## 2 Materials and Methods
### Statistical Analysis  
**Data processing:** Data collection in scientific research often runs the risk of including missing and atypical data, which underscores the importance of beginning any statistical analysis with a comprehensive data analysis protocol. One of the most popular methods for processing data in multivariate samples is using Mahalanobis distances. This technique allows us to quantify how many standard deviations an observation is from the mean of a distribution. This tool is particularly useful for identifying observations that deviate significantly from the norm since it can distinguish between regular and outlier data. From a geometric point of view, unlike the Euclidean distance, which measures the shortest path between two points without taking into account the correlation between the variables, the Mahalanobis distance incorporates this correlation in its calculation [23; 24], offering a most precise measure of the distance between a specific point \(\mathbf{x}\in\mathbb{R}^{p}\), generated by a multivariate probability distribution \(f_{\mathbf{X}}(. )\), and the average \(\mu=E(\mathbf{X})\) of said distribution. Assuming that the \(f_{\mathbf{X}}(. )\) distribution has finite second-order moments, the covariance matrix \(\sum=E(\mathbf{X}-\mu)\) can be established, thus allowing the precise definition of the Mahalanobis distances as:  
\[D(\mathbf{X},\mu)=\sqrt{(\mathbf{X}-\mu)^{T}\Sigma^{-1}(\mathbf{X}-\mu)} \tag{2}\]  
**Confirmatory factor analysis:** Once the processed data are available, it is possible to perform a Confirmatory Factor Analysis (CFA) to assess the validity and reliability of the instrument used. CFA is a method of analyzing variables that make up a construct, analyzing the dependency relationship of each variable on the factor to which it is associated. This technique is particularly effective in the treatment of ordinal variables, allowing us to evaluate the extent to which a vector of dimensional responses \(p\times 1\), made up of observable variables, manages to represent one or more latent variables, called factors \(\eta\). Through this approach, it is established that each observed variable contributes to the explanation of the behavior of the latent variable that is intended to be measured and estimated. In this context, a vector of observed responses is used \(Y_{i}\) to explain the latent variable \(\xi\) through the applied model. \[\mathbf{Y}=\mathbf{\Lambda}_{\xi}^{x}+\epsilon, \tag{3}\]  
Figure 2: Shallow Neural Network used to calculate the Interaction Index. In the context of the presented model, \(\mathbf{Y}\) is defined as a dimension vector \(p\times 1\) composed of observed random variables, while \(\xi\) refers to unobserved latent variables, and \(\mathbf{\Lambda}\) denotes a dimension matrix \(p\times k\), being \(k\) the number of involved unobserved latent variables. Considering that \(\mathbf{Y}\) comprises a set of latent variables \(\xi\), the model incorporates an error \(\epsilon\). The resolution of this model is facilitated by using the maximum likelihood (ML) estimation technique, which is based on the iterative minimization of the function:  
\[F_{ML}=ln|\mathbf{\Lambda}\mathbf{\Omega}\mathbf{\Lambda}^{{}^{\prime}}+\mathbf{I}-diag(\mathbf{ \Lambda}\mathbf{\Omega}\mathbf{\Lambda}^{{}^{\prime}})|+tr(R(\mathbf{\Lambda}\mathbf{\Omega} \mathbf{\Lambda}^{{}^{\prime}}+\mathbf{I}-diag(\mathbf{\Lambda}\mathbf{\Omega}\mathbf{\Lambda}^{{} ^{\prime}})^{-1}))-\ln(R)-p, \tag{4}\]  
where \(\mathbf{\Lambda}\mathbf{\Omega}\mathbf{\Lambda}^{{}^{\prime}}\) symbolizes the variance-covariance matrix derived from the proposed factorial model, while \(\mathbf{R}\) is identified with the observed variance-covariance matrix. In more detail, the parameters associated with the model in the context of the CFA are estimated by minimizing the discrepancy between the theoretical variance-covariance matrix of the model and the observed matrix [20, 25, 26, 27, 28, 29].