## 2 Materials and Methods
### Methods
#### 2.2.3 Classification Model  
Parallel to the feature selection procedure, we compared the classification performances of three different algorithms: General Linear Model, Random Forest and Neural Networks. These models have been fed with the 52 considered features (27 heavy metals concentration + 25 polymorphisms). We started with a linear hypothesis and then we applied machine learning models to improve the classification performances. To increase the robustness of our findings, we used two very different machine learning algorithms as neural networks based on the artificial neuron as fundamental unit and Random Forest composed of an ensemble of decision trees. These algorithms are very versatile and not require any a priori assumption. The Generalized Linear Model (GLM) expands and, in a sense, completes the perspective and the applications field of the linear regression model [39]. It was formulated by [PERSON] and [PERSON] as a way to uniform different statistical models within a single model including the linear model, logistic regression and Poisson regression [40]. While in the classical linear model it is assumed that the dependent variable has a normal distribution, in the GLM the dependent variable can be distributed as any variable belonging to exponential family: binomial, poissonian, gamma, etc... The linear model presumes that the expected value of the dependent variable \(\mathbf{y}\) is provided by a linear combination of the independent variables \(\mathbf{x}\). This introduces a limitation of linearity that restricts the practical application field. Instead, GLM introduces a linearizing link function which transforms the expectation value of the dependent variable [41]. In this way, non-normal and discrete distributions of \(\mathbf{y}\) can also be described through this model [42]. Specifically, GLM is composed of three components [43]:  
1. A random component that specifies the conditional distribution of the dependent variable \(y=y_{i},\ldots,y_{n}\) composed of \(n\) independent observations in relation to the values of the independent variables of the model. 2. A linear function of regressors. \[v_{i}=\alpha_{1}x_{i1}+\alpha_{1}x_{i2}+\ldots+\alpha_{1}x_{ik}+\gamma\] (1)
3. A linearizing link function \(g()\) that converts the expectation of \(y_{i}\), \(\chi_{i}\) in \(v_{i}\). \[g(\chi_{i})=v_{i}=\alpha_{1}x_{i1}+\alpha_{1}x_{i2}+\ldots+\alpha_{1}x_{ik}+\gamma\] (2)Random Forest (RF) is a machine learning algorithm which exploits a set of decision trees built through resampling with repetitions (bootstrapping) of the training dataset [44]. In the training phase, at each node of a RF tree, a subset of features is randomly selected so that the RF trees are poorly correlated with each other. Then, the classification task is performed through a majority vote procedure; each tree yields its own prediction and the final decision is obtained by majority voting. In general, RF has some features that make it a suitable choice for many classification problems: (i) RF is easy to tune with only two parameters to tune, the number of trees \(t\) used to grow the forest and the number of features \(m\) sampled to grow each leaf within a tree; (ii) it is robust to overfitting; (iii) using an out-of-bag procedure, RF can compute an unbiased estimation of the generalization error. Node impurity is measured by the Gini index, in supervised classification tasks, this metric allows the forest to learn the optimal cuts and how to optimally separate the available classes [45]. In our analysis, in order to obtain results independent from the model tuning, we adopted a standard configuration with each forest grown with \(t=300\) trees and \(m=f/3\) (\(f\) being the total number of features). Artificial neural networks (ANNs) are constituted by computational networks reproducing the human nervous system that can learn from known examples and generalize to unknown cases.