Square Root Transformation

In the past, application of the square root transformation of GA area for assessment of GA growth rate has been associated with a number of benefits, such as (1) it reduces test–retest variability, (2) it creates uniformity in intergrader differences across a range of lesion sizes, and (3) it reduces baseline GA size dependency from the GA growth rate. Furthermore, the square root transformation is not affected by zeroes and extremely small values.

Yehoshua et al.23 investigated the reproducibility of GA area measurements and enlargement rate of GA, including usefulness of the square root transformation. They found that it eliminated GA baseline size dependency from the GA growth rate. The correlation between lesion size and test–retest standard deviations was significant with respect to original GA area (Pearson's r = 0.60, P < 0.001; Spearman's ρ = 0.73, P < 0.001). However, when a square root transformation of the lesion area measurements was performed prior to test–retest standard deviation calculations, the correlation between baseline lesion size and test–retest standard deviations was no longer apparent (Pearson's r = 0.07, P = 0.72; Spearman's ρ = 0.12, P = 0.51).

Pfau et al.24 quantified lesion progression using a linear mixed-effects model with two-level random effects (i.e., eye- and patient-specific effects) and shape-descriptive factors. The authors normalized the variables for the lesion area, perimeter, and circularity using the square root transformation.24 Monés and Biarnés25 assessed the progression of GA and its baseline using the square root transformation, with both Pearson's r and Spearman's ρ in the assessment. They plotted the relationships using linear regression with locally weighted scatterplot smoothing curves; one plot compared GA area growth (mm²/year) against baseline GA area (mm²), whereas the other plotted radial growth (mm/year) against the square root transformed baseline. They found the correlation between radial growth and square root–transformed baseline area was negative (Pearson's r = –0.30, P = 0.0005; Spearman's ρ = –0.25, P = 0.0042), which suggests that as lesions grow larger, the progression rate starts decreasing.25 Domalpally et al.26 studied a parameter—the Geographic Atrophy Circularity Index (GACI)—in the assessment of GA progression. They used regression analysis to assess the relationship between baseline characteristics and annual progression rates of GA. Similar to Monés and Biarnés,25 they found statistically significant correlations between GACI and growth rate in mm² (r = –0.31, P < 0.001) and GACI and square root–transformed measurements (r = –0.39, P < 0.001).

In this study, we investigated six regression models against two outcomes: (1) the original scaled, untransformed total GA area (with a unit of mm²/year); and (2) the square root–transformed GA area (with a unit of mm/year). We used the following criteria to determine the strength of the square root transformation for the cohort: (1) whether the square root transformation normalized the distribution of residuals from the regression models tested; (2) if the transformation linearized the growth rates, as would be expected from this type of transformation; and (3) if the transformation significantly improved the fit of the model as compared with its untransformed, original-scaled counterpart. It is important to note that, in regression analysis, the assumption of normality applies to the residuals only. The distribution of independent and dependent variables can be skewed if the residuals of the regression model are normally distributed.