## 4 Proposed Hierarchical Models
The combination of both results in a Poisson regression, as indicated below:  
\[\text{Noise distribution}:\ y_{i,s}\ \sim\ pdf(\mu_{i,s}\,\ [parameters])\ \sim\ poison(\mu_{i,s}) \tag{9}\]  
\[\text{Inverse link function}:\mu_{i,\ s}=f(\ link(x_{i,\ s}),[ parameters])\ \sim\ exp(lin\ (x_{i,s}) \tag{10}\]  
As can be seen, a Poisson regression is a type of generalized linear model (GLM) that enables a non-negative integer response, that is, a natural number, to be modelled against a linear predictor via an exponential link function. The exponential link function allows us to transform the expected values of the response found on the scale of (0, \(\infty\)) into the scale of the linear predictor, that is (\(-\infty\), \(\infty\)). To mitigate the limitations due to overdispersion in the data, three additional models have been tested. **Hierarchical negative binomial regression model. ** The negative binomial model is parameterized in terms of the mean (\(\lambda_{i}\)) and the scale factor (\(r\)) [33]. It can be likened to a hierarchical two-stage process in which the response is modelled against a Poisson distribution whose expected recount is in turn modelled by a Gamma distribution with a mean \(\lambda_{i}\) and a constant scale parameter (\(r\)). \[p(y_{i})=\frac{\Gamma(y_{i}+r)}{\Gamma(r)y!}\times\frac{\lambda_{i}^{y_{i}}*r ^{r}}{(\lambda_{i}+r)^{\mu_{i}+r}} \tag{11}\]  
where the expected value of \(y_{i}\) is given by \(\mu_{i}\) and the variance is \(\frac{\mu_{i}+r_{i}^{2}}{\omega}\). The negative binomial model is appropriate for situations of over-dispersion caused by clustering (due to the influence of other factors that have not been measured). JAGS is Just Another Gibbs Sampler, a program for analysis of Bayesian hierarchical models using MCMC) that uses a parametrization of the negative binomial distribution based on the parameters \(p\) and \(r\). Direct estimation of the parameters \(p\) and \(r\) of the binomial distribution usually implies a bad autocorrelation of the MCMC chains. To avoid this problem, we will use a reparameterization in which we will set priors for the mean \(\lambda_{s}\) and the parameter \(r\), while the variance and the parameter p will be obtained from \(\lambda_{s}\) and the parameter \(r\). \[y[i]\ \sim NB(p_{s},r)\ ;\] \[p_{s}=\frac{r}{r+\lambda_{s}}\ ;\ var_{s}=\ \frac{r*(1-p_{s})}{p_{s}*p_{s}}\ ; \tag{12}\] \[a=\ \theta K,\ b=(1-\vartheta)K\ \text{with}\ K=\frac{\vartheta(1-\vartheta)}{ Var}-1 \tag{5}\]0  
**Hierarchical zero-inflated regression model regression. ** This model is appropriate when the overdispersion is due to a higher-than-expected number of zeros in a Poisson distribution [34]. Zero-inflated models combine a binary logistic regression model with a Poisson regression. **Hierarchical normal quadratic regression with variance proportional to the number of flights**. This last model combines the advantages of Poisson distributions with a normal model [35; 36]. This allows a variance proportional to the number of SMIs to be combined with the effect of stable confidence intervals, especially if the number of incidents is high, which gives rise to a quadratic regression. The data points \(y_{i,\ s}\) are assumed to derive from a normal distribution. The model establishes a different variance for each ATM system proportional to the number of flightshandled by each organization. A Gaussian noise is added to the regression to account for cases where the variance is constant. The hierarchical structure of parameters is similar to that of the previous models. Table 1 summarizes the mathematical formulation of the models developed.