**Purpose**:
One of the difficulties in modeling visual field (VF) data is the sometimes large and correlated measurement errors in the point-wise sensitivity estimates. As these errors affect all locations of the same VF, we propose to model them as global visit effects (GVE). We evaluate this model and show the effect it has on progression estimation and prediction.

**Methods**:
Visual field series (24-2 Full Threshold; 15 biannual VFs per patient) of 125 patients with primary glaucoma were included in the analysis. The contribution of the GVE was evaluated by comparing the fitting and predictive ability of a conventional model, which does not contain GVE, to such a model that incorporates the GVE. Moreover, the GVE's effect on the estimated slopes was evaluated by determining the absolute difference between the slopes of the models. Finally, the magnitude of the GVE was compared with that of other measurement errors.

**Results**:
The GVE model showed a significant improvement in both the model fit and predictive ability over the conventional model, especially when the number of VFs in a series is limited. The average absolute difference in slopes between the models was 0.13 dB/y. Lastly, the magnitude of the GVE was more than three times larger than the measureable factors combined.

**Conclusions**:
By incorporating the GVE in the longitudinal modeling of VF data, better estimates may be obtained of the rate of progression as well as of predicted future sensitivities.

^{1}One of the difficulties in modeling VF data is the large measurement variability of VFs, partially due to the inherent subjective nature of such a test.

^{2–4}This large variability means that in clinical practice, repeated measurements are performed to confirm real progression.

^{2,5–7}Furthermore, test–retest studies have shown that variability is dependent on defect depth and test location.

^{2}

^{8}In addition, this effect may differ between the first and second eye at the same visit. The number of false-negative answers have been shown to be higher in eyes with field loss.

^{9}It has also been shown that there is an inverse relationship between variability and sensitivity.

^{10}That is, there is a large amount of variability in eyes with severe damage.

^{11,12}Although these factors are statistically significant, they are rather small, and hence only explain a small part of the observed global variation in VFs. Junoy Montolio et al.

^{11}modelled the visit effect with these known factors. However, we speculate that other transient factors, such as fatigue, lack of concentration, or delayed reaction time may play a more important role. An example of the importance of these factors can be seen in Figure 1, where all locations have a drastic decrease in sensitivity in one of a series of visits. From the longitudinal profiles, it is evident that this decrease is caused by something that affected all VF measurements of that visit, rather than by actual damage.

**Figure 1**

**Figure 1**

^{13,14}All data is available through the Rotterdam Ophthalmic Data Repository at http://rod-rep.com. In brief, the patients were followed up approximately twice per year. The VFs were tested by using the Humphrey Field Analyzer (Carl Zeiss Meditec, Dublin, CA, USA) with the 24-2, white-on-white test strategy by means of the Full Threshold algorithm. The response variables of interest were the sensitivity estimates from the 52 VF points (excluding the 2 points that correspond to the blind spot). All patients gave their written informed consent for participation. The research procedures followed the tenets set forth in the Declaration of Helsinki. We excluded VFs with unknown reliability as indicated by the instrument. Additionally, to simplify the evaluation of the statistical models, we excluded individuals with less than 15 measurements (in either eye). For those individuals with more than 15 measurements, only the first 15 measurements were included in the analysis. The resulting data set consisted of 250 eyes from 125 individuals, resulting in 3750 VFs and 195,000 location-specific sensitivity estimates. Descriptive statistics can be found in Table 1.

**Table 1**

^{15}Hence, the analysis was done by using a Bayesian hierarchical mixed-effects model.

^{16–18}We modeled the hierarchical structure of the data using four levels, namely, (1) the individual, (2) the eye, (3) the hemisphere, and (4) the location. An example of the mixed-effects model for the four level data structure can be seen in Figure 2A. Furthermore, censoring was taken into account at 0 dB,

^{19}due to the limitation of the device.

^{20}We will refer to this model as the conventional model. To account for the visit-dependent offset at all locations, or GVE, we included a parameter in the model to capture the offset at every visit for each eye within each individual. Hence, this effect accounts for factors that affect all measurements belonging to the same eye at each visit. The impact of this additional parameter is demonstrated in Figure 2B. This model will be referred to as the GVE model.

**Figure 2**

**Figure 2**

^{21}which allowed us to simplify the computation by splitting the hierarchical model at the individual level. Hence, individuals were analyzed independently before combining them at the population level. Figure 3 illustrates the hierarchical structure divided into the two stages. A full description of the models and the computational procedure is given by Bryan et al.

^{22}

**Figure 3**

**Figure 3**

^{th}percentile of the absolute errors, which is the value below which 95% of the absolute prediction errors may be found, was also computed to compare the models. Nonparametric Wilcoxon (matched paired when applicable) tests were performed to determine whether the differences between the models were significant.

^{th}percentile of the absolute errors. An example of the fits for one eye can be seen in Figure 4.

**Figure 4**

**Figure 4**

**Figure 5**

**Figure 5**

^{th}percentile of the absolute errors.

**Figure 6**

**Figure 6**

^{11}showed that the time of day, season, reliability indices (number of fixation losses, false negatives, and false positive), technical experience, and follow-up period have a clinically relevant influence on the MD test results. To determine the magnitude and importance of the GVE, we compared it with these factors. We excluded technical experience and follow-up period due to the lack of data on these factors. Hence, we focussed on the time of day, season, and the reliability indices. For time of day, the tests were stratified into four categories: performed before 10 AM, between 10 AM and 12 PM, between 12 PM and 2 PM, and after 2 PM. For season, the tests were also stratified into four categories, of 3 months each (March–May, June–August, September–November, December–February), based on the annual variation of retinal sensitivity.

^{11}The reliability indices were treated as continuous variables. Reliability indices include the percentage of fixation losses, the proportion of false positives, as well as the proportion of false negatives. We will refer to this model as the fixed-factor model. An example of the model fits for one location can be seen in Figure 7. We compared the model fits using the RMSE, MAE, and 95

^{th}percentile of the absolute errors. Furthermore, we determined the magnitude of the GVE compared with the factors by calculating their absolute means. A limitation of the two-stage approach occurs when there is sparse data, such as the season or time of day. Because each individual was analyzed separately, information could not be borrowed from the data set as a whole as done in the classical one-stage approach. Due to this limitation, we used the classical one-stage approach including 50 randomly selected individuals for this analysis.

**Figure 7**

**Figure 7**

^{th}percentile of the absolute errors for the models, showing that by incorporating the GVE there is an improvement in the model fit. Both the squared errors and absolute errors were significantly smaller for the GVE model than for the conventional model (

*P*< 0.001, matched paired).

**Table 2**

*P*< 0.001, matched paired). Figure 8 shows the distribution of the differences, including the mean and 95% CI.

**Figure 8**

**Figure 8**

^{th}, 12

^{th}, and 15

^{th}measurement, respectively). The GVE model showed a significant improvement in the predictions compared with the conventional model, irrespective of how many measurements were used (

*P*< 0.001, matched paired). However, the difference between the models predictive abilities decreased as more measurements were included. For the conventional model, there was a significant difference between including three and six measurements (

*P*< 0.001) and between including six and nine measurements (

*P*< 0.001). For the GVE model, these differences were not significant (

*P*= 0.08 and

*P*= 0.47, respectively).

**Table 3**

^{th}percentile of the absolute errors for each model. By including the factors, the model fit was slightly improved compared with the conventional model (

*P*< 0.001, matched paired). The improvement in the fit was much larger for the GVE model compared with both the conventional (

*P*< 0.001, matched paired) and the fixed-factor model (

*P*< 0.001, matched paired).

**Table 4**

**Table 5**

^{11}Of the known factors, we found season to have the largest effect, with a mean absolute value of 0.13 dB. In agreement with our results, Junoy Montolio et al.

^{11}concluded that the number of false positive answers has the largest, or most severe, effect out of the reliability indices. In their study, the MD was determined to be overestimated by 1 dB per 10% of false positive answers. Time of day was found to have a mean absolute value of 0.09 dB. It has, however, been shown that the 24-hour IOP rhythm differs between eyes in glaucoma patients.

^{23}Because the patient's IOP may affect each eye differently at the same time of day, and will vary between individuals, it may be of more interest to determine this factor at an eye-specific level, as done with the GVE. Although the GVE takes both measureable and unmeasurable factors into account, including the known factors allows for some explanation of the variation. Hence, a combination of both the GVE and the known factors may be beneficial in the modeling of VF data.

^{20}The 85

^{th}percentile is used to compensate for effects by for instance cataract, which would lead to a general reduction of retinal sensitivity throughout the VF. Hence, the entire VF height is adjusted to the 85

^{th}percentile. However, with diffuse loss, the entire VF height tends to be overcorrected whenever the 85

^{th}percentile becomes significantly affected.

^{24}The two approaches differ in how the visit effects are expressed. Namely, the percentile correction is treated as a fixed term, while the GVE is treated as a random effect. Because we treat the GVE as a random effect, which has a distribution with mean zero, it is forced to fluctuate around zero. The fixed term of the percentile model does not impose any constraint, and hence accounts for both the visit effect and the rate of progression/slope. Hence, in contrast to pattern deviation analysis, the GVE model allows us to estimate the visit effect without disrupting the estimation of the progression.

^{25}Our model is in line with these findings and confirms that results, which deviate from what is expected may be due to unknown factors that affect the VF measurements on that specific visit rather than representing actual damage.

**S.R. Bryan**, None;

**P.H.C. Eilers**, None;

**E.M.E.H. Lesaffre**, None;

**H.G. Lemij**, Carl Zeiss Meditec (C);

**K.A. Vermeer**, None

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