## 3 Results
### Three Variables
#### 3.2.3 Corn Extrudate Fortified with [PERSON] et al. [37] studied the optimization of the extrusion characteristics of corn-yam extrudates. Their variables were \(\mathrm{x_{1}}\), yam flour contents, \(\mathrm{x_{2}}\), moisture content, and \(\mathrm{x_{3}}\), screw speed. The four responses included \(\mathrm{y_{BD}}\), bulk density, \(\mathrm{y_{RER}}\), radial expansion ratio, \(\mathrm{y_{WAL}}\), water absorption index, and \(\mathrm{y_{HD}}\), hardness. The authors reported that their RSM equations were quadratic polynomial models. Then, the quadratic polynomial equations were used to make the contour and response surface plots, and the effects of variables on these responses were observed in the two types of plots.
The results of modern regression indicated that the x\({}_{1}\)\({}^{2}\) and x\({}_{3}\)\({}^{2}\) variables were valid, and only the x\({}_{2}\) variables had a linear relationship with the response y\({}_{\rm RER}\). 3. y\({}_{\rm WAI}\), water adsorption index  
The complete equation is  
\[\rm{+0.00000483x_{3}{}^{2}+0.000113x_{1}x_{2}+0.000000501x_{1}x_{3-}0.0000201\ x_{2}x_{3}}, \tag{45}\]0  
\[\rm{+0.00000483x_{3}{}^{2}+0.000113x_{1}x_{2}+0.000000501x_{1}x_{3-}0.0000201\ x_{2}x_{3}}, \tag{45}\]1  
\[\rm{+0.00000483x_{3}{}^{2}+0.000113x_{1}x_{2}+0.000000501x_{1}x_{3-}0.0000201\ x_{2}x_{3}}, \tag{45}\]2  
\[\rm{+0.00000483x_{3}{}^{2}+0.000113x_{1}x_{2}+0.000000501x_{1}x_{3-}0.0000201\ x_{2}x_{3}}, \tag{45}\]3  
The adequate equation is  
\[\rm{+0.00000483x_{3}{}^{2}+0.000113x_{1}x_{2}+0.000000501x_{1}x_{3-}0.0000201\ x_{2}x_{3}}, \tag{45}\]4  
\[\rm{+0.00000483x_{3}{}^{2}+0.000113x_{1}x_{2}+0.000000501x_{1}x_{3-}0.0000201\ x_{2}x_{3}}, \tag{45}\]5  
\[\rm{+0.00000483x_{3}{}^{2}+0.000113x_{1}x_{2}+0.000000501x_{1}x_{3-}0.0000201\ x_{2}x_{3}}, \tag{45}\]6  
The normality test and constant variance tests were passed. One influential data point was found, the fifth (t\({}_{\rm i}\) = 2.539). According to the modern regression, the x\({}_{1}\) variables did not significantly affect the y\({}_{\rm WAI}\) response. Only the x\({}_{2}\) variable had a quadratic relationship with the y\({}_{\rm WAI}\) response. 4. y\({}_{\rm HD}\), hardness  
The complete equation is  
\[\rm{+0.00000483x_{3}{}^{2}+0.000113x_{1}x_{2}+0.000000501x_{1}x_{3-}0.0000201\ x_{2}x_{3}}, \tag{45}\]7  
\[\rm{+0.00000483x_{3}{}^{2}+0.000113x_{1}x_{2}+0.000000501x_{1}x_{3-}0.0000201\ x_{2}x_{3}}, \tag{45}\]8  
\[\rm{+0.00000483x_{3}{}^{2}+0.000113x_{1}x_{2}+0.000000501x_{1}x_{3-}0.0000201\ x_{2}x_{3}}, \tag{45}\]9  
\[\rm{(1.392)\ \ \ (5.395)\ \ \ \ (0.200)\ \ \ \ \ \ \ \ \ (-4.795)}\]0  
\[\rm{(1.392)\ \ \ (5.395)\ \ \ \ (0.200)\ \ \ \ \ \ \ \ \ (-4.795)}\]1  
The adequate equation is  
\[\rm{(1.392)\ \ \ (5.395)\ \ \ \ (0.200)\ \ \ \ \ \ \ \ \ (-4.795)}\]2  
\[\rm{(1.392)\ \ \ (5.395)\ \ \ \ (0.200)\ \ \ \ \ \ \ \ \ (-4.795)}\]3  
\[\rm{(1.392)\ \ \ (5.395)\ \ \ \ (0.200)\ \ \ \ \ \ \ \ \ (-4.795)}\]4  
The normality test and constant variance tests were passed. One influential data point was found (ninth, t\({}_{\rm i}\) = \(-3.102\), DFFITS\({}_{\rm i}\) = \(-2.403\)). For the y\({}_{\rm HD}\) response, x\({}_{2}\) and x\({}_{3}\) have a linear relationship, and only x\({}_{1}\) has a quadratic form (curves). The authors used the complete models to produce the contour and response plots [37]. However, the ANOVA tables in their report indicated that some parameters did not significantly affect the response. In comparing the adequate equations with the complete equations proposed by the authors, the importance of using regression analysis correctly cannot be overstated.