**Purpose.**:
To reduce variability and improve measurements of true change signal in visual field (VF) assessments through the use of filters that combine functional and structural test results.

**Methods.**:
Humphrey VF data (Swedish Interactive Thresholding Algorithm [SITA] Standard, 24-2) and confocal scanning laser ophthalmoscopy (Heidelberg Retina Tomograph [HRT]) data from 1057 eyes of 637 participants were used to derive a filter. Another dataset, consisting of VF and HRT data from 112 eyes of 62 participants each with ≥5 visits, was used to test the filter. At each VF location per eye, the trend over time was modeled by a linear model (LM), and a nonlinear model (NLM), using filtered or unfiltered data, but with the last visit excluded. The SD of residuals from the trends, and prediction errors (PE) for the last visit were compared between filtered and unfiltered data. The filter was reconstructed and analyses were repeated after truncating VF data so that thresholds < 19 dB were replaced by 19 dB to reduce noise.

**Results.**:
The SD of the residuals at all 52 VF locations for all analyses was reduced by filtering (*P* < 0.001). The PE was reduced by filtering at 43 and 47 VF locations (*P* < 0.05) for LM analyses on observed and truncated data, and all 52 VF locations (*P* < 0.05) for both NLM analyses. Truncating data before filtering reduced variability (*P* < 0.01) at 41 and 40 VF locations for LM and NLM analyses.

**Conclusions.**:
Filtering can reduce variability about trends in longitudinal sequences of VF data, and improves the accuracy of predicting the next test result.

^{ 1 }demonstrated that early medical treatment can delay or prevent the onset of primary open-angle glaucoma (POAG) among high risk individuals with statistically raised IOP. In addition, it is known that eyes with damaged visual fields are more likely to display more rapid progression.

^{ 2,3 }Therefore, it is important that glaucoma is identified at an early stage and managed appropriately.

^{ 4–11 }but sometimes functional damage can occur without apparent structural damage assessed using conventional methods.

^{ 7,12,13 }However, this observed lack of concordance may be due to variability in test results rather than a true time lag alone.

^{ 14 }Although the association between functional damage (as measured in SAP) and structural damage (with RNFL thickness and rim area [RA] measurements) is modest at best,

^{ 7,15,16 }it is important as clinicians put great emphasis on mutually confirming functional and structural test results.

^{ 17–20 }Variability obscures the identification of true glaucomatous progression in VF and makes detection of progression challenging. Therefore, reducing variability is particularly important

^{ 19 }as the change in a glaucomatous VF must be greater than the noise about the measurement before it becomes statistically distinguishable.

^{ 21,22 }Reducing variability may be possible through improved methods of data acquisition, or by postprocessing currently available data.

^{ 23 }One attractive feature of postprocessing of existing data is that it requires no additional testing time. One approach to processing existing data is to exploit the associations known to exist between sensitivities at neighboring VF locations, and between regions of the ONH and certain VF locations through a process of filtering.

^{ 24,25 }and Fitzke et al.

^{ 26 }first applied this technique in a novel fashion to VF data in the mid-1990s. Unlike the pixel values in digital images, VF sensitivities have associations that are shaped by the anatomy of the RNFL axon paths. Consequently, Gaussian filters, which ignore retinal anatomy, can give misleading results.

^{ 23,27 }A more novel filter that was designed to reduce noise without obscuring true defects through mimicking the physiological relationship between VF test locations was proposed by Gardiner et al.

^{ 23 }In that study, the filter was constructed as a weighted average of the actual and predicted sensitivity at each VF test location, where the weightings were determined by the predictive power of each test location through examining the covariances between sensitivities among all pairs of test locations. The filter appeared to reduce the noise in glaucomatous VF data, and improved the sensitivity and specificity of determining VF changes through simulating progressing and stable visual fields, respectively. Another filter proposed by Strouthidis et al.

^{ 28 }incorporated structural data into the construction of a spatial filter, which is based on a multiple regression predictive model. The model predicted scaled functional correlations at each test locations from the angular distance between all possible pairings of test points within the retina and the optic nerve head using the anatomic map described by Garway-Heath et al.

^{ 29 }Strouthidis et al.

^{ 28 }reported that this technique resulted in a similar filter to that derived by Gardiner et al.

^{ 23 }The current work proposes a filter to incorporate data from structural and functional tests, while also using techniques to remove spurious associations between locations to simplify the filter and better reflect the true anatomy of the RNFL projections to the optic nerve head.

^{ 30 }suggested using 30° ONH sectors with the first sector centered on the fovea-ONH axis for general clinical use, since it is robust to the effects of small inaccuracies in mapping individual VF locations to the ONH, and imperfections in imaging and VF measurements.

*j*(

*y˜*) was constructed in three steps. Firstly, a linear regression model using the cross-sectional dataset was fitted to predict sensitivity at location

_{j}*j*, namely (

*ε*∼

_{j}*N*(0,

^{31–33}on structural and functional tests have suggested that the change in sensitivity when expressed on a logarithmic scale in decibels (dB) appears to be proportional to the logarithm of the percentage loss of retinal ganglion cells, for which rim area is a structural surrogate. All 12 sectors of the optic disc were used, instead of restricting the analysis to one-half of the optic disc (as was done for the VF locations), to allow for the possibility that VF locations do not necessarily correspond solely with one half of the optic disc, in particular when near the horizontal raphe

^{34}or when between the optic disc and the fovea.

^{35}The regression coefficients

*β*s were constrained to be non-negative based on the assumption that if sensitivity at one particular location is high then it should not predict that the sensitivity at another location will be low. The estimation of

_{i}*β*, denoted by

_{i}*β̄*, was determined by least absolute shrinkage and selection operator (LASSO).

_{i}^{36}The LASSO often is used for automatic variable selection when predictors are highly dimensional and correlated. The method shrinks the coefficients of noninformative variables to zero by penalizing on the sum of the absolute size of the regression coefficients. To simplify the filter and avoid overfitting, a subset of the eight best predictors was selected through LASSO for each VF location.

*j*(

*ŷ*) was defined as where

_{j}*ᾱ*and

*β̄*were obtained from the filter constructed on the cross-sectional data and predictors

_{i}s*j*was defined as the average of the observed and predicted sensitivity, such that

*y˜*= 0.5(

_{j}*y*+

_{j}*ŷ*).

_{j}*j*using the first to second last visits (visits 1: [

*n*− 1]), and

*t*is the time for each corresponding visit. Then, the LM and NLM analyses were repeated for filtered data for each eye. Use of a nonlinear model for trend analyses is supported by a recent study that found that mean deviation in dB appears to decline exponentially over time.

^{37}The SDs of the residuals from the trends over time served as a measure of variability that is unaffected by the possibility of progression over time for that eye. This was compared between observed and filtered data sequences. Prediction error (PE), which is defined as the deviation between the observed sensitivity at the last visit (visit

*n*) and the sensitivity that would be predicted by extrapolating the trend from visits 1 to [

*n*− 1], also was compared. The PE has been used widely as a surrogate of the performance of new techniques to detect change in series of VF measures and was first suggested by McNaught et al.

^{38}As a formal comparison, a Wilcoxon matched-pairs one-sided test between the filtered and unfiltered data was performed on the SD of the residuals and the absolute values of the PE (|PE|).

^{ 39 }concluded that sensitivities estimated to be between 0 and approximately 15 to 19 dB during SAP perimetry are unreliable, and are only weakly correlated with true functional status within that range. Therefore as a secondary analysis, we used 19 dB as the lower limit of the reliable stimulus range of standard perimetry to improve the reliability of sensitivities. The filter was reconstructed using the cross-sectional dataset after setting thresholds < 19 dB equal to 19 dB. Then all analyses were repeated after applying this change to the longitudinal dataset in an additional effort to reduce noise.

**Table 1**

**Table 1**

Longitudinal Dataset | Cross Sectional Dataset | |||||

Mean | SD | Range | Mean | SD | Range | |

Series length, n tests | 8.1 | 3.1 | 5–16 | 1.8 | 1.2 | 1–8 |

Age | 61.0 | 11.3 | 34.5–88.5 | 45.1 | 15.4 | 12–96 |

Sensitivities from SAP, dB | 30 | 4 | <0–45 | 27 | 7 | <0–50 |

Rim area from HRT, μm^{2} | 1,540,000 | 363,010 | 476,000–2,620,000 | 1,270,000 | 475,000 | 193,000–4,150,000 |

^{ 40 }

**Figure 1**

**Figure 1**

**Figure 2**

**Figure 2**

*P*< 0.001). The mean and 95% confidence limits (in brackets) of SDs for the filtered and unfiltered data though LM analysis are 1.06 (0.25, 3.42) and 1.53 (0.37, 4.68). Overall the SDs were reduced by 0.46 by filtering on average, and the 95% confidence limits for the difference was (−0.27, 1.74). Similarly, boxplots (Supplementary Figs. S3.1–S3.3) which compared SD of the residuals between filtered and unfiltered data from a NLM analysis, and between filtered and unfiltered data when the filter was constructed using truncated data using either LM or NLM analyses, showed that the SD of residuals always were significantly smaller at all VF locations after filtering. A formal comparison using Wilcoxon signed-rank tests of matched pairs of SD between filtered and unfiltered data shown in Table 2 suggests that filtering reduces variability of residuals about the trend at all 52 VF locations (

*P*< 0.001 for all analyses).

**Figure 3**

**Figure 3**

**Table 2**

**Table 2**

Hypothesis | Base Data | Model | Wilcoxon Signed-Rank Test |

SD of residuals of filtered data | Observed data | LM | 52 of 52 P < 0.001 |

< | NLM | 52 of 52 P < 0.001 | |

SD of residuals of unfiltered data | Truncated data | LM | 52 of 52 P < 0.001 |

NLM | 52 of 52 P < 0.001 |

*P*< 0.05). Through LM analysis, the mean and 95% confidence limits (in brackets) of PE for the filtered and unfiltered data are 1.67 (0.05, 7.12) and 1.86 (0.05, 7.55). The overall PE were reduced by 0.20 dB on average by filtering, and the 95% confidence limits for the difference was (−1.28, 2.02 dB). Similarly, analyses using truncated data suggested that PE were reduced at 47 of 52 VF locations after filtering (Supplementary Fig. S4.1; Table 3). Using NLM analyses, with either observed or truncated data, PE were significantly reduced (

*P*< 0.05) at all 52 VF locations after filtering (Supplementary Figs. S4.2, S4.3; Table 3).

**Figure 4**

**Figure 4**

**Table 3**

**Table 3**

Hypothesis | Base Data | Model | Wilcoxon Signed-Rank Test |

|PE| of filtered data | Observed data | LM | 43 of 52 P < 0.05 |

< | NLM | 52 of 52 P < 0.05 | |

|PE| of unfiltered data | Truncated data | LM | 47 of 52 P < 0.05 |

NLM | 52 of 52 P < 0.05 |

^{ 23 }(SG) and Strouthidis et al.

^{ 28 }(NS), and a function-only (F) filter, which was derived similarly but using functional data only rather than combining structural and functional data. Results of comparison based on the reduction of SDs of residuals from the LM analysis and the PE were shown in Table 4. It is seen that including structural measures in the filter (i.e., SF versus F) improves fit at 27 locations and PE at 6 locations. Model fits and PE were not made worse by applying filtering at even a single VF location. Perhaps more interestingly, it is seen that while the filters of SG and NS may provide smaller residuals (indicating that that they fit the data better) at many locations, the new filter generally has smaller PE, which implies that our new filter is not merely fitting the noise, but is revealing more of the true change signal.

**Table 4**

**Table 4**

SF vs. SG | SF vs. NS | SF vs. F | ||||

SG | SF | NS | SF | F | SF | |

SD | 33/52 | 11/52 | 34/52 | 13/52 | 0/52 | 27/52 |

PE | 1/52 | 47/52 | 3/52 | 17/52 | 0/52 | 6/52 |

*P*< 0.05.

*P*< 0.01) at 41 and 40 VF locations for the LM and NLM analyses, respectively.

**Figure 5**

**Figure 5**

**Table 5**

**Table 5**

Hypothesis | Base Data | Model | Wilcoxon Signed-Rank Test |

SD of residuals of truncated data | Filtered | LM | 41 of 52 P < 0.05 |

< SD of residuals of observed data | NLM | 40 of 52 P < 0.05 |

^{ 41–43 }that integrated structural and functional test results to improve the detection of glaucoma tended to adopt a Bayesian approach. Unlike the conventional frequentist approach of using ordinary least squares (OLS) to perform VF trend analyses, the Bayesian approach has the flexibility of adding more variables to an inference model and strengthens posterior beliefs (the reader's belief in the parameters conditional on the data) when prior beliefs (the reader's belief in an uncertain parameter before evaluating the results of a study) are informative. Medeiros et al.

^{ 41 }used a joint multivariate mixed-effects model within a Bayesian hierarchical modeling framework to integrate information from the SAP VF index (VFI) and average RNFL thickness from scanning laser polarimetry (SLP) for a cohort of participants. Their study used noninformative priors to model the random effects and updated the posterior estimate in each patient through information from the whole sample. It found that the method improved detection of glaucoma progression compared to standard OLS linear regression of VFI measurements alone. Russell et al.

^{ 42 }applied a Bayesian normal error linear regression on a sample that consisted of patients with ocular hypertension (OHT). The model used individual imaging measurements of global neuroretinal rim area from HRT, as an informative prior, to predict changes in perimetric mean sensitivity over time. Compared to the OLS linear regression approach for mean sensitivity, the Bayesian method suggested VF progression rates can be estimated more accurately. However, both methods used global indices/measurements, such as VFI and mean sensitivity for trend analyses, and did not attempt to exploit the spatial correlations between VF locations. VF evaluations based on global indices are more likely to overlook focal damage than evaluations based on pointwise data. Zhu et al.

^{ 43 }used RNFLT from SLP to predict pointwise VF sensitivities from SAP using a BRBF (radial Basis function customized under a Bayesian framework) approach, which improved predictions of pointwise VF sensitivities compared to OLS linear regression. Nonetheless, those Bayesian approaches rely on structural measurements that relate closely to function. The filtering method proposed in this study, which leverages the physiological relations between VF locations and ONH regions, appears to be advantageous in reducing variability and improving predictability, even when the structure–function association appears poor.

**Figure 6**

**Figure 6**

^{ 30,44,45 }have used a computational model that combines an individual eye's structural and functional test data with simple biometric data to derive individually customized maps relating each VF location to the ONH. It is possible to refine the filter developed in this paper to a “personalized” filter based on individual customized mapping of VF locations to the ONH. However, it recently has been reported that the benefit of such personalization on the structure–function relation is marginal (Ganeshrao S, et al.

*IOVS*2014;155:ARVO E-Abstract 962). Another possible reason for the variability in how different eyes benefit from filtering could be that some individuals in our longitudinal dataset have more follow-up tests than others. To evaluate the impact of longitudinal sequence length on filtering, two groups of data were created, one with sequence length between five and seven tests, the other with eyes having sequence length greater than seven (to balance the number of eyes within each group, so 61 eyes have shorter sequence and 51 eyes have longer sequence). The overall mean of the SDs of residuals for the shorter sequences was 1.61 using unfiltered data and 1.13 using filtered data, while for longer sequences these values were 1.44 and 0.99, respectively. Notably, filtering improved the results regardless of sequence lengths. Another possible explanation for the differences between individuals is that perhaps some individuals are worse test takers than others, providing less reliable measurements.

^{ 27 }An example of a small nasal step defect is shown in Supplementary Figure S7. It can be seen in this particular example that our filter was able to capture VF deterioration occurring within a small, localized defect; however, there may be cases in which change occurring within a very small defect is “blurred out,” especially if the defect is surrounded by healthy, stable VF locations. Therefore, it is recommended that filtered and unfiltered data be examined, especially when the two approaches produce inconsistent results with respect to glaucomatous VF progression.

^{ 23–26,28 }and provides better prediction of change in series of pointwise VF sensitivities when compared to using unfiltered or raw VF data alone. However, the true impact of filtering applied to series of VF data still remains untested. Filtering only improves predictions “on average.” Most VF series are very noisy, so any dampening of this noise also will improve the prediction on average. Since most VF series gathered during clinical testing are relatively stable with few rapidly deteriorating cases,

^{ 46 }filtering is likely to yield clinical benefits by reducing the likelihood that a glaucomatous visual field will falsely be deemed progressing.

**L. Deng**, None;

**S. Demirel**, None;

**S.K. Gardiner**, Allergan (C)

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