## 2 Methodology
### Statistical Methods  
In factor analysis--FA [44]--the original variables are written as a linear combination of the common and specific factors. Let \(X_{j}\) be original variable \(j\), for \(j\in\{1,\dots,p\}\). Suppose \(f_{k}\) is common factor \(k\), for \(k\in\{1,\dots,q\}\), and \(u_{j}\) is a specific factor \(j\). Then, we have that  
\[X_{j}=\lambda_{j1}f_{1}+\lambda_{j2}f_{2}+\dots+\lambda_{jq}f_{q}+d_{j}u_{j}, \quad j\in\{1,\dots,p\}. \tag{1}\]  
The coefficients \(\lambda_{jk}\) and \(d_{j}\), for \(j\in\{1,\dots,p\}\) and \(k\in\{1,\dots,q\}\), are known as loadings. In addition, we have the following model assumptions with orthogonal factors: (i) \(\forall l\
eq m\), \(\text{Cor}(f_{h},f_{m})=0\); (ii) \(\forall h\
eq m\), \(\text{Cor}(u_{h},u_{m})=0\); and (iii) \(\forall h\) and \(\forall m\), \(\text{Cor}(f_{h},u_{m})=0\). Furthermore, all factors are assumed to be standardized. Note that the expression given in (1) can be formulated in matrix form as  
\[\mathbf{X}=\mathbf{F}\mathbf{\Lambda}^{\top}+\mathbf{E}, \tag{2}\]  
where \(\mathbf{X}=(x_{ij})\) is the data matrix of order \(n\times p\), \(\mathbf{F}=(f_{ik})\) is the score matrix of order \(n\times q\), \(\mathbf{\Lambda}=(\lambda_{jk})\) is the factor matrix of order \(p\times q\), and \(\mathbf{E}=(e_{ij})\) is the error matrix of order \(n\times p\). In (1), \(d_{j}u_{j}\) is considered as an error term, that is, \(e_{j}=d_{j}u_{j}\). The FA has as its main goal to find and interpret the factor matrix \(\mathbf{\Lambda}\) stated in (2). To achieve a dimension reduction, we need \(q<p\). There are some algorithms to minimize the residual sum of squares represented as  
\[\min_{\mathbf{F},\mathbf{\Lambda}}\|\mathbf{E}=\mathbf{X}-\mathbf{F}\mathbf{\Lambda}^{ \top}\|^{2},\] subject to: \[\mathbf{\Lambda}^{\top}\mathbf{\Lambda}=\mathbf{I}_{q}.\]  
Some of these algorithms are the maximum likelihood, least squares, and PCA methods. We use the SPSS software to compute the matrix \(\mathbf{\Lambda}\). Contrary to what happens in FA, in PCA, each component is written as a linear combination of the original variables. For a detailed discussion of PCA, see [44]. Let \(X_{j}\) be original variable \(j\) and \(c_{l}\) be component \(r\) (\(r<p\)). Then, we get  
\[c_{l}=b_{1l}X_{1}+b_{2l}X_{2}+\cdots+b_{pl}X_{p},\quad l\in\{1,\ldots,r\}. \tag{3}\]  
The coefficients \(b_{jl}\), for \(j\in\{1,\ldots,p\}\) and \(l\in\{1,\ldots,r\}\), are known as loadings. Calculation of components formulated in (3) is conducted sequentially. For the first component, its variance is maximized. For the second one, its variance is maximized, and it must be orthogonal to the first component. For the third one, its variance is maximized, and it must be orthogonal to the two previous components, and so on. In general, the variance of each component is maximized, and the constraint that this new component must be orthogonal to all previous components is added. The model assumes that the components are unitary and independent. In matrix terms, the PCA is modeled by  
\[\mathbf{X}=\mathbf{A}\mathbf{B}^{\top}+\mathbf{E}, \tag{4}\]  
where \(\mathbf{X}=(x_{ij})\) is the \(n\times p\) data matrix, \(\mathbf{A}=(a_{il})\) is the \(n\times r\) score matrix containing the coordinates of the \(n\) individuals in the reduced dimension space, \(\mathbf{B}=(b_{jl})\) is the \(p\times r\) loading matrix, and \(\mathbf{E}=(e_{ij})\) is the \(n\times p\) error matrix. In PCA, we must calculate and interpret the matrix \(\mathbf{B}\) stated in (4). There are some algorithms, based on the singular value decomposition of matrices, to solve the optimization problem formulated as  
\[\min_{\mathbf{A},\mathbf{B}}\|\mathbf{E}=\mathbf{X}-\mathbf{A}\mathbf{B}^{\top}\|^{2},\] subject to: \[\mathbf{B}^{\top}\mathbf{B}=\mathbf{I}_{r}.\]  
We employ the R software to calculate the components of the PCA. Interpreting the matrix \(\mathbf{B}\) means that we can unambiguously determine how the components group the original variables.