**3. Results**

Demographic data for a total of 621 eyes are listed in Table 2. AL ranged from 21.41 to 30.60 mm, with a mean of 24.08 ± 1.54 mm; ACD ranged from 2.02 to 4.29 mm, with a mean of 3.20 ± 0.41 mm; and K ranged from 40.30 to 49.28 diopter, with a mean of 44.12 ± 1.42 diopter. ELP ranged from 3.67 to 8.76 mm, with a mean of 5.16 ± 0.63 mm. The three preoperative parameters and ELP in the four subgroups are listed in Table 3.

**Table 2.** Demographic data in this study.


Figure 1 shows the relationship between AL and the other two variables used in structural equation models. When all the 621 eyes were analyzed at once, AL and ACD showed a positive correlation, and AL and K showed a negative correlation (r = 0.588; *p* < 0.001 and −0.362; *p* < 0.001, respectively). However, in the subgroup analysis, both parameters showed significant correlations when AL was 24.5 mm or shorter (all *p* < 0.001). R<sup>2</sup>

Figure 2 demonstrates structural equation models with the highest value among 24 cases in four subgroups. When AL was shorter than 23.0 mm, the model where both

K and ACD acted as the moderating factor (Model 2 from PROCESS macro) showed the highest R<sup>2</sup> value (0.217, *p* < 0.001, Figure 2a). When AL ranged from 23.0 to 24.5 mm, the R<sup>2</sup> value (0.217, *p* < 0.001) was highest with Model 15 (K as a mediating variable, ACD as a moderating variable in both the process from AL to ELP and the process from K to ELP, Figure 2b). In the range of AL between 24.5 mm and 26.0 mm, unlike the above, when ACD acts as a mediating variable and K acts as a moderating variable in the processes of influencing ELP, the R<sup>2</sup> value (0.220, *p* < 0.001) is highest (Figure 2c). Figure 2d shows the model when AL was longer than 26.0 mm. The predictability is highest (R<sup>2</sup> value = 0.401, *p* < 0.001) with K as a mediating variable and ACD as a moderating variable.


**Table 3.** Demographic data in 4 subgroups classified according to preoperative axial length (AL).

ACD = anterior chamber depth; K = mean corneal dioptric power; ELP = effective lens position; D = diopter.

**Figure 1.** *Cont*.

**Figure 1.** The relationships between axial length (AL) and other variables for structural equation models. (**a**) Total 621 eyes; (**b**) AL ≤ 23.0 mm; (**c**) 23.0 mm < AL ≤ 24.5 mm; (**d**) 24.5 mm < AL ≤ 26.0 mm; (**e**) AL > 26.0 mm. K = mean corneal dioptric power; ACD = anterior chamber depth.

Table 4 shows regression formulas derived from a multiple linear regression analysis using AL and ACD in a total of 621 eyes and conditional process analysis.

The mean ELP prediction error and the predictive accuracy from the above two analysis methods are listed in Table 5. The results from conditional process analysis yielded lower standard deviation (SD) of mean ELP prediction error, lower SD of mean prediction error, lower median absolute error and lower mean absolute error compared with results from a multiple regression

and ±1.00 diopter.

 test. It also produced higher

**Figure 2.** The structural equation models for the prediction of effective lens position (ELP) in each range of axial length (AL). (**a**) AL ≤ 23.0 mm; (**b**) 23.0 mm < AL ≤ 24.5 mm; (**c**) 24.5 mm < AL ≤ 26.0 mm; (**d**) AL > 26.0 mm. K = mean corneal dioptric power; ACD = anterior chamber depth.

**Table 4.** Regression formulas for prediction of effective lens position according to preoperative axial length.


ACD = anterior chamber depth; AL = axial length; K = mean corneal dioptric power; ELP = effective lens position.

percentages

 within ±0.25, ±0.50,


**Table 5.** Predictive outcomes derived from the Haigis formula and conditional process analysis.

ELP = effective lens position; D = diopter.

## **4. Discussion**

The results of this study demonstrated that the optimal structural equation models, consisting of preoperative parameters for the prediction of ELP, were different according to preoperative AL. The regression equations derived from the conditional process analysis could be developed into an IOL calculation formula with high predictive accuracy.

Recently, the Barret Universal II formula, the EVO (Emmetropia Verifying Optical) formula, the Hill-RBF (radial basis function) formula, and the Kane formula have been introduced, and the accuracy of these new formulas has been reported to be improved compared to existing ones [17]. Unfortunately, the detailed mechanism of these new formulas is not known. In particular, both the Hill-RBF and the Kane formula are well known for using artificial intelligence algorithms. In addition, the possibility of IOL calculation formulas using multilayer perceptron, which is another form of artificial intelligence, has been suggested [18]. Even if the design mechanism of artificial intelligence is clearly disclosed, artificial intelligence algorithms usually have multiple hidden layers, so surgeons cannot understand the detailed calculation process [19]. This effect is called the "black box effect" and has been pointed out as a disadvantage in equations through artificial intelligence. Of course, the accuracy of IOL power calculation through artificial intelligence is already high, and there is no doubt that it will develop further in the future. However, through the results of this study, we would emphasize that the accuracy of the formula can be improved through conditional process analysis, and that the information on the detailed calculation process can be clearly provided to anyone.

The formula that produces high accuracy for postoperative refractive outcomes differs according to preoperative AL. When the AL is markedly short or long, the accuracy of the IOL calculation formula is lower than that in eyes with AL in the normal range. The Hoffer Q formula was more accurate than the other formulas in cases of eyes with short AL (AL < 22.0 mm) [20,21]. Wang et al. advocated the use of the Haigis formula for the determination of IOL power in myopia with long AL [22], so we divided a total of 621 eyes into four subgroups in 1.5 mm increments according to AL.

The correlation between postoperative refraction error and AL and K has also been studied in patients that underwent cataract surgery after refractive surgery. Recently, an advanced lens measurement approach (ALMA) was proposed to improve the accuracy of postoperative refraction error by Rosa et al. [23]. They showed the improvement of R Factor [24] and ALxK methods [25] by applying ALMA, which is a mixed theoretical regression method based on the SRK-T formula.

Almost all theoretical formulas for IOL power calculation are based on the use of a simplified eye model with a thin cornea and an IOL model. [26]. With this approach, the power of the IOL can be easily calculated using the Gauss equation in paraxial optics. [27]. ELP is back-calculated by "predicting" the effective ACD value with the actual postoperative refraction of a given data set. Therefore, ELP is formula-dependent and does not need to consider the real postoperative IOL position in terms of the eye's anatomy [28]. Models based on statistically analyzed relationships between some or all of the previously men-

tioned preoperative measurements of the eye and postoperative IOL position have been used to predict ELP in preoperative settings. In 1975, Fyodorov et al. [29] derived an equation based on the individual eye's keratometry and AL to estimate ELP. Third-generation formulas, including the Hoffer Q, [27] Holladay 1, [28] and SRK/T formulas, [30] use AL and K to predict ELP and IOL power calculation, and the main difference among these formulas is the predicted value of ELP. As ACD values can be measured accurately after the development of slit-scan technology, a fourth-generation formula, the Haigis formula, was developed to estimate ELP with the AL and ACD values, [31,32]. The commonality of the various formulas used from the past to the present is that the AL is considered to be the most important factor in ELP prediction. Therefore, in this study, the AL was set as a constant independent variable in the ELP prediction process.

Sheard et al. concluded that the SRK/T formula has non-physiologic behavior that contributes to IOL power prediction errors [9]. Specifically, Reitblat et al. found that the SRK/T formula induced myopic results in eyes with a mean K greater than 46.0 diopter, and hyperopic results in eyes with a mean K lower than 42.0 diopter [33]. In contrast, the Haigis formula, which does not consider corneal steepness during ELP calculation, causes myopic outcomes in flat corneas. This tendency has also been proven in large-scale research by Melles et al. [6]. However, previous studies have concluded that there is no significant association between mean K and postoperative IOL position [11,24]. In this study, we attempted to investigate the effects of K on ELP and to determine why the conclusions of the above-mentioned studies are controversial. We found a highly predictable model by setting K as variables that mediate or moderate the action of AL.

In general, ACD is positively correlated with AL, and this was reconfirmed in this study. In other words, ACD and AL inevitably have the problem of multicollinearity. However, the results of this study documented the correlation between the two parameters varies depending on the range of AL. ACD, like K, also acted as a mediator or a moderator depending on the AL and has been found to be an essential element in ELP prediction.

There are some limitations in this study. We did not evaluate other factors such as lens thickness (LT) or corneal diameter. Recently, there has been a growing interest in the thickness of the crystalline lens and LT is considered in newly developed IOL power calculation formulas. Norrby et al. concluded that LT was not an essential factor and ACD alone would predict the postoperative IOL position accurately [11]. In the above study, they used a partial least squares (PLS) regression test and LT, as an independent variable, may not be as effective. However, it could act as a factor that mediates or modulates the effect of ACD, and it is also expected that this will further improve the accuracy of the equation. Corneal diameter was significantly correlated with postoperative IOL position in another study [34]. Therefore, it is thought that corneal diameter can act as a mediator in the process from AL to ELP by itself (direct effect), or can act as a factor that mediates or moderates the effect of K (indirect effect). Corneal asphericity is another candidate. The prediction error from modern IOL calculation formulas was influenced by corneal asphericity [35,36]. Corneal asphericity could be a mediating or moderating variable in the process where K or corneal diameter affects ELP. Various models have already been introduced that can handle many variables using conditional process analysis. If new variables are included in conditional process analysis, the predictive accuracy of the equation would be further improved. A second limitation is the relatively small population. The main problem that can arise from the small population is the overfitting of the derived equation. This "overfitting" problem would be solved by increasing the number of the study populations in future studies. In addition, ideal models were found by classifying four groups in 1.5 mm increments in this study. If the number of populations is sufficient, we can find more optimized models by reducing the units of AL and increasing the number of subgroups. Lastly, the analysis of refractive outcomes based on postoperative refraction could be affected by the bias in the preoperative measurement of AL, as shown by the decrease in AL measured using an IOLMaster after cataract surgery reported by De Bernardo M et al. [37].

In conclusion, depending on the preoperative AL, the ideal structural equation model for ELP prediction derived from conditional process analysis differs. Conditional process analysis can be an alternative to conventional multiple linear regression analysis in ELP prediction and IOL power calculation.

#### *4.1. What Was Known*


#### *4.2. What This Parer Adds*


**Author Contributions:** Conceptualization, Y.-S.Y. and W.-J.W.; methodology, Y.-S.Y. and W.-J.W.; software, W.-J.W.; validation, Y.-S.Y. and W.-J.W.; formal analysis, W.-J.W.; investigation, W.-J.W.; resources, W.-J.W.; data curation, Y.-S.Y. and W.-J.W.; writing—original draft preparation, Y.-S.Y. and W.-J.W.; writing—review and editing, Y.-S.Y. and W.-J.W.; visualization, Y.-S.Y. and W.-J.W.; supervision, W.-J.W.; project administration, W.-J.W.; funding acquisition, Y.-S.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2021R1I1A1A01056094).

**Institutional Review Board Statement:** The study was conducted in accordance with the Declaration of Helsinki and approved by the Institutional Review Board (IRB #SC20RASI0071) for Human Studies at Yeouido St. Mary's Hospital.

**Informed Consent Statement:** Patient consent was waived due to the design of the present study (retrospective study) by IRB of Yeouido St. Mary's Hospital.

**Data Availability Statement:** Data collected for this study, including individual patient data, will not be made available.

**Conflicts of Interest:** The authors declare no conflict of interest.
