## 3 Results
### Three Variables  
#### 3.2.1 Extruded African Breadfruit-Corn-Soy  
Nwabueze [33] reported the effect of three variables, x\({}_{1}\), feed composition, x\({}_{2}\), feed moisture, and x\({}_{3}\), screw speed, on three responses, y\({}_{\text{TIA}}\), trypsin inhibitor activity, TIA, \(Y_{\text{phytic acid}}\), phytic acid, and y\({}_{\text{tan}}\), tannin content, with a CCD experimental design. The replicates were performed at center points, and the total number of samples was twenty-five.  
The \(p\)-value was used to justify the significant effect of the coefficients of the parameters for three responses. If the \(p\)-value was higher than 0.05, the parameter was removed. The RSM models of this response are recorded according to the estimated regression values of the remaining parameters [33].  
Typical results of the ANOVA for y\({}_{\text{TIA}}\) presented in the study are listed in Table 5. This table lists the \(p\)-values of x\({}_{1}\), x\({}_{2}\), x\({}_{3}\), x\({}_{1}\)x\({}_{2}\), x\({}_{1}\)x\({}_{3}\), x\({}_{2}\)x\({}_{3}\), x\({}_{1}\)\({}^{2}\), x\({}_{2}\)\({}^{2}\), and x\({}_{3}\)\({}^{2}\), and then compares them with the cutoff value, \(p<\)0.05.  
In Table 5, only the coefficients of b\({}_{3}\) and b\({}_{11}\) significantly affected the response. The researchers then left these two variables alone and deleted all other variables at once. The estimated values of b\({}_{3}\) and b\({}_{11}\) in this table were used as the final estimated values.  
By the elimination-at-once method, the authors reported that the RSM equations in the study are [33]  
\[\text{y}_{\text{TIA}}=-2.980433+0.071086\text{x}_{3}+0.00427\text{x}_{1}\text{ }^{2} \tag{35}\]  
With the same technique, the other equations are  
\[Y_{\text{phytic acid}}=436.2951+0.022895\text{x}_{1}\text{ }^{2} \tag{36}\]  
\[\text{y}_{\text{tan}}=3.51248-0.0000186\text{x}_{3}\text{ }^{2} \tag{37}\]The authors' regression technique involved deleting all variables whose \(p\)-values were >0.05 at once, but the interaction effect of these variables was not considered.  
Another question arises about the form of these equations [33]. The \({\rm x_{1}}^{2}\) and \({\rm x_{3}}^{2}\) variables are derived from the \({\rm x_{1}}\) and \({\rm x_{3}}\) variables for the polynomial equations. Suppose that the \({\rm x_{i}}^{2}\), \({\rm x_{j}}^{2}\), or \({\rm x_{i}}{\rm x_{k}}\) variables are effective parameters. In this case, the \({\rm x_{i}}\) and \({\rm x_{j}}\) variables are validated parameters; the \({\rm x_{i}}\) and \({\rm x_{j}}\) variables should be included in this regression because the \({\rm x_{i}}^{2}\), \({\rm x_{j}}^{2}\), or \({\rm x_{i}}{\rm x_{j}}\) variables are derived from the \({\rm x_{i}}\) and \({\rm x_{j}}\) variables.  
The selection steps for \({\rm y_{TIA}}\) are listed in Appendix A.2. The adequate equations evaluated by modern regression analysis are  
\[{\rm y_{TIA}}= 1.574+0.0622{\rm x_{1}}-0.00307{\rm x_{3}}+0.173{\rm x_{1}}^{2}+0. 168{\rm x_{3}}^{2} \tag{38}\] \[{\rm y_{Dyptic\,acid}}= 101.215-1.753{\rm x_{1}}-0.984{\rm x_{2}}+5.186{\rm x_{1}}^{2}-8.4 41{\rm x_{2}}^{2}\] (39) \[{\rm y_{tan}}= 103.957-0.984{\rm x_{2}}-8.273{\rm x_{2}}^{2} \tag{40}\]  
The normality and constant variance tests were passed. \({\rm y_{TIA}}\), \({\rm y_{Dyptic\,acid}}\), and \({\rm y_{tan}}\) have one, two, and two influential data points, respectively.  
The study presented three 3D response surface plots. These plots are plotted with the complete models involving \({\rm x_{1}}^{2}\), \({\rm x_{2}}^{2}\), and \({\rm x_{3}}^{2}\)[33]. However, the RSM equations proposed by the authors (Equations (35)-(37)) are not complete equations. The curves of these figures did not present appropriate results for the relationship among the three responses and variables. Adequate RSM equations are essential for providing helpful information for researchers.