Background Bootstrapping is a modern technique mostly used in statistics to evaluate
the robustness of model parameters. The purpose of this study was to develop a method
for evaluation of formula constant uncertainties and the effect on the prediction
error (PE) in intraocular lens power calculation with theoretical-optical formulae
using bootstrap techniques. Methods In a dataset with N = 888 clinical cases treated
with the monofocal aspherical intraocular lens (Vivinex, Hoya) constants for the Haigis,
the Castrop and the SRKT formula were optimised for the sum of squared PE using nonlinear
iterative optimisation (interior point method), and the formula predicted spherical
equivalent refraction (predSEQ) and the PE were derived. The PE was bootstrapped NB
= 1000 times and added to predSEQ, and formula constants were derived for each bootstrap.
The robustness of the constants was calculated from the NB bootstrapped models, and
the predSEQ was back-calculated from the NB formula constants. Results With bootstrapping,
the 90% confidence intervals for the a0/a1/a2 constants of the Haigis formula were
-0.8317 to -0.5301/0.3203 to 0.3617/0.1954 to 0.2100, for the C/H/R constants of the
Castrop formula they were 0.3113 to 0.3272/0.1237 to 0.2149/0.0980 to 0.1621, and
for the A constant of the SRKT formula they were 119.2320 to 119.3028. The back-calculated
PE from the NB bootstrapped formula constants standard deviation for the mean/median/mean
absolute/root mean squared PE were 5.677/5.735/0.401/0.318 e-3 dpt for the Haigis
formula, 5.677/5.735/0.401/0.31829 e-3 dpt for the Castrop formula and 14.748/14.790/0.561/0.370
e-3 dpt for the SRKT formula. Conclusion We have been able to prove with bootstrapping
that nonlinear iterative formula constant optimisation techniques for the Haigis,
the Castrop and the SRKT formulae yield consistent results with low uncertainties
of the formula constants and low variations in the back-calculated mean, median, mean
absolute and root mean squared formula prediction error.
