## 3 Statistical Methodology
### Screening Stage  
The proposed method (see Table 1) conducts a screening phase to identify the characteristic parameters that generate significant differences in either high-fidelity or multi-fidelity variables. In this analysis stage, the focus is on direct and linear effects rather than possible quadratic or nonlinear effects. Analysis of variance (ANOVA) is a versatile method for this screening phase as it allows the exclusion of nonlinear terms from the model, thus increasing its power. Thus, non-significant terms that usually correspond to interaction effects can be specifically eliminated to improve the significance of the model by including only the main results. ANOVA proves to be a valuable technique to screen input parameters and thus reduce the extension of subsequent numerical [45] or experimental campaigns [46] for optimization.  
This analysis of variance (ANOVA) is based on multiple comparisons of the means of the high and multi-fidelity variables at different levels of the characteristic input parameters. In the ANOVA model, the F-statistic and its corresponding \(p\)-value are critical in identifying the characteristic parameters that cause significant differences in the high-fidelity and multi-fidelity variables. This F-statistic is calculated for the main effect of each characteristic parameter as well as for the residual variability of the model. The F-statistic follows a Fisher-Snedecor distribution, assuming the null hypothesis (\(H_{0}\)) that different levels of the characteristic parameters do not produce significant differences in the highand multi-fidelity variables. The F statistic represents a ratio between two measures of variability weighted by degrees of freedom (mean squares): the variability explained by each characteristic parameter relative to the residual variability of the model. Thus, the F statistic represents the variability explained by each characteristic parameter relative to the variability not explained (residual) by the model.  
The variability explained by different levels of characteristic parameters can be calculated as the Explained Sum of Squares (ESS):  
\[ESS=\sum n_{i}\frac{(\bar{Y}_{i}-\bar{Y})^{2}}{l-1} \tag{1}\]  
where \(\bar{Y}_{i}\) represents the sample mean of the high or multi-fidelity variable at each level of the characteristic parameter, \(\bar{Y}\) represents the overall mean of this variable, and \(l\) denotes the number of levels of the parameter.  
The residual (intra-level) variability (Residual Sum of Squares) is calculated as follows:  
\[RSS=\sum_{ij}\frac{(Y_{ij}-\bar{Y}_{i})^{2}}{N-l} \tag{2}\]  
where \(Y_{ij}\) represents the \(j\)-th simulation at each of the \(i\)-th level of the high or multi-fidelity variable, and \(\bar{Y}\) represents the mean value of that variable at each level of the characteristic parameter, \(N\) represents the total sample size, and \(l\) represents the number of levels.  
Using both terms, the F statistic of each parameter characteristic can be computed as the ratio between the variability explained by different levels and the (unexplained) residual variability, within levels of characteristic parameters.  
\[F=\frac{ESS}{RSS} \tag{3}\]  
Along with this F-statistic, \(p\)-values can be obtained for identifying significant effects (\(p\)-value \(<\)\(\alpha\)). These \(p\)-values are based on the comparison between calculated F-statistics and critical values of the Fisher - Snedecor distribution. The degrees of freedom for the numerator and denominator of this distribution are calculated by combining the number of levels and the screening sample size. A 95% confidence level is usually considered for assessing the significance of \(p\)-values, although more restrictive approaches (99% confidence level). In this screening stage, some low-level interactions (double or even triple) may show significant effects. The retention of these interaction terms is at the discretion of the researcher, although in most cases it is not interesting for research.