**Purpose**:
The primary objective of this research is to examine how precision in intraocular lens calculation formulas can be impacted by zeroing the mean error through adjustments in the effective lens position value. Additionally, the study aims to evaluate how this modification influences outcomes differently based on the source of the prediction error.

**Methods**:
In order to analyze the impact of individual variables on the standard deviation, the study maintained all variables constant except for one at a time. Subsequently, variations were introduced to specific parameters, such as corneal curvature radius, keratometric refractive index, axial length, and predicted implant position.

**Results**:
According to our findings, when zeroing the mean error is applied to correct for inaccuracies in corneal power estimation, it results in a significant and exponential rise in standard deviation, thus adversely affecting the formula's precision. However, when zeroing is employed to compensate for prediction errors stemming from axial length measurements or predicted implant position, the effect on precision is minimal or potentially beneficial.

**Conclusions**:
The study highlights the potential risks associated with the indiscriminate but necessary zeroing of prediction errors in implant power calculation formulas. The impact on formula precision greatly depends on the source of the error, underscoring the importance of error source when analyzing variations in the standard deviation of the prediction error after zeroing.

**Translational Relevance**:
Our study contributes to the ongoing effort to enhance the accuracy and reliability of these formulas, thereby improving the surgical outcomes for cataract patients.

_{k}), typically set at 1.3375, also referred to as the Javal Index. This method of computing corneal power has found utility in various applications, including IOL power estimations. The continued usage of the Javal Index is primarily due to its compatibility with established IOL constants in most calculation formulas. However, the Javal Index may overestimate the corneal power, resulting in inaccuracies in the calculated IOL power.

^{3}

^{–}

^{8}Some IOL power calculation formulas use a specific value for the refractive index (Haigis: 1.3315, Holladay: 4/3). Similarly, systematic errors can arise when a new or different device or technique is used for ocular biometry measures, such as estimating the axial length using the group refractive index or sum of segments to optimize axial length (AL).

^{9}Any of these variations lead to potential errors contributing to the sign and magnitude of the average prediction error by IOL power calculation formulas, whose constants may need to be adjusted to restore their accuracy.

^{10}provided convincing arguments that PE's SD of the prediction error is the single most accurate measure of outcomes. This metric provides an accurate assessment and forecasts other measures like the percentage of cases within a given range (e.g., ±0.50), the mean absolute deviation, and the median. The computation of the SD is performed after constant optimization to cancel any systematic bias.

^{11}The SD of the error is based on the distance of each data point from the mean, not the actual value of the mean. Hence, the impact of the zeroization of the mean prediction error on the SD of the error is not predictable without additional information because the SD depends on the individual values of the error, not the mean of the error.

^{12}From this work, we derived analytical methodology from identifying relationships and predicting the impact of systematic errors on keratometry and AL measurement, the keratometric index estimation, and the effective implant position prediction. We focused on how these factors influenced the change in SD following the zeroization of the predicted error.

*ELP*) so that the refractive change produced (Δ

*R*) is such that

_{i}*R*) incurred by an incremental change in ELP (Δ

_{i}*ELP*) can be estimated using a differential calculation and a thick lens pseudophakic eye model:

*P*is the IOL power, and

_{i}*K*is the corneal power of a given eye of the data set.

_{i}*ELP*

_{0}) necessary to cancel the systematic bias \(\bar{E}\) is given by

*P*

_{i}^{2}+ 2

*K*) over the

_{i}P_{i}*N*eyes of the data set.

^{2}= (

*A*+

*B*+

*C*), where

*P*

_{i}^{2}+ 2

*K*), and

_{i}P_{i}*E*and (

_{i}*P*

_{i}^{2}+ 2

*K*).

_{i}P_{i}^{13}was less than 0.05 D. This ensures that the mean prediction error and its SD are negligible at baseline. This configuration is assumed to correspond to the situation obtained thanks to a “perfect” theoretical calculation formula (negligible prediction error) postoperatively. In such a situation, a systematic change in any of the numerical values of the variables or indices used by the formula would alter the value of the predicted refraction of each eye. This variation in predicted refraction can be likened to a prediction error caused by an inaccurate estimation or measurement of one of these variables and enables computation of its specific impact on the formula's accuracy and precision before and after zeroization.

_{k}) and the effective lens position (ELP)—are examined. Systematic variations in the ELP are performed to investigate the influence of lens design changes, such as a constant shift in haptic position relative to the IOL's optic. Additionally, by introducing incremental adjustments to the initial keratometric index (n

_{k}= 1.3315) utilized in the Haigis formula, prediction errors are generated for each alteration. This approach allows us to evaluate the theoretical impact of systematic shifts on the accuracy of the implant power prediction formula and gain valuable insights into the precision of the formula concerning changes in this particular variable, which directly affects corneal power estimation.

*P*

_{i}^{2}+ 2

*K*) and its covariance with (

_{i}P_{i}*E*) to the postoptimization total variance of the PE.

_{i}*n*= 4042), and group 2 comprised the eyes with an absolute value of the PE < 0.05 D (

*n*= 541). The Mann–Whitney

*U*test was used to compare the distributions of the main variables of interest. A

*P*value of less than 0.05 indicated a statistically significant difference between the groups (see Table).

_{k}was adjusted in increments that ranged from −0.005 to +0.005, with each step involving a change of ±0.001.

_{k}) cause a minor increase in the SD prior to neutralization, yet this grows significantly after zeroization. Conversely, positional variables (ELP, AL) show a reversed trend.

^{14}These design variations cause a displacement of the principal planes of the implant and are therefore analogous to an ELP prediction error. We found that zeroization was an important step to reduce the impact of optical design changes, as it induced a marked reduction in the value of the SD. Our results show that the beneficial effect of zeroization cannot, however, be generalized to all sources of prediction errors. Introducing a systematic error of the AL produces an increase in the SD, which is minimally influenced by the zeroization of the mean prediction error.

*P*

_{i}^{2}+ 2

*K*). The term C appears to calculate a form of covariance between (

_{i}P_{i}*E*) and (

_{i}*P*

_{i}^{2}+ 2

*K*). Covariance measures how two variables move together—if they tend to increase and decrease together, the covariance is positive, and if one tends to increase while the other decreases, the covariance is negative. So if mean PE is positive, and

_{i}P_{i}*P*and

*K*are always positive, the covariance could be positive if higher values of (

*E*) often correspond to higher values of (

_{i}*P*

_{i}^{2}+ 2

*K*) and negative if lower values of (

_{i}P_{i}*E*) often correspond to lower values of (

_{i}*P*

_{i}^{2}+ 2

*K*).

_{i}P_{i}*E*) and (

_{i}*P*

_{i}^{2}+ 2

*K*). has the same sign as the mean error (C < 0) or not (C > 0). In our simulations, the sign of C was negative for the variables tested. This is explained by the impact of the systematic change in the variables on the sign of the mean prediction error and that of the covariance between (

_{i}P_{i}*E*) and (

_{i}*P*

_{i}^{2}+ 2

*K*). For the variables studied, the greater the covariance between (

_{i}P_{i}*E*) and (

_{i}*P*

_{i}^{2}+ 2

*K*), the more negative the value of C, which reduces the impact of the increase in variance related to B.

_{i}P_{i}*E*, and (

_{i}*P*

_{i}^{2}+ 2

*K*), where

_{i}P_{i}*P*represents the implant power and

*K*the corneal power. The SD of (

*P*

_{i}^{2}+ 2

*K*) remains invariant for a given data set. Despite this, it is challenging to link the covariance to the correlation between (

_{i}P_{i}*E*) and (

_{i}*P*

_{i}^{2}+ 2

*K*) as the variation of the SD of (

_{i}P_{i}*E*) cannot be easily anticipated. While we can generally anticipate some correlation between covariance and the correlation coefficient, it is essential to understand that this is not a direct relationship. Covariance measures the degree to which two variables move together, but it does not account for the individual variances of the variables as the correlation coefficient does. Thus, although both statistical measures describe the relationship between two variables, they are not directly interchangeable, and any presumptions between them should be made with a comprehensive understanding of their unique characteristics and limitations.

_{i}*E*) and (

_{i}*P*

_{i}^{2}+ 2

*K*) for the eyes of our data set as a function of the variable causing the error. The sign of the covariance and the intensity of the correlation between these variables illustrate the observed trends.

_{i}P_{i}^{15}in 2021 revealed that frequently used IOL formulas in contemporary clinical practice were unable to consistently generate precise postoperative refractive outcomes in eyes with a history of CRS. With the majority of formulas accessible on the American Society of Cataract and Refractive Surgery (ASCRS) IOL calculator, less than 70% of the eyes achieved refractive prediction errors within ±0.5 D. Recently, advanced technologies and methods such as total keratometry, ray tracing, intraoperative aberrometry, and machine learning–based algorithms have shown substantial potential in enhancing IOL power computations in eyes previously subjected to corneal refractive surgery. This demonstrated improvement is particularly evident when compared to traditional vergence-based formulas like SRK/T, Haigis, Holladay, Hoffer Q, Potvin–Hill, and Barrett.

^{16}

^{–}

^{19}These conventional methods, which estimate corneal power based on the anterior corneal radius and keratometric index, appear less effective in comparison. In our study, we strategically chose to keep all variables constant, with the exception of the one. This strategy was specifically devised to facilitate the examination of its impact on the standard error, considering both variance and bias components. This approach is especially beneficial when dealing with intricate models such as pseudophakic eyes with thick lenses, where numerous variables interact. By simplifying the model this way, we aimed to isolate the influence of some variables of interest. Understanding the effects of individual variables can yield valuable insights before introducing further complexity. The approach outlined in our research provides a solid foundation for future studies. A subsequent study could harness the power of computational tools like Monte Carlo simulations. This technique, known for its robustness in addressing complex stochastic systems, would be especially valuable in exploring the intertwined effects of multiple variables. It would enable us to examine the interactive behavior of the variables in a probabilistic framework, providing a more nuanced understanding of their impact on the outcome. A recent study used a Monte Carlo simulation approach to analyze a large data set of preoperative IOLMaster 700 (Jena, Germany) biometric measurements, consisting of 16,667 observations, to assess the uncertainty in the predicted refractive outcome (REFU) following cataract surgery.

^{20}This uncertainty was evaluated in light of measurement uncertainties in modern optical biometers. The study employed the Haigis and Castrop formulas, using error propagation strategies to derive REFU. Results showed that without considering lens power labeling tolerances, the median REFU was 0.10 dpt for the Haigis formula and 0.12 dpt for the Castrop formula. The REFU systematically increased for short eyes or high-power IOLs. The largest contributor to REFU, particularly in long eyes and low-power IOLs, was the uncertainty in measuring the corneal front surface radius when not considering lens power labeling tolerances. The study concluded that the uncertainty of biometric measures contributes to about one-third to one-half of the SD compared with the published data on the formula prediction error of refractive outcome after cataract surgery.

*E*) and (

_{i}*P*

_{i}^{2}+ 2

*K*), the more zeroization is likely to degrade the formula's accuracy through an increase in SD. Still, Equations (12) and (13) can be used in a real clinical situation to estimate the impact of zeroization on the formula's precision. Calculating the covariance between the prediction error and the IOL power as well as comparison of the relative values of B and C could provide interesting insight into the origin of the variables causing the prediction error. In essence, the prediction of a marked increase of the SD after zeroization suggests that the source of the prediction errors could be mainly caused by inaccurate estimation of the corneal power or any other variable causing prediction errors that are not correlated with (

_{i}P_{i}*P*

_{i}^{2}+ 2

*K*). Several studies suggest that the value of the keratometric index (n

_{i}P_{i}_{k}) used by the formula for computing the IOL power is overestimated, leading to an overestimation of corneal power.

^{3}

^{–}

^{8}Compensation for all or part of this error can be accomplished by adjusting constants, but our results suggest that this comes at the cost of degradation in the precision of the formula. This trend is likely to increase for eyes undergoing corneal refractive surgery, for which the correct estimation of corneal power raises a challenge.

*P*

_{i}^{2}+ 2

*K*) could exhibit substantial fluctuations within a specific subgroup, and this variation could potentially distort the findings and interpretations. Therefore, a thorough comprehension of these intricacies is essential for accurately evaluating the performance and practicality of various implant power calculation formulas.

_{i}P_{i}**D. Gatinel**, BVI Medical (C);

**G. Debellemanière**, None;

**A. Saad**, None;

**A. Wallerstein**, None;

**M. Gauvin**, None;

**R. Rampat**, None;

**J. Malet**, None

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*Acta Ophthalmol*. Published online June 23, 2023, doi:10.1111/aos.15726.