## 4 Results and Discussion
### Comparative Analysis of Results Using Statistical Methods and Selection of the Final Model  
Comparative analysis compares the delay prediction results of three ML models, each of which was selected as the best solution in one of the three observed approaches to optimizing the input set of variables. The main goal is to determine the statistical significance of the differences between the prediction results, which is the reason for testing the null hypothesis:  
**Hypothesis 0 (H0). **\(\mu_{1}\) = \(\mu_{2}\) = \(\mu_{3}\), where \(\mu_{1}\), \(\mu_{2}\) and \(\mu_{3}\) are the arithmetic means of delay prediction values for \(k\)-NN models selected as the best solutions in the approach based on the RReleiff algorithm, Backward selection via the recursive feature elimination algorithm, and the Pareto 80/20 rule, respectively. In other words, this hypothesis represents the assumption that there are no significant statistical differences in the arithmetic means of the delay prediction results for the three observed models. In contrast, the alternative hypothesis can be stated as follows:  
**Hypothesis 1 (H1). **_There are significant statistical differences in the prediction results between at least two models, i.e., two optimization approaches._  
The parametric statistical test that tests the null hypothesis is ANOVA with Repeated Measures [49]. However, it is first necessary to test one of the basic conditions for the application of this test, which is the normality of the distribution of the dependent variable in groups. The results of the Kolmogorov-Smirnov normality test for the observed models are given in Table 5.  
The obtained significance value of the Sig. test for all three cases has the same value (Fig. = 0.000). It means that the assumption about the normality of the distribution of the dependent variable in groups can be rejected. This conclusion can be confirmed graphically on the basis of the Q-Q plots shown in Figure 13. On the diagrams, it is obvious that there are significant deviations of the points from the line representing the normal distribution. Given the non-fulfillment of the conditions from the aspect of normality of distribution, it is necessary to test the hypotheses with the Friedman test, which is a non-parametric alternative to ANOVA with Repeated Measures. Table 6 shows the results of the Friedman test performed in IBM SPSS Statistics [50]. In addition to the sample size (N), a statistical test (Chi-Square), degree of freedom (df) and significance level (Asymp. Sig.) are given in the table. Based on the value of Asymp. Sig., which is less than the \(\alpha=0.05\) level, it is concluded that there are statistically significant differences in the prediction results for the three models, i.e., for three approaches to optimizing the set of input variables. The results given in Table 6 do not show the information for which pair of combined optimization techniques there is a significant statistical difference. The answer to this question is obtained with a Post Hoc statistical test. Table 7 shows the results of the Wilcoxon signed-rank post hoc test with the value of \(Z\) and Asymp. Sig. for each of the three combinations of approaches. In order to interpret the results obtained, it is necessary to calculate the adjusted Bonferroni level of significance as the ratio of level \(\alpha=0.05\) and the number of pairs being compared, which as a result provides a value of 0.017. Given that Asymp. Sig. < 0.017 \begin{table}
\begin{tabular}{c c} \hline N & 31,143 \\ Chi-Square & 268.019 \\ df & 2 \\ Asymp. Sig. & 0.000 \\ \hline \end{tabular}
\end{table} 
Table 6: Results of the Friedman test.