REGRESSION MODELS: STRENGTHS AND LIMITATIONS

As highlighted, regression splines demonstrated promising predictive performance, particularly in their ability to model nonlinear effects in biometric parameters. However, we also observed that random forest regression suffered from overfitting, limiting its generalizability. While machine learning techniques such as random forests can excel in capturing complex relationships within the training set, their performance deteriorates when applied to independent datasets. This underscores the importance of balancing predictive power with model robustness. Thank you for raising this important point regarding the limitations of random forest models in our study. We acknowledge that while random forest regression exhibited strong in-sample performance, its generalizability to out-of-sample predictions was suboptimal, highlighting the classical challenge of overfitting. This is particularly relevant when working with smaller datasets or datasets that do not fully represent the entire biometric spectrum. However, our study aimed to compare the performance of existing regression-based models rather than to develop new ensemble learning techniques. While methods such as Bayesian frameworks, dropout regularization, or hybrid approaches could theoretically mitigate overfitting, their implementation would require a separate line of investigation. Furthermore, hybrid models incorporating decision trees alongside theoretical optics-based approaches have not yet demonstrated clear superiority over current best-performing formulas in clinical settings. Instead of solely focusing on random forest models, we prioritized exploring the role of nonlinear regression splines, which provided superior out-of-sample performance while maintaining interpretability. Our findings suggest that regression splines can serve as a more practical and clinically relevant alternative to random forest regression in the context of IOL power calculation. Future research could indeed explore advanced ensemble approaches, particularly in cases where large-scale, diverse datasets are available to mitigate overfitting risks.

EXTERNAL VALIDATION AND FORMULA ACCESSIBILITY

Regarding the absence of external validation, we agree and explicitly named this among our limitations. While external validation is always desirable, our study prioritized rigorous formula constant optimization, which was performed with careful attention to minimizing bias. Many previous studies may have optimized formula constants on their test dataset rather than employing a dedicated training set, leading to biased conclusions and a lack of reproducibility. We would also like to emphasize that the IOL model itself—particularly how its power steps are realized—may act as a confounding factor. Although the labeled power steps are uniformly spaced (eg, in 0.5 D increments), the optical or structural modifications used to achieve these steps, such as changes in curvature, central thickness, or haptic design, may follow nonlinear patterns. This can affect the lens behavior in ways that are not immediately apparent from the nominal power and may pose additional challenges for external validation, especially when comparing different calculation philosophies.

Importantly, our study did not propose a new formula nor aim to provide a direct calculation tool. Rather, it was designed as a theoretical comparison of different underlying calculation principles.

Additionally, we do not believe it is necessary to validate proprietary formulas that are not fully disclosed. The authors of newer-generation formulas have deliberately chosen not to make their methodologies publicly available, thereby precluding their use for scientific comparisons. Our study protocol made significant efforts to standardize formula constant adaptation, but when formula developers explicitly decide against transparency, exclusion from comparative studies becomes inevitable.

PERFORMANCE OF NEW-GENERATION FORMULAS IN LONG VS SHORT EYES

Recent advancements in IOL power calculation have significantly improved predictions for long eyes, reducing the historical challenges associated with myopic axial lengths. However, short eyes continue to pose difficulties, as predictive models require sufficient data coverage across extreme biometric ranges. Any dataset with gaps in these regions will inevitably lead to decreased prediction accuracy. This highlights the importance of comprehensive datasets when training empirical models, as missing data in edge cases directly translates to limitations in predictability.

HYBRID APPROACHES AND EMPIRICAL COMPONENTS

Our study discusses the limitations of hybrid approaches in our section on study limitations. In fact, there is no purely theoretical-optical strategy of lens power calculation! Based on assumptions (eg, refractive indices) or models (consideration of corneal power from conversion factors such as keratometer index) all calculations based on a pseudophakic model eye (all vergence formulae as well as raytracing) require (empirical) prediction of the axial IOL position. This applies to modern formulas like PEARL-DGS and Kane as well, which are not significantly more “hybrid” than traditional vergence-based formulas. For instance, PEARL-DGS follows a disclosed structural framework, making it more transparent than other newer-generation formulas. Whether any performance advantage stems from the hybrid approach itself, the nature and quality of the training datasets, or from specific empirical fine-tuning remains unclear and cannot be definitively answered based on current evidence.

ROLE OF FORMULA CONSTANT OPTIMIZATION

A key focus of our study was to rigorously investigate the impact of formula constant optimization. We systematically examined how the composition of the optimization group influences final outcomes, comparing:

• a)
  

  An optimization group based on a random subset.

  
  
• b)
  

  An optimization group with ‘normal’ axial lengths.

  
  
• c)
  

  An optimization group enriched with short eyes.

  
  
• d)
  

  An optimization group enriched with long eyes.

To our knowledge, no prior study has performed such a comparative analysis. As previously noted, many published studies introduce bias by optimizing their test dataset rather than applying adjustments to a separate training set, leading to artificially favorable results. Avoiding such biases is crucial for ensuring the validity of predictive performance assessments.

CONCLUSION

We appreciate the opportunity to further elaborate on these aspects and welcome continued discussion on the evolution of IOL power calculation methodologies. Ensuring methodological rigor and prioritizing transparency remain central to advancing this field.

CRediT authorship contribution statement

Liam D. Redden: Validation. Birgit Grubauer: Validation. Peter Hoffmann: Supervision. Achim Langenbucher: Validation. Kamran M. Riaz: Supervision. Damien Gatinel: Validation. Helga Wagner: Validation. Jascha A. Wendelstein: Writing – original draft, Supervision, Project administration.