The Long-Term Fluctuation of the Visual Field in Stable Glaucoma

Abstract

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.

Progressive visual field loss in primary open-angle glaucoma (POAG) is a fundamental indicator for change in the therapeutic management of the patient. However, the separation of true progression from the inherent fluctuation in the threshold estimate between any two consecutive visual field examinations currently represents a major clinical dilemma. 

The outcome of any given single visual field examination can be influenced by many factors including pupil size, ^{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). 

Knowledge of the LF has been considered a prerequisite for quantitative comparison of visual fields. 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). 

The Humphrey Field Analyser (HFA; Humphrey Systems, Dublin, CA.) has become a standard for automated perimetry. Two algorithms of the system (Full Threshold and Fastpac) undertake a double determination of threshold (Programs 30-2 or 24-2) at each of 10 specified stimulus locations within 21° eccentricity. The SF is calculated from these 10 specified locations, but the corresponding components of LF have never been specified. The two HFA algorithms also undertake additional double determinations of threshold in cases of decreased patient reliability or in cases of irregularity of the field loss between adjacent locations. ³² The effect on the magnitude of the LF components when these additional double determinations of sensitivity are included is also unknown. 

The overall purpose of the study was to determine, in stable POAG, the magnitude of the LF between two consecutive visual field examinations over a routine (long) follow-up and over a confirmatory (short) follow-up, thereby aiding the identification of progressive loss from fluctuation in sensitivity. The detailed goals were threefold. First, to compare the magnitudes and distributions of the LF(Ho) and LF(He) components between a pair of successive examinations over each of the two types of follow-up. Second, to determine the degree of similarity in the magnitudes of the components derived from the succeeding pair of examinations. Third, to determine the relationship between the magnitude of the LF components and the summary visual field indices. 

Methods

Two experimental protocols were used to determine the components of the LF. The first protocol was concerned with the derivation of the LF from pairs of consecutive fields separated by a routine (long) follow-up. The second protocol was concerned with the derivation of the LF from pairs of consecutive fields separated by a confirmatory repeat (short) follow-up. The subjects were drawn from a database of 118 patients. The research followed the tenets of the Declaration of Helsinki, informed consent was obtained from each subject after the nature and possible consequences had been explained, and the study had approval from the ethics committee of the Toronto Western Hospital. 

Long Follow-up

The protocol for the derivation of the LF over the long follow-up involved a retrospective study. The sample comprised 30 patients with stable POAG drawn from the Glaucoma Service of the Toronto Western Hospital and was selected to provide the broadest possible range of field loss. Three consecutive visual field examinations (HFA Program 30-2; stimulus size III), each separated by 6 to 9 months, were selected from the randomly assigned eye of each patient. The three fields were chosen on the basis that they were stable and had been preceded by at least two earlier examinations (to minimize the effects of learning) and that the optic nerve head appearance, intraocular pressures, and therapeutic modality remained unchanged for each patient during the period in question. All patients had a distance visual acuity of 6/12 or better during the selected period. Patients with clinically significant cataract, receiving pilocarpine therapy, or with ocular history in addition to glaucoma were excluded from the study. The sample was not controlled for the type of ocular medical therapy between patients, with the exception of pilocarpine. The mean age of the sample was 64.8 ± 11.4 (SD) years. The mean deviation (MD) at the initial presentation ranged from −0.2 to −23.7 dB (mean,− 8.3 ± 7.3 dB [SD]) and the pattern standard deviation in full (PSD) from 1.3 to 16.5 dB (mean, 6.7 ± 3.4 dB). The group mean intraocular pressure over the three examinations was 15.0 ± 2.3 mm Hg (SD). The controlled intraocular pressure did not vary by more than 3 mm Hg for any patient over the study course. The mean interval between each of the three examinations was 7.9 ± 1.3 months. 

The stability of the optic nerve head appearance had been evaluated using stereo observation by a single experienced clinician (GET) in terms of the occurrence of new or increased notching, new or increased saucerization, increased cup-to-disc area ratio, or new hemorrhage. The nerve head evaluation was undertaken with the investigator masked to the outcome of the visual field examinations. 

The usual protocol for examination of the visual field had been strictly followed, and all fields were deemed to be reliable in terms of the incidence of false responses to the catch trials. Stability of the visual field over each of the three examinations was confirmed from inspection of the Overview, Change Analysis, and Glaucoma Change Probability Analysis Statpac (Humphrey) printouts by one of the authors, experienced in visual field interpretation (JMW) who was masked to the outcome of the optic nerve head evaluation and statistically in terms of the stability of the visual field indices MD, corrected PSD (CPSD), and SF. 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). 

The LF(Ho) and LF(He) components were calculated by a two-factor analysis of variance (ANOVA) with replications that uses the sensitivities from the double determination of threshold at each of a given number of stimulus locations that are common to the two examinations. ^{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). 

Although the true values of the LF components can only be positive or zero, the formulae generally used to estimate these components allow negative values, and negative values of both components have been reported. 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. 

Short Follow-up

The protocol for the derivation of the LF over the short examination interval involved a prospective study. The sample comprised 30 patients with POAG selected from the database of the Glaucoma Service of the Toronto Western Hospital. Each patient was individually matched, as closely as possible, to a patient in the long follow-up on the basis of the appearance of the field defect in the designated eye (i.e., the number, severity, and location of the pattern deviation probability symbols and the magnitudes of the MD and CPSD; 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. 

Each patient underwent three field threshold tests (HFA Program 30-2; stimulus size III) in the designated eye over a period of 6 weeks with an interval of 3 weeks between each examination. The short follow-up examinations followed a protocol identical with that of the long follow-up and were performed by one experienced perimetrist. All fields were deemed to be reliable in terms of the incidence of false responses to the catch trials. 

The assessment of the visual field stability of the three fields over the 6-week period of the short follow-up was performed as described for the long follow-up by one author (JMW), who was masked to the study arm from which the fields were drawn and to all other clinical information. The respective ANCOVAs showed that the three separate indices were each stable over the three examinations (MD 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. 

Analysis

Stability of the LF(Ho) and LF(He) components and of the error term between the three successive examinations of each arm of the study were determined using a separate repeated measures analysis of covariance (ANCOVA) for each component. Age was considered as a covariate, examination order as a within-subject factor, and type of study as a between-subjects factor. The magnitudes of the LF(Ho) and LF(He) and of the error term were not significantly different over the three examinations in either arm of the study: LF(Ho) P = 0.940; LF(He) P = 0.426; error P = 0.652. 

Results

The group means, medians, SDs, and ranges for each component of the LF derived from the 10 locations featuring a standard double determination of threshold are shown in Table 1 for each arm of the study. The difference in the means between examinations was within ±0.35 dB for both components. The 95% limits of agreement for the between-examination difference in the LF(Ho) were± 2.31 and ±2.39 dB for the short and long follow-ups respectively, ± 2.36 and ±2.09 dB for the LF(He), and ±0.94 and ±2.40 dB for the aggregate error. The LF(Ho) was greater in the short follow-up than in the long follow-up (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). 

Little relationship was present for either follow-up between the LF(Ho) component derived from the 10 locations featuring a double determination of threshold and the modulus difference in each of the three types of MD for the corresponding pair of fields. The coefficients of determination (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%. 

Only 14 of the 30 patients in the long follow-up exhibited additional double determinations of threshold at stimulus locations that were common to the first and second examinations, and only 16 patients had additional double determinations at locations common to the second and third examinations. The corresponding figures for the short follow-up were 29 and 25 patients, respectively. The mean number of additional double determinations of threshold across pairs of examinations was 2.2 in the long follow-up group and 2.9 in the short follow-up group. The LF(Ho) and LF(He) components derived from all available double determinations were, in general, higher in magnitude than the corresponding components derived from the 10 standard double determinations of threshold (Table 2) . The higher group mean for the LF components derived from all available double determinations of threshold is to be expected, inasmuch as a second threshold determination is undertaken at a stimulus location exhibiting either decreased patient reliability and/or irregularity of the field loss between adjacent locations. 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). 

Little relationship was present for either follow-up between the LF(Ho) component derived from the 10 locations featuring a double determination of threshold and any of the three MD indices expressed as a mean of the corresponding two fields. The coefficients of determination ranged from 5.4% to 32.5% for the three types of MD. Little relationship was present between the LF(He) and any of the three CPSD indices expressed as a mean of the corresponding two fields: the coefficients of determination ranged from 14.5% to 32.7%. The aggregate error across the two examinations correlated moderately well with the mean SF across the two examinations for both follow-ups: the coefficients of determination ranged from 48.6% to 60.6%. Little relationship was present between the weighted or unweighted SF and either LF(Ho) or LF(He). The coefficients of determination ranged from 1.2% to 9.8% across the two follow-ups for the LF(Ho) component and from 5.5% to 42.9% for the LF(He) component. 

Discussion

The LF can be calculated for any combination of visual fields. The LF was calculated between pairs of consecutive visual fields because the identification of visual field progression is particularly difficult when the second field of any given pair of successive fields shows an apparent deterioration compared with that of the preceding field. The deterioration may be due to the LF or to change resulting from disease progression. Similarly, when the second of any given pair of fields exhibits an apparent improvement compared with that of the preceding field, the validity of either field is called into question. The calculation of numerous values of LF based on comparisons of multiple pairs of visual fields must be treated with caution, because each comparison has a finite probability of being incorrect. The overall effect of such probabilities becomes more apparent as the number of comparisons increases and, as such, the comparisons should be used sparingly to reduce the possibility of an incorrect inference. The most important comparison is always that of the current field with the immediate previous field. 

The mean values of the LF(Ho) and LF(He) were larger than those reported previously. The early theory relating to the components of the LF was based on results from Program JO of the Octopus automated perimeter (Interzeag, Schlieren, Switzerland), which determined the threshold twice at each of 49 stimulus locations within 26° eccentricity. With this perimeter, the values for the LF(Ho) and LF(He) were 0.5 and 0.2 dB, respectively, in normal subjects; 0.9 and 0.4 dB in those with suspected glaucoma, and 1.2 and 0.5 dB in those with glaucoma. 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  

The variance estimates were obtained using established methodology ^{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. 

Calculation of the LF for the standard algorithm of the HFA (Program 30-2 or 24-2) is effectively limited to the 10 standard double determinations of threshold situated within 21° eccentricity. The use of 10 stimulus locations merely samples the extent of the fluctuation occurring across the field on the assumption that it is representative of the field as a whole and potentially ignores the influence of the relatively greater variance at the more peripheral locations. 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. 

The LF(Ho) may appear similar to the difference in the MD between two successive examinations, because both measures represent descriptions of the change in sensitivity occurring across the entire field. The difference in the MD provides a measure of alteration in the overall height of the visual field and represents the difference between two linear averages of measured values distorted from the true average by a weighting function applied to the MD (adjusting for the variability of the threshold at each stimulus location). 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. 

The stability of the two LF components over both the short and long follow-ups, together with there being no relationship with the severity of field loss, suggests that the confidence limits associated with each LF component could be used as a reliability index of the relationship between any two consecutive fields. Such an index would be based on the variability inherent in the stable glaucomatous visual field, rather than on the basis of the variability inherent in the normal field. It can be argued that having confirmed the presence of glaucomatous field loss by reference to the properties of the normal visual field, glaucomatous progression should be evaluated by reference to the properties of the stable glaucomatous visual field. This latter approach has been adopted in the Glaucoma Change Probability Analysis of the HFA Statpac for Windows. ^{39 40}  

The resultant values for each component of the LF were therefore separately bootstrapped 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. 

It is proposed that the LF as a parameter for reliability between two examinations could be used in conjunction with those indicating reliability within each examination, particularly that of the SF. The specific type of between-examination variability—that is, general (LF(Ho) or location- and shape-specific LF(He)—would be indicated by the corresponding LF component lying outside the confidence limit. In the present study the most conservative estimate of the 90% confidence limit for the LF(Ho) was 3.3 dB and for the LF(He), 3.6 dB (Table 3) . The utility of the technique is illustrated in Figure 1 . Assuming that the outcome at the second field of a given pair (Fig. 1) occurs with equal frequency, knowledge of the LF would halve the required number of confirmatory fields. In all other outcomes, the LF would provide a basis from which to clinically classify fluctuation or progression. 

In conclusion, the LF derived from patients with stable glaucoma provides a statistical indicator for the necessity of a confirmatory follow-up field in glaucoma management. The procedure facilitates separation of fluctuation from true progression in the determination of progressive glaucomatous visual field loss. The various perimeter manufacturers should be encouraged to incorporate such a measure in their visual field printout. 

Appendix 1

The model for estimating the components of the LF ²¹ was described by  

\[Y_{jkl}{=}{\mu}{+}L_{j}{+}V_{k}{+}LV_{jk}{+}E_{jkl}\]

where 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 . 

The mean square estimates (MSE) for V _k and LV _jk were then reduced to their constituent variances to remove the accompanying error variance E _jkl by the hypothesis ³⁴  

\[MSE(V_{k}){=}jl{\sigma}_{k}^{2}{+}l{\sigma}_{jk}^{2}{+}{\sigma}^{2}\]

 

\[MSE(LV_{jk}){=}l{\sigma}_{jk}^{2}{+}{\sigma}^{2}\]

 

\[MSE(E_{jkl}){=}{\sigma}^{2}\]

 

The LF(He) component (σ _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 . 

Treatment of Negative Components of LF(Ho) and LF(He)

In cases in which either or both of the variance estimates,σ _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, σ̂². The resultant root mean square of σ̂² 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  

\[{\hat{{\sigma}}}^{2}{=}\ \frac{SS_{\mathrm{error}}{+}SS_{x}}{df_{\mathrm{error}}{+}df_{x}}\]

where σ̂² 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 σ² becomes σ̂², the interaction variance, σ̂, was recalculated by substitution into equation 2 . Becauseσ ² = σ̂², then  

\[{\hat{{\sigma}}}_{jk}^{2}{=}\ \frac{MSE(LV)-{\hat{{\sigma}}}^{2}}{l}\]

 

Therefore, σ̂ _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. 

 Supported by Medical Research Council of Canada Grant 11023 (JGF, GET).

 Submitted for publication September 14, 1999; revised March 2, 2000; accepted March 22, 2000.

 Commercial relationships policy: R (JW); F (JF); N (NH, MH, GT).

Table 1.

 

 

Group Data for LF Components

Table 1.

 

Group Data for LF Components

	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

 Data are in decibels and are for each of the LF components between the designated first pair of fields (examinations n and n + 1 of the series) and the designated second pair of fields (examinations n + 1 and n + 2 of the series) in the long follow-up and short follow-up.

Table 2.

 

 

Group Data for Double Determinations

Table 2.

 

Group Data for Double Determinations

	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

 Data are mean decibels ± SD for each LF component for those patients exhibiting additional determinations of threshold common between the designated first pair of fields (examinations 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.

Table 3.

 

 

Bootstrapped 90% Confidence Limits

Table 3.

 

Bootstrapped 90% Confidence Limits

	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

 Data are mean ± SD of the 90% confidence limits associated with the distribution of the LF(Ho) and LF(He) components for the designated first pair (examinations n and n + 1) and the designated second pair (examinations n + 1 and n + 2) of visual fields in the long and short follow-up.

Figure 1.

 

Flow diagram illustrating the utility of either component of the LF between any designated pair of consecutive fields for separating progressive loss from increased LF and for indicating the necessity of a confirmatory follow-up visual field examination. When the LF falls within the magnitude expected in a stable glaucomatous population, a confirmatory follow-up field is unnecessary. In all other cases, knowledge of the LF values provides a basis for the clinical identification of fluctuation or true progression.