Improved prediction of rates of visual field loss in glaucoma using empirical Bayes estimates of slopes of change

Abstract

Purpose: To describe and test a new methodology for estimation of rates of progressive visual field loss in glaucoma.

Methods: This observational cohort study enrolled 643 eyes of 368 patients recruited from the Diagnostic Innovations in Glaucoma Study, followed for an average of 6.5±2.0 years. The visual field index was used to evaluate degree of visual field loss in standard automated perimetry. Growth mixture models were used to evaluate visual field index changes over time. Empirical Bayes estimates of best linear unbiased predictions (BLUPs) were used to obtain slopes of change based on the first 5 visual fields for each eye. These slopes were then used to predict future observations. The same procedure was done for ordinary least squares (OLS) estimates. The mean square error of the predictions was used to compare the predictive performance of the different methods.

Results: The growth mixture model successfully identified subpopulations of nonprogressors, slow, moderate, and fast progressors. The mean square error was significantly higher for OLS compared with the BLUP method (32.3 vs 13.9, respectively; P<0.001), indicating a better performance of the BLUP method to predict future observations. The benefit of BLUP predictions was especially evident in eyes with moderate and fast rates of change.

Conclusions: Empirical Bayes estimates of rates of change performed significantly better than the commonly used technique of OLS regression in predicting future observations. Use of BLUP estimates should be considered when evaluating rates of functional change in glaucoma and predicting future impairment from the disease.

Figure 1

Distributions of empirical Bayes estimates (BLUPs) and ordinary least squares (OLS) slopes of VFI change.

Figure 2

Relationship between differences in absolute values of residuals for OLS and BLUP predictions and time. Positive values indicate that residuals for the OLS method were greater than those from the BLUP method.

Figure 3

Scatterplot with marginal distributions of observed minus predicted values of VFI according to BLUP and OLS methods for moderate and fast progressors. Negative values indicate that predicted values were higher (i.e., more normal) than observed values and, therefore, underestimated visual function loss. A reference (45°) line is also plotted.

Figure 4

VFI results for an eye followed over time. Black and gray continuous lines show OLS and BLUP regression lines, respectively, calculated from the first 5 VFI measurements. The eye was classified as non-progressor. The OLS slope suggests worsening over time, whereas the BLUP slope is close to zero. Dashed lines show predictions from the calculated slopes of change. Predictions from BLUP were closer to actual observed values than those from OLS.

Figure 5

VFI results for an eye followed over time. Black and gray continuous lines show OLS and BLUP regression lines, respectively, calculated from the first 5 VFI measurements. The eye was classified as moderate progressor. The OLS slope suggests improvement over time, whereas the BLUP slope suggests worsening. Dashed lines show predictions from the calculated slopes of change. Predictions from BLUP were closer to actual observed values than those from OLS.