## 5 Evaluation of the Models
### Model 1  
Figures 6 and 7 give the overall result and precision of Model 1. Figure 6 shows the values predicted by Model 1 (in red) and the real values (in black). It also gives the confidence intervals at 2.5% and 97.5%, respectively (red lines), and the mean value of the prediction (black line). It is clear from Figure 4 that the 95% interval does not contain all of the SMIs and that the prediction is less optimal for high values of flight hours. Figure 7 shows the residuals of the model, the predicted values versus residuals, and the Q-Q plot. The residual plot shows a high density of points close to the origin and a low density of points away from the origin. It is symmetric about the origin, and there are not any patterns in the value of the residuals as we move along the \(x\)-axis. A small number of cases have notable errors. There are nine measurements with errors greater than 50. These correspond to ANSPs 9, 11, 22, 28, 8, 20, and 12. Figure 8 comprises caterpillar plots giving the distributions of each hyperparameter \(K_{js}\) of the model (K, K1, K2, K3, and K4). Each distribution is indicated by a horizontal line representing the 95% interval and a dot showing the mean value. Each provider is non-dimensionalized by assigning a number to it. The vertical red line represents the global mean of the means of the posterior distributions. Each \(K_{js}\) parameter is a measure of the dispersion of the regression model coefficients for each service provider. The higher the value of \(K_{js}\), the lower the variance and, therefore, the nearer the value of the parameter for an ANSP to the mean value of the \(\hat{\theta}_{j}\) parameters at a European level. Conversely, the smaller the value of \(K_{js}\), the greater the dispersion. It should be borne in mind that all values of \(\hat{\theta}_{j}\) (\(\vartheta,\ \hat{\theta}_{1},\ \hat{\theta}_{2},\ \hat{\theta}_{3}\) and \(\hat{\theta}_{4}\) ) are common for all ANSPs. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Model 1: Hierarchical Poisson Regression** & **Model 2: Hierarchical Negative Binomial Regression** & **Model 3: Hierarchical Zero-Inflated Regression** & **Model 4: Hierarchical Normal Quadratic Regression with Variance Proportional to the Number of Flights. ** \\ \hline Response distribution & \(y_{i,o}\sim\text{\emph{poisson}}(\mu_{s})\) & \(y_{i,s}\sim\text{\emph{NB}}(p_{i,s})\) & \(y_{i,s}\sim\text{\emph{ZIP}}(\lambda_{s},\hat{\theta}_{s})\) & \(y_{i,o}\sim\text{\emph{NB}}(\mu_{s},\tau_{s}\ast\text{\emph{x1}}_{i,  
Figure 6: Model 1: Predicted values (Red) vs. actual values (Black), and 2.5% and 97.5% confidence intervals in Model 1. Figure 7: Model 1: Residuals, predicted values vs. residuals, and Q–Q plot for Model 1. The numbers show that the mean values of \(K_{js}\) are generally high. This is in agreement with the hypothesis that the coefficients of the regression model are of the same order for the ANSPs and similar to the corresponding \(\hat{\theta}_{j}\) value. However, in each case, there is a set of service providers whose behavior deviates significantly from the mean. Extreme values of \(K_{js}\), that is, values that are unusually high or low compared to the mean values, can be helpful in identifying ANSPs with atypical rates of incidents. For different ANSPs, the parameters that show the most variation are K1, K2, and K3. In other words, the variability in response between ANSPs is greater for the variables x1 \"number of flight hours\", x2 \"Complexity Score\", and x3 \"Adjusted Density\", in that order. Finally, the values of \(\hat{\vartheta}_{j}\) (\(\hat{\theta},\ \hat{\theta}_{1},\ \hat{\theta}_{2},\ \hat{\vartheta}_{3},\ \text{and}\ \hat{\vartheta}_{4}\) ) for this model are summarized in Table 2.