## 6 Comparison of the Models  
The four hierarchical models in the study use the following probability density functions: Poisson, negative binomial, zero-inflated, and normal. We will use deviation information criteria (DIC) to compare the models [39] via MCMC simulations. This parameter calculates the deviation as the mean of the posterior predictive function:  
\[DIC=-2\ln\bigl{(}p\bigl{(}y\bigl{|}\theta^{-}\bigr{)}\bigr{)}+2 pD\text{ where }D=-2\int\ \ln\bigl{(}p\bigl{(}y\bigl{|}\theta^{-}\bigr{)}\bigr{)}d\theta \tag{13}\]  
This criterion is a measure of the goodness of fit of the data and, at the same time, introduces a term to evaluate the complexity of the model. This criterion is asymptotically the same as performing cross-validation using part of the data to estimate the data and the rest to calculate the goodness of the predictions [40]. The lower the calculated DIC value, the more accurate the model's predictions. One model can be considered superior to another when its DIC is more than five points below that of its competitor [41]. According to the values obtained in the simulation, Model 4 appears to be the most efficient and Model 2 the least efficient. This coincides with the analysis carried out in the previous sections. The table confirms that Models 1 and 4 are better at predicting SMIs. For small values of SMIs, the Poisson model (Model 1) is more efficient, while for high values of SMIs, the normal model (Model 4) is more accurate. For intermediate values, all models behave in a similar manner. The DIC values for Models 1 and 3 are similar. The hierarchical relationship framework underlying all the models includes parameters \(\vartheta_{j}\) and \(K_{js}\). Each of the models includes a parameter or set of parameters \(\vartheta_{s}\) that synthesizes the general level of safety of the EU ATM system. Parameter \(\vartheta_{s}\) represents the mean or overall influence of variable \(xj\) on the predicted variable. Parameter \(K_{js}\) is a measure of the dispersion of an ANSP around parameter \(\vartheta_{j}\). These parameters could be used in further studies to benchmark among different regions in the world (USA, Middle East, etc. ), providing similar data set are available for all the regions. If the values of \(K_{js}\) are high and similar to each other for all ANSPs, this means that there is a central tendency around the European mean, since \(K_{js}\) is inversely proportional to the variance of \(\vartheta_{j}\). Conversely, small values of \(K_{js}\) indicate that an ANSP has an unusually low or high number of SMIs, compared to the European average. Figure 18: Model 4: Distributions of parameters \(\varepsilon_{s}\) (**a**) and \(\sigma_{s}\) (**b**) of Model 4. A value of \(K_{js}\) that is significantly different to that of other ANSPs will alert us to those ANSPs whose behavior deviates from the ideal. To illustrate the meaning of \(K_{js}\), let us analyze in detail an example. Let us look at parameter K4 in Model 1 (Figure 8d). The estimated values of K4 for each of the 29 ANSPs are presented in Table 4. K4 is related to the influence of the variable \(x4\), \"complexity index\", in the number of SMIs. K4 indicates the deviation of each ANSP with respect to the influence of complexity index in the number of SMIs from the overall European ATM system. It can be observed that the values of K4 are higher than 900 for almost all ANSPs, except for ANSP-12 and ANSP-9. \(K_{js}\) is inversely proportional to the variance of \(\vartheta_{j}\). For those ANSPs with elevated K4 values, the influence of complexity index in the occurrence of SMIs is similar and close to that of the overall European ATM system. However, ANSPs 12 and 9 exhibit lower values of K4, 78 and 335, respectively, indicating that for these two providers the dependency of the SMIs with the variable complexity index does not behave as the overall EU ATM system.