## 3 Results and Discussion
### Analysis of the Statistical Design of Experiments
#### 3.6.2 Graphical Analysis  
The Pareto chart of factors shows the greater or lesser influence of each study variable on the target variable, as well as the statistical significance of each variable. Therefore, it can be considered, to some extent, as a graphical representation of the ANOVA test. The horizontal bars represent the effect of each variable, namely the concentration of Fe\({}^{2+}\), the concentration of Fe\({}^{3+}\), the concentration of H\({}_{2}\)O\({}_{2}\), and the concentration of polyphenols, as well as all their combinations. Figure 8a shows the Pareto chart for the elimination of chrysoidine. The vertical line seen in Figure 8a corresponds to a \(p\)-value of 0.05. Factors whose bars exceed this line are considered to have a statistically significant effect on the dye elimination efficiency with a probability of 95%. As indicated earlier, these variables appear in the ANOVA test table (Table S2) with a value below 0.05. Additionally, the graph shows the type of influence exerted by each considered variable, i.e., whether its effect on the elimination of chrysoidine is positive or negative. Variables represented by gray bars have a positive effect on the target variable, while those represented by blue bars have a negative effect on the target variable. Note that factors that have a positive influence on the dye elimination efficiency appear in Equation (10) preceded by a \"+\" sign, while factors that have a negative influence on the response variable are preceded by a \"\(-\)\" sign in the equation. The Pareto chart allows for a more direct interpretation of the results discussed earlier in the ANOVA test. The five variables with the greatest influence and the degree of each variable demonstrate that an appropriate concentration of hydrogen peroxide has a very significant influence on the target variable. However, an excess of this concentration (CC bar) achieves the opposite effect. This is because the function of hydrogen peroxide in the elimination of dye lies in carrying out Fenton or Fenton-like radical processes. As indicated above, when the concentration of hydrogen peroxide is too high, a large amount  
Lastly, although they have a lesser influence, the concentrations of iron salts are of great importance as they both exhibit a minimum elimination and higher values as their concentrations increase or decrease. This is because both Fe\({}^{3+}\) and Fe\({}^{2+}\) can initiate Fenton-like reactions and lead to the formation of iron nanoparticles. These two processes generate a synergistic effect, as will be discussed later. The reason for the need to increase or decrease the concentration of one of the iron salts is to maintain an excess of the other. One possible explanation for this observation lies in the fact that the formation of nanoparticles is somewhat favored for Fe\({}^{3+}\) cations, but these cations give rise to \"Fenton-like\" reactions, which are less effective in dye elimination due to their lower oxidation power and slower kinetics. On the other hand, Fe\({}^{2+}\) cations contribute to the development of the Fenton process, which is effective in the destruction of the dye [86]. Another graph of great interest in the analysis of experimental design data is the interaction plot (Figure 8c). Studying the presence (or absence) of interactions between the working variables is one of the key advantages of statistical experimental design and justifies, from a methodological point of view, its use over the traditional method of modifying \"one variable at a time\". This method is valid if it is assumed that there are no interactions between variables, meaning that a change in one variable does not affect how the other variables influence the target variable.