Langenbucher Achim, Wendelstein Jascha, Cayless Alan, Olsen Thomas, Hoffmann Peter, Szentmáry Nóra
Department of Experimental Ophthalmology, Saarland University, Homburg, Saar, Germany.
Department of Ophthalmology, Ludwig-Maximilian University Munich, Munich, Germany.
Acta Ophthalmol. 2025 Sep;103(6):e422-e433. doi: 10.1111/aos.17522. Epub 2025 May 15.
The purpose of this study was to develop a method for evaluating intraocular lens (IOL) formula constant uncertainties using two modern statistical techniques-jackknife and bootstrap resampling.
Using two datasets (dataset 1: 888 eyes treated with the aberration correcting Hoya Vivinex IOL, dataset 2: 821 eyes with the spherical Alcon SA60AT/SN60AT IOL), formula constant uncertainties for the SRK/T (Aconst), Hoffer-Q (pACD), Holladay 1 (SF), simplified Haigis (a0) with preset a1/a2, Haigis (triplet a0/a1/a2), Castrop (triplet C/H/R) and Olsen formula (ACD) were evaluated. All input parameters were jackknife and bootstrap (N = 1000) resampled, and formula constants for each sample derived using nonlinear iterative optimisation techniques.
In single constant formulae where the constant acts directly on the effective lens position (Hoffer-Q, Holladay 1, simplified Haigis, Olsen), the formula constant in each case showed a standard deviation (SD) of about 0.01 with both jackknife and bootstrap sampling. The SRK/T Aconst showed a SD of about 0.018, and the Haigis and Castrop formulae with constant triplets showed large variations in the 3 constants (a0/a1/a2 about 0.036/0.005/0.002, C/H/R about 0.001/0.011/0.012). Direct formula reversion and solving for the formula constant yielded systematically larger SD values (Aconst/pACD,SF/a0/ACD = 0.586/0.395/0.403/0.324/0.304) with highly skewed distributions.
The distributions of formula constants with relevant benchmarks such as SD or confidence intervals can be derived with jackknife and bootstrap resampling techniques, offering potential advantages over direct formula reversion which yields skewed distributions, making central metrics such as the formula constant distribution mean unsuitable for constant optimisation.
本研究的目的是开发一种使用两种现代统计技术——刀切法和自助重抽样法来评估人工晶状体（IOL）公式常数不确定性的方法。
使用两个数据集（数据集1：888只接受了像差校正的Hoya Vivinex人工晶状体治疗的眼睛，数据集2：821只植入了球面Alcon SA60AT/SN60AT人工晶状体的眼睛），对SRK/T（Aconst）、Hoffer-Q（pACD）、Holladay 1（SF）、预设a1/a2的简化Haigis（a0）、Haigis（三联体a0/a1/a2）、Castrop（三联体C/H/R）和Olsen公式（ACD）的公式常数不确定性进行评估。对所有输入参数进行刀切法和自助法（N = 1000）重抽样，并使用非线性迭代优化技术得出每个样本的公式常数。
在常数直接作用于有效晶状体位置的单常数公式（Hoffer-Q、Holladay 1、简化Haigis、Olsen）中，使用刀切法和自助抽样法时，每种情况下的公式常数标准差（SD）约为0.01。SRK/T Aconst的标准差约为0.018，具有常数三联体的Haigis和Castrop公式在3个常数中显示出较大变化（a0/a1/a2约为0.036/0.005/0.002，C/H/R约为0.001/0.011/0.012）。直接公式反转并求解公式常数会产生系统上更大的标准差数值（Aconst/pACD、SF/a0/ACD = 0.586/0.395/0.403/0.324/0.304），且分布高度偏态。
通过刀切法和自助重抽样技术可以得出具有相关基准（如标准差或置信区间）的公式常数分布，与产生偏态分布的直接公式反转相比具有潜在优势，使得诸如公式常数分布均值等中心指标不适用于常数优化。
