## 3 Results and Discussion
### Analysis of the Statistical Design of Experiments
#### 3.6.1 Numerical Analysis  
The analysis of the results in an experimental design consists of two parts: numerical analysis and graphical analysis. The numerical analysis can be divided into three sections: (i) analysis of variance; (ii) derivation of a regression equation, with the corresponding analysis of its correlation coefficients; and (iii) determination of the experimental conditions that result in an optimum for the objective variable. The analysis of variance, commonly known as ANOVA (from its initials in English), allows for testing the significance of each of the effects or variables in the experimental matrix. To do this, the parameter \(p\) is analyzed. If \(p\) is less than 0.05, then the operational variable in question has a statistically significant effect on the objective variable with a 95% probability. If it is greater than 0.05, then that effect is not statistically significant at that confidence level. Table S2 summarizes the result of the analysis of variance. From this Table, several highly relevant facts can be observed concerning the removal of chrysoidine. Firstly, it stands out that there are five effects with a \(p\)-value of less than 0.05, indicating that they have a statistically significant influence on the target variable with a 95% confidence level. Secondly, the R-squared statistic, obtained through a linear fit of the experimental \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Experiment**} & \multicolumn{4}{c}{**Coded Values**} & \multicolumn{4}{c}{**Removal Efficiency (\%)**} \\ \cline{2-7}  & **Fe (II)** & **Fe (III)** & **H\({}_{2}\)O\({}_{2}\)** & **Polyphenols** & **Chrysoidine** & **Methylene Blue** \\ \hline
11 & 0 & 0 & 0 & 0 & 88.4 & 86.1 \\ \hline
12 & \(-1\) & 1 & 1 & \(-1\) & 96.4 & 93.5 \\ \hline
13 & \(-1\) & \(-1\) & 1 & 1 & 93.7 & 55.6 \\ \hline
14 & \(-1\) & \(-1\) & 1 & \(-1\) & 94.9 & 84.1 \\ \hline
15 & 0 & 0 & 0 & 0 & 89.1 & 80.4 \\ \hline
16 & 0 & 0 & 0 & 0 & 88.6 & 80.0 \\ \hline
17 & 1 & 1 & \(-1\) & 1 & 82.7 & 40.5 \\ \hline
18 & 2 & 0 & 0 & 0 & 89.3 & 78.1 \\ \hline
19 & 0 & 0 & 0 & 2 & 81.3 & 50.4 \\ \hline
20 & \(-1\) & 1 & \(-1\) & \(-1\) & 82.8 & 94.1 \\ \hline
21 & 0 & 0 & \(-2\) & 0 & 61.5 & 53.0 \\ \hline
22 & \(-1\) & 1 & 1 & 1 & 94.5 & 83.1 \\ \hline
23 & 0 & 0 & 0 & 0 & 89.7 & 87.9 \\ \hline
24 & 0 & \(-2\) & 0 & 0 & 95.2 & 63.5 \\ \hline
25 & 1 & \(-1\) & \(-1\) & \(-1\) & 84.2 & 94.0 \\ \hline
26 & 0 & 0 & 0 & 0 & 88.3 & 84.8 \\ \hline
27 & 0 & 0 & 0 & 0 & 89.0 & 85.0 \\ \hline
28 & 1 & 1 & 1 & 1 & 94.6 & 94.8 \\ \hline
29 & 1 & \(-1\) & \(-1\) & 1 & 81.0 & 24.7 \\ \hline
30 & 0 & 0 & 0 & 0 & 89.4 & 84.8 \\ \hline
31 & 0 & 0 & 0 & 0 & 88.9 & 85.2 \\ \hline
32 & 1 & \(-1\) & 1 & 1 & 90.7 & 69.6 \\ \hline
33 & 0 & 0 & 0 & 0 & 88.7 & 86.8 \\ \hline
34 & 1 & \(-1\) & 1 & \(-1\) & 95.6 & 96.2 \\ \hline
35 & \(-2\) & 0 & 0 & 0 & 94.3 & 61.2 \\ \hline
36 & \(-1\) & 1 & \(-1\) & 1 & 86.3 & 45.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._values compared to those obtained from the experimental design, indicates that the model resulting from the analysis of the experimental design can explain 88.22% of the variability in the percentage of chrysoidine elimination. Another determined parameter is the estimated standard error, which shows the standard deviation of the residuals. In this specific case, its value is 3.02, with the residual value being the difference between the observed value and the predicted value by the experimental design. Additionally, the calculation of the mean absolute error (1.59) was performed, which is defined as the average value of the residuals used to determine the previous parameter. Lastly, the Durbin-Watson (DW) statistic is noteworthy, as it determines if there is a significant correlation among the data presented in Table 2, based on the order in which the data appear in the file. Since the \(p\)-value of this statistic is greater than 5.0%, it can be concluded that there is no autocorrelation in the residuals.