Table 1.
Descriptive statistics of the entire dataset with mean, standard deviation (SD), median, minimum and maximum, 5%, and 95% quantiles (90% confidence intervals).
Table 2.
Prediction error (×100) as the difference between achieved spherical equivalent and formula predicted spherical equivalent for the formulae under test.
Fig 1.
Prediction error PE as the deviation of achieved spherical equivalent from the formula predicted refraction.
Formula constants have been optimized on the training dataset (N = 1017) and cross-validated on the test set (N = 435). The upper part of the plot shows the approximated kernel density distribution for the 6 formulae under test, and the lower part the scatter of the PE together with the median (circle), the quartiles (black line), and the 90% confidence interval (5% and 95% quantile, blue line). We see that after optimization of the formula constants on the training data none of the formulae shows a systematic offset in the prediction error. The blue lines in the lower plot representing the 90% confidence interval show that the variation on the test data is largest with the Hoffer-Q formula and lowest with the Castrop formula.
Table 3.
Absolute value of the prediction error (×100) as the difference between achieved spherical equivalent and formula predicted spherical equivalent for the formulae under test.
Fig 2.
Absolute value of the prediction error absPE as the deviation of achieved spherical equivalent from the formula predicted refraction.
Formula constants have been optimized on the training dataset (N = 1017) and cross-validated on the test set (N = 435). The upper part of the plot shows the approximated kernel density distribution for the 6 formulae under test, and the lower part the scatter of the absPE together with the median (circle), the quartiles (black line), and the 90% confidence interval (5% and 95% quantile, blue line). The blue lines in the lower plot representing the 90% confidence interval show that the absPE is largest with the Hoffer-Q formula and lowest with the Castrop and the SRKT formula.
Fig 3.
Performance of formula outcome in terms of absolute prediction error absPE.
Formula constants have been optimized on the training dataset (N = 1017) and cross-validated on the test set (N = 435). The lines indicate the number of cases within the limits of absolute prediction error for the 6 different formulae under test, the closer the lines to 1 the more cases within limits. The ticks on the x axis indicate the typical thresholds of ±0.25, ±0.5, ±1.0 dpt as found in most publications. The plot indicates that between the limits of around 0.3 to 0.8 dpt of absolute prediction error the performance of the formulae under test show some differences, where the Castrop formula (solid line in red) appears to have a better performance compared to the Hoffer-Q formula (dashdotted line in dark green).
Fig 4.
Ratio of cases within limits of absolute prediction error absPE for the 6 formulae under test.
Formula constants have been optimized on the training set (N = 1017) and cross-validated on the test set (N = 435). The thresholds of ±0.25, ±0.5, ±1.0 dpt are used in accordance with the majority of publications on formula performance. For all 3 thresholds shown in this plot, the ratio of cases within limits shows a maximum difference of 5% between all 6 formulae.