## 3 Results  
### Two Variables  
#### 3.1.1 Extrusion Process for Producing High-Antioxidant Instant Amaranth Flour  
In Study [29], the process variables are x\({}_{1}\), temperature, and x\({}_{2}\), screw speed, and the response variables are y\({}_{\rm ORAC}\), antioxidant capacity (ORAC), and y\({}_{\rm WSI}\), water solubility index (WSI)--used in a central composite design including 13 runs formed by five central points. The proposed equations for the y response are complete models; that is, y\({}_{\rm ORAC}\) = b\({}_{0}\) + b\({}_{1}\)x\({}_{1}\) + b\({}_{2}\)x\({}_{2}\) + b\({}_{11}\)x\({}_{1}\)\({}^{2}\) + b\({}_{22}\)x\({}_{2}\)\({}^{2}\) + b\({}_{12}\)x\({}_{1}\)x\({}_{2}\) and y\({}_{\rm WSI}\) = c\({}_{0}\) + c\({}_{1}\)x\({}_{1}\)\({}_{+}\) c\({}_{2}\)x\({}_{2}\) + c\({}_{11}\)x\({}_{1}\)\({}^{2}\) + c\({}_{22}\)x\({}_{2}\) + c\({}_{12}\)x\({}_{1}\)x\({}_{2}\). Contour plots and response surface plots show the effect on y\({}_{\rm ORAC}\) and y\({}_{\rm WSI}\) of x\({}_{1}\) and x\({}_{2}\). The study presents all the curved relationships [29]. 1. The y\({}_{\rm ORAC}\) response  
The experimental data are listed in the study [29]; the multiple regression results for y\({}_{\rm ORAC}\) are  
\[y_{\rm ORAC}=1481.845+24.670\rm{x}_{1}-0.490\rm{x}_{2}-0.0262\rm{x}_{1}{}^{2} +0.0217\rm{x}_{2}{}^{2}-0.0725\rm{x}_{1}\rm{x}_{2} \tag{20}\]  
\[(4.601)\quad(-0.10)\quad(-4.730)\quad(2.065)\quad(-2.396)\]  
\[\rm{R}^{2}=0.859,\rm{R}^{2}_{\rm adj}=0.758,\rm{s}=122.503,\rm{PRESS}=162,8 11,173\]  
The numeric values in parentheses below the estimated values of parameters are the t-values of the estimated values of parameters. The normality test is passed (\(p\) = 0.285), and the constant variance test is passed (\(p\) = 0.295). The estimated values of each independent variable and its criteria are listed in Table 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Coefficient** & **Estimated** & **Standard** & & & **Standard** & \\  & **Value** & **Error** & **t-Value** & \(p\)**-Value** & **Coefficient** & **VIF** \\ \hline Constant & 1481.845 & 571.865 & 2.591 & 0.036 & & \\ x1 & 24.670 & 5.362 & 4.601 & 0.002 & 8.922 & 186.628 \\ x2 & \(-0.490\) & 4.456 & \(-0.110\) & 0.916 & \(-0.108\) & 48.234 \\ x1\({}^{2}\) & \(-0.0262\) & 0.00555 & \(-4.730\) & 0.002 & \(-4.705\) & 49.088 \\ x2\({}^{2}\) & 0.0217 & 0.0105 & 2.065 & 0.078 & 1.498 & 26.095 \\ x1x2 & \(-0.0725\) & 0.0302 & \(-2.396\) & 0.048 & \(-4.320\) & 161.328 \\ \hline \end{tabular}
\end{table} 
Table 3: The experimental values of each independent variable and its criteria for the y\({}_{\rm ORAC}\) response. Because the \({\rm x_{1}}^{2}\), \({\rm x_{2}}^{2}\), and \({\rm x_{1}x_{2}}\) variables are derived from \({\rm x_{1}}\) and \({\rm x_{2}}\), the five variables' variance inflation factor (VIF) is \(>\)10. This indicates the multicollinearity problems of these variables, and the backward elimination procedure is suitable for the RSM equations. In a comparison of the t-values of \({\rm x_{1}}^{2}\), \({\rm x_{2}}^{2}\), and \({\rm x_{1}x_{2}}\), the variable \({\rm x_{2}}^{2}\) has the lowest t-value, and its \(p\)-value is 0.078 (\(>\)0.05). The term \({\rm x_{2}}^{2}\) is deleted, and the regression equation is recalculated. \[{\rm y_{ORAC}}=974.020+25.670{\rm x_{1}}+6.138{\rm x_{2}}-0.0285{\rm x_{1}}^{2 }-0.0725{\rm x_{1}x_{2}} \tag{21}\]  
\[(4.039)\quad(1.673)\quad(-4.415)\quad(-2.019)\]  
In a comparison of the t-values of \({\rm x_{1}}^{2}\) and \({\rm x_{1}x_{2}}\), the \({\rm x_{1}x_{2}}\) variable has a lower t-value, and its \(p\)-value is 0.078. The variable of \({\rm x_{1}x_{2}}\) is deleted, and the regression equation is recalculated. \[{\rm y_{ORAC}}=2079.174+14.550{\rm x_{1}}-1.109{\rm x_{2}}-0.0285{\rm x_{1}}^{2} \tag{22}\]  
\[(3.927)\quad(-1.257)\quad(-3.811)\]  
The t-value of variable \({\rm x_{2}}\) is \(-1.2557\), and its \(p\)-value is 0.240 (\(>0.05\))  
The variable \({\rm x_{2}}\) is deleted, and the regression equation is recalculated.