## 3 Results and Discussion
### Design of Experiment (DOE)
#### 3.2.4 Mathematical Model and Analysis of Variance  
Source Phase  
An analysis of variance (ANOVA) was conducted to examine the reliability of the model. ANOVA shows whether the model-related variations are significant compared to the variations in experimental results. According to the MINITAB software, the following regression equation was obtained:  
\[ \begin{split}\text{S}=87.033+2.645\text{ [Cr(VI)] + 0.129 [Aliquat-336] - 0.967 T + 0.077 [Cr(VI)] * [Aliquat-336] + 0.530 [Cr(VI)] * T + 3.667 [Aliquat-336] * T - 3.530 [Cr(VI)] * [Aliquat-336] * T.}\end{split}  \tag{1}\]  
The individual effects, interaction effects, and model coefficients are presented in Table 6. The main effects and interaction effects of each factor having a significance probability (\(P\)-value) \(\leq\) 0.05 and a high [PERSON]'s exact test (F-Value) means that this factor is significant. Table 7 summarizes the variance analysis. It reports both the significance probability and the [PERSON]'s exact test. The results confirm that the factors [Aliquat-336], T, and interactions T * [Cr (VI)] and [Cr(VI)] * [Aliquat-336] had insignificant effects on the response, which was the extraction yield of Cr(VI) in the source phase. On the other hand, [Cr (VI)] and interactions [Aliquat-336] * T and [Cr(VI)] * [Aliquat-336] * T had significant effects. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Term** & **Effect** & **Coefficient** \\ \hline Constant & & 87.033 \\
[Cr(VI)] & 5.289 & 2.645 \\
[Aliquat-336] & 0.257 & 0.129 \\ T & \(-\)1.935 & \(-\)0.967 \\
[Cr(VI)] * [Aliquat-336] & 0.155 & 0.077 \\
[Cr(VI)] * T & 1.059 & 0.530 \\
[Aliquat-336] * T & 7.333 & 3.667 \\
[Cr(VI)] * [Aliquat-336] * T & \(-\)7.060 & \(-\)3.530 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Factor effects and their coefficients. Figure 11: Factors of interactions effects plots for receiving phase. After analysis of variance and by eliminating nonsignificant factors, the mathematical model obtained at the beginning becomes:  
\[\text{S}=87.033+2.645\text{ [Cr(VI)]}+3.667\text{ [Aliquat-336] }^{*}\text{T}-3.530\text{ [Cr(VI)] }^{*}\text{ [Aliquat-336] }^{*}\text{T}. \tag{2}\]  
Figure 12 compares the experimental results of the Cr(VI) extraction performance with those predicted, showing a good agreement between experimental and predicted yields. The model shows a first-order linear regression with a high coefficient of determination, R\({}^{2}\) = 97.77% (R\({}^{2}\)corr = 94.26), indicating validation of the mathematical model. Receiving Phase  
The MINITAB software allowed the processing of results by the following regression equation:  
\[ \begin{split}\text{R}=43.872+7.945\text{ [NaCl]}+1.717\text{ [Aliquat-336] }+5.329\text{ T}+3.397\text{[NaCl] }^{*}\text{ [Aliquat-336] }^{*}\text{T}\\ -2.674\text{[NaCl] }^{*}\text{T}-0.921\text{[Aliquat-336] }^{*}\text{T}+0.676\text{[NaCl] }^{*}\text{ [Aliquat-336] }^{*}\text{T}.\end{split}  \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c} \hline
**Source** & **DF \({}^{1}\)** & **Adj SS \({}^{2}\)** & **Adj MS \({}^{3}\)** & **F-Value** & **P-Value** \\ \hline Regression & 7 & 273.116 & 39.017 & 29.14 & 0.001 \\ \(\text{[Cr(VI)]}\) & 1 & 55.956 & 55.956 & 41.80 & 0.001 \\ \(\text{[Aliquat-336] }\) & 1 & 0.132 & 0.132 & 0.10 & 0.766 \\ T & 1 & 7.487 & 7.487 & 5.59 & 0.064 \\ \(\text{[Cr(VI)] }^{*}\text{[Aliquat-336] }\) & 1 & 0.048 & 0.048 & 0.04 & 0.858 \\ \(\text{[Cr(VI)] }^{*}\text{T}\) & 1 & 2.243 & 2.243 & 1.68 & 0.252 \\ \(\text{[Aliquat-336] }^{*}\text{T}\) & 1 & 107.552 & 107.552 & 80.34 & 0.000 \\ \(\text{[Cr(VI)] }^{*}\text{[Aliquat-336] }^{*}\text{T}\) & 1 & 99.698 & 99.698 & 74.47 & 0.000 \\ T & & & & & & \\ \hline \end{tabular} \({}^{1}\) DF: Degree of Freedom; \({}^{2}\) SS: Sum of Square; and \({}^{3}\) MS: Mean Square.  
\end{table} 
Table 7: Analysis of variance. Figure 12: The predicted values of the response versus experimental values. The individual, interaction.