## 4 Results  
Once the internal consistency of the discriminant analysis has been verified, it is possible to reformulate Equation (1) with the coefficients obtained for the five factors considered: (\(\alpha\)) the percentage of the state's population with a life-limiting chronic disease, (\(\beta\)) the average education level (one basic and five superior) of the state's population, (\(\gamma\)) the percentage of the state's population that has support from someone, (\(\delta\)) the percentage \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**(A)**} & \multicolumn{6}{c}{**(B)**} \\ \hline \multirow{2}{*}{Function} & \multicolumn{2}{c}{Wilks\({}^{\prime}\) Lambda} & \multirow{2}{*}{Chi-Square} & \multirow{2}{*}{Significance} & \multirow{2}{*}{Function} & \multirow{2}{*}{Eigenvalue} & \multirow{2}{*}{\% variance} & \multirow{2}{*}{
\begin{tabular}{c} Canonical \\ correlation \\ \end{tabular} } \\ \hline
1 & 0.015 & 207 & \textless{}0.001 & 1 & 64.951 \({}^{\text{a}}\) & 100 & 0.992 \\ \hline \multicolumn{6}{c}{**(C)**} & \multicolumn{6}{c}{**(D)**} \\ \hline \multirow{2}{*}{Step} & \multicolumn{2}{c}{Number of Factors} & \multirow{2}{*}{Lambda} & \multirow{2}{*}{Significance} & \multirow{2}{*}{Box M} & \multirow{2}{*}{67} \\ \hline
1 & 1 & 0.036 & \textless{}0.001 & F Statistic & Approximate & 4 \\ \hline
2 & 2 & 0.025 & \textless{}0.001 & & df1 & 15 \\ \hline
3 & 3 & 0.019 & \textless{}0.001 & & df2 & 10,887 \\ \hline
4 & 4 & 0.017 & \textless{}0.001 & & Significance & \textless{}0.001 \\ \hline
5 & 5 & 0.015 & \textless{}0.001 & & & \\ \hline \end{tabular} \({}^{\text{(*)}}\) the first canonical discriminant function was used in the analysis.  
\end{table} 
Table 3: Discriminant analysis statistics. (**A**) Global (Chi-Squared) Test of the Significance of the Discriminant Functions; (**B**) Test (Eigenvalue) of the Function’s Ability to Discriminate Groups Appropriately; (**C**) Test (Wilks’ Lambda) of the Differences Between the Factors of Each Group; (**D**) Test (Box M) of the Null Hypothesis of Equal Covariance Matrices. of the state's population that performs unhealthy work, and (\(\varepsilon\)) the percentage of the male population. By applying the coefficients of importance of each of these factors, it is possible to construct the multidimensional depression diagnostic index through the following discriminant function:  
\[\phi_{\kappa}=1.029\alpha+0.487\beta+0.363\gamma+0.456\delta-0.623\varepsilon \tag{2}\]  
where \(\phi_{\kappa}\) is the composite indicator score for the state \(\kappa\) resulting from the product of two coefficients of the discriminant function \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), and \(\varepsilon\) based on the values of their respective factors in each state \(\kappa\). The negative sign of \(\varepsilon\) in Equation (2) confirms the current literature indicating that a positive diagnosis of depression is more common in women than in men [73; 74]. The analysis of the two normalized coefficients of Equation (2) also shows that having a limiting chronic illness and being male are the most determining factors when separating individuals with a positive diagnosis of depression from individuals with a negative diagnosis. By applying Equation (2), it is possible to determine the multidimensional depression diagnosis composite indicator score, as represented by the two histograms in Figure 1. These histograms show that the multidimensional depression diagnostic index scores of individuals with a self-reported positive depression diagnosis are well separated from the scores of individuals with a self-reported negative depression diagnosis. This separation reveals that the composite indicator scores are compatible with the dependent categorical variable.