultant LF would be more representative of the LF for the field as a whole but would be acquired at the expense of an increased examination time. The longer examination duration would produce a greater fatigue effect,

^{ 16 }particularly in the second eye examined, with the consequent reduction in the accuracy of the estimate of the true threshold. Inclusion of all double determinations of threshold consistent between any two consecutive fields would theoretically seem to offer an opportunity for a more representative estimate of the various LF components. However, these supplementary double determinations are seldom common between the second field of the given pair and the following field and, in any case, did not yield LF components that were statistically different in magnitude than those based only on the 10 double determinations.

^{ 32 }However, the LF(Ho) component derived by ANOVA is an expression of the variability of the measured height of the field based on a linear model involving an interaction term that allows for the inherent error in the measurement of threshold. Similarly, the LF(He) may appear similar to the difference in the CPSD between the two successive fields. However, the difference between the CPSDs provides a measure of alteration in the overall shape of the visual field, whereas the LF(He) component represents the difference in the variability of the measured shape of the visual field. That there is no relationship between the LF(Ho) component and the difference in the MD and also between the LF(He) and the difference in the CPSD emphasizes the distinction between the change in measurement (i.e., the index) and the variability present across the visual field between examinations. The magnitude of the LF components might also be expected to change as a function of the severity of field loss, because of ceiling and floor effects of the measured sensitivity; however, this was not the case. Because the LF cannot be inferred from the magnitude of the visual field indices, the importance of the LF in identifying progressive visual field loss cannot be overstated.

^{ 39 }

^{ 40 }

^{ 41 }for each pair of successive fields to identify the error distribution associated with the 90th percentiles (i.e., the empiric confidence limits). Bootstrapping is a statistical technique of sampling with replacement and increases the maximum information content from a given sample size. A sampling size of 27 was drawn and replaced 200 times for each LF component, and the 90% confidence limits were determined (Table 3) . The construction of a 95th percentile based on a sample of 30 patients was thought to be statistically inappropriate.

^{ 21 }was described by

*Y*is the threshold determination at each individual location; μ is the overall mean value of sensitivity, which is related to MD by an aggregate constant;

*L*

_{ j }is the effect of the examination locations;

*V*

_{ k }is the effect of each visual field examination

**;**

*LV*

_{ jk }is the interaction of

*L*

_{ j }and

*V*

_{ k }; and

*E*

_{ jkl }is the experimental error. The integers

*j*,

*k*, and

*l*represent, respectively, the number of considered stimulus locations with a double determination of threshold (

*j*= 1,2, … … … . .

*n*), the number of examinations (

*k*= 2), and the number of determinations of threshold at each considered location (

*l*= 2). The LF(Ho) component can be attributed to

*V*

_{ k }, the LF(He) component can be attributed to the interaction term

*LV*

_{ jk }, and the short-term variance across both examinations can be attributed to the experimental error (

*E*

_{ jkl }) normally distributed with an expected value of zero, such that

*E*

_{ jkl }is not correlated with any other error terms and is independent of the independent variables

*L*

_{ j },

*V*

_{ k }, and

*LV*

_{ jk }.

*V*

_{ k }and

*LV*

_{ jk }were then reduced to their constituent variances to remove the accompanying error variance

*E*

_{ jkl }by the hypothesis

^{ 34 }

_{ jk }) was calculated by substituting the mean square error of the error termσ

^{2}from equation 3 into equation 2 . The LF(Ho) component (σ

_{ k }) was calculated in a similar manner by substituting equation 2 into equation 1 .

_{ k }

^{2}andσ

_{ jk }

^{2}, corresponding to the LF(Ho) and LF(He) components, respectively, were found to be negative, the mean square estimate

*V*

_{ k }and

*LV*

_{ jk }of eitherσ

_{ k }

^{2}orσ

_{ jk }

^{2},was combined with the mean square estimate of

*E*

_{ jkl }to give a corrected error variance, σ̂

^{2}. The resultant root mean square of σ̂

^{2}was considered to be more representative of the actual aggregate error. This corrected error variance was achieved by adding the sums of squares of the original error estimate to the sums of squares of the source of the negative value

^{2}is the corrected aggregate error;

*SS*

_{error}and

*df*

_{error}are the sums of squares and degrees of freedom of the error variance, respectively; and

*SS*

_{ x }and

*df*

_{ x }are the sums of squares and degrees of freedom of the source of the negative value (i.e., either

*LV*or

*V*). Because the estimate of σ

^{2}becomes σ̂

^{2}, the interaction variance, σ̂, was recalculated by substitution into equation 2 . Becauseσ

^{2}= σ̂

^{2}, then

_{ jk }

^{2}was a more accurate assessment of σ

_{ jk }

^{2}from which to calculate the LF(He). Because the magnitude of the aggregate error variance masks the statistical effect of the negative value, the corrected aggregate error variance is the most representative value of the negative effect. If bothσ

_{ k }

^{2}andσ

_{ jk }

^{2}were found to be negative, then the combined error estimate calculation (equation 4) included the sums of squares of both

*V*

_{ k }and

*LV*

_{ jk }from which they are derived.

Long Follow-up | Short Follow-up | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Mean ± SD | Median | Range | Mean ± SD | Median | Range | |||||

LF(Ho) | ||||||||||

First pair | 1.72 ± 1.06 | 1.47 | 0.08–4.79 | 2.19 ± 1.04 | 2.18 | 0.38–5.29 | ||||

Second pair | 1.50 ± 0.75 | 1.32 | 0.37–2.96 | 1.95 ± 0.91 | 2.08 | 0.19–3.39 | ||||

LF(He) | ||||||||||

First pair | 2.02 ± 1.23 | 1.51 | 0.70–5.05 | 2.00 ± 1.27 | 1.76 | 0.33–5.32 | ||||

Second pair | 1.70 ± 1.03 | 1.50 | 0.41–5.16 | 2.05 ± 1.08 | 1.82 | 0.49–5.48 | ||||

Error | ||||||||||

First pair | 1.93 ± 0.87 | 1.74 | 1.00–4.79 | 2.31 ± 0.97 | 2.09 | 1.21–5.29 | ||||

Second pair | 1.80 ± 0.98 | 1.58 | 0.77–5.98 | 2.35 ± 0.84 | 2.20 | 1.24–5.48 |

Long Follow-up | Short Follow-up | |||||
---|---|---|---|---|---|---|

LF(Ho) | LF(He) | LF(Ho) | LF(He) | |||

First pair | ||||||

Ten | 2.00 ± 1.21 | 2.65 ± 1.48 | 2.05 ± 1.26 | 2.24 ± 1.02 | ||

All | 1.83 ± 1.37 | 2.75 ± 1.21 | 1.99 ± 1.11 | 2.21 ± 1.07 | ||

Second pair | ||||||

Ten | 1.74 ± 0.71 | 1.96 ± 1.19 | 1.76 ± 0.87 | 1.99 ± 1.06 | ||

All | 2.16 ± 1.19 | 2.54 ± 1.11 | 2.12 ± 1.03 | 2.78 ± 1.27 |

*n*and

*n*+ 1 of the series) and the designated second pair of fields (examinations

*n*+ 1 and

*n*+ 2 of the series) for the long and short follow-up.

Long Follow-up | Short Follow-up | |||||
---|---|---|---|---|---|---|

LF(Ho) | LF(He) | LF(Ho) | LF(He) | |||

First pair | 2.95 ± 0.38 | 3.70 ± 0.49 | 3.30 ± 0.26 | 3.60 ± 0.44 | ||

Second pair | 2.43 ± 0.36 | 3.00 ± 0.56 | 2.97 ± 0.23 | 3.40 ± 0.55 |

**Figure 1.**

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