## 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 authors used the coefficient of determination \(\mathrm{R}^{2}\) and lack of fit as criteria to evaluate significant effects for all quadratic equations. However, their ANOVA table in the study showed an insignificant effect of some variables at \(p\) < 0.01 and \(p\) < 0.05 [37]. That is, the authors misunderstood the meaning of a significant test. The modern regression technique evaluates the adequate equation. The experimental data were listed in the study [37]. The results are listed as follows:\(\rm{y_{BD}}\), bulk density  
The complete equation is  
\[\rm{y_{BD}=0.0449-0.00192x_{1}+0.00657x_{2-}0.000122x_{3}+0.0000221x_{1}{}^{2}+0. 0000599\ x_{2}{}^{2}}\]  
\[\rm{(-2.833)\ \ \ (3.258)\ \ \ (-0.559)\ \ \ (2.543)\ \ \ (1.104)}\]  
\[\rm{+0.00000483x_{3}{}^{2}+0.000113x_{1}x_{2}+0.000000501x_{1}x_{3-}0.0000201\ x_{2}x_{3}}, \tag{45}\]  
\[\rm{(1.392)\ \ \ (5.395)\ \ \ \ (0.200)\ \ \ \ \ \ \ \ \ (-4.795)}\]  
\[\rm{R^{2}=0.996,\ R^{2}_{adj}=0.988,\ s=0.002,\ PRES=1.031\times 10^{-4}}\]  
The adequate equation is  
\[\rm{y_{BD}=-0.0127-0.00171x_{1}+0.00825\ x_{2}+0.000177\ x_{3}+0.0000205x_{1}{} ^{2}}\]  
\[\rm{(-3.741)\ \ \ (6.206)\ \ \ (2.975)\ \ \ \ (2.374)}\]  
\[\rm{+0.000113x_{1}x_{2}-0.0000201x_{2}x_{3}}, \tag{46}\]  
\[\rm{(5.586)\ \ \ \ \ (-4.787)}\]  
\[\rm{R^{2}=0.993,\ R^{2}_{adj}=0.988,\ s=0.002,\ PRES=7.71\times 10^{-5}}\]  
The normality test was failed (\(p=0.047\)), and the constant variance test was passed (\(p=0.281\)). The influential data were the 8 th data point (\(\rm{DFFITS_{i}}\) =2.338) and the 13 th (\(\rm{t_{i}=-3.737}\)). The adequate equation only involved the \(\rm{x_{1}}^{2}\) variable. In other words, \(\rm{x_{1}}^{2}\) was the only quadratic variable that influenced the \(\rm{y_{BD}}\) response. \(\rm{x_{2}}^{2}\) and \(\rm{x_{3}}^{2}\) did not significantly affect the \(\rm{y_{1}}\) response. An inappropriate equation could induce an incorrect result when producing response surface plots. The datasets did not pass the normality test, and two influential data points were found. Further study needs to be performed. 2. \(\rm{y_{REF}}\) radial expansion ratio  
The complete equation is  
\[\rm{(-2.833)\ \ \ (3.258)\ \ \ (-0.559)\ \ \ (2.543)\ \ \ (1.104)}\]0  
\[\rm{(-2.833)\ \ \ (3.258)\ \ \ (-0.559)\ \ \ (2.543)\ \ \ (1.104)}\]1  
\[\rm{(-2.833)\ \ \ (3.258)\ \ \ (-0.559)\ \ \ (2.543)\ \ \ (1.104)}\]2  
\[\rm{(-2.833)\ \ \ (3.258)\ \ \ (-0.559)\ \ \ (2.543)\ \ \ (1.104)}\]3  
\[\rm{(-2.833)\ \ \ (3.258)\ \ \ (-0.559)\ \ \ (2.543)\ \ \ (1.104)}\]4  
The adequate equation is  
\[\rm{(-2.833)\ \ \ (3.258)\ \ \ (-0.559)\ \ \ (2.543)\ \ \ (1.104)}\]5  
\[\rm{(-2.833)\ \ \ (3.258)\ \ \ (-0.559)\ \ \ (2.543)\ \ \ (1.104)}\]6  
\[\rm{(-2.833)\ \ \ (3.258)\ \ \ (-0.559)\ \ \ (2.543)\ \ \ (1.104)}\]7  
\[\rm{(-2.833)\ \ \ (3.258)\ \ \ (-0.559)\ \ \ (2.543)\ \ \ (1.104)}\]8  
\[\rm{(-2.833)\ \ \ (3.258)\ \ \ (-0.559)\ \ \ (2.543)\ \ \ (1.104)}\]9The normality test was passed (\(p\) = 0.442), and the constant variance test was passed (\(p\) = 0.620). Two influential data points were found, the fourth (t\({}_{\rm i}\) = 2.966, DFFITS\({}_{\rm i}\) = 4.494) and sixth (t\({}_{\rm i}\) = 2.657, DFFITS\({}_{\rm i}\) = 2.366).