Langenbucher Achim, Szentmáry Nóra, Cayless Alan, Wendelstein Jascha, Hoffmann Peter
Department of Experimental Ophthalmology, Saarland University, Homburg, Germany.
Dr. Rolf M. Schwiete Center for Limbal Stem Cell and Aniridia Research, Saarland University, Homburg, Germany.
Acta Ophthalmol. 2023 May;101(3):e264-e274. doi: 10.1111/aos.15277. Epub 2022 Oct 26.
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.
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.
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.
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.
自举法是一种现代技术，主要用于统计学中评估模型参数的稳健性。本研究的目的是开发一种方法，使用自举技术评估理论光学公式在人工晶状体屈光力计算中公式常数的不确定性及其对预测误差（PE）的影响。
在一个包含888例使用单焦点非球面人工晶状体（Vivinex，豪雅）治疗的临床病例的数据集中，使用非线性迭代优化（内点法）对Haigis公式、Castrop公式和SRKT公式的常数进行优化，以获得平方和PE，并得出公式预测的球镜等效屈光度（predSEQ）和PE。对PE进行1000次自举抽样，并将其加到predSEQ上，然后为每次自举抽样得出公式常数。根据1000个自举抽样模型计算常数的稳健性，并根据1000个公式常数反算出predSEQ。
通过自举法，Haigis公式的a0/a1/a2常数的90%置信区间为-0.8317至-0.5301/0.3203至0.3617/0.1954至0.2100；Castrop公式的C/H/R常数的置信区间为0.3113至0.3272/0.1237至0.2149/0.0980至0.1621；SRKT公式的A常数的置信区间为119.2320至119.3028。根据1000个自举抽样公式常数反算出的PE的标准差，对于Haigis公式，平均/中位数/平均绝对/均方根PE分别为5.677/5.735/0.401/0.318×10⁻³屈光度；对于Castrop公式，分别为5.677/5.735/0.401/0.31829×10⁻³屈光度；对于SRKT公式，分别为14.748/14.790/0.561/0.370×10⁻³屈光度。
我们通过自举法证明，对Haigis公式、Castrop公式和SRKT公式进行非线性迭代公式常数优化技术，可产生一致的结果，公式常数的不确定性较低，反算出的平均、中位数、平均绝对和均方根公式预测误差的变化也较小。
