## 3 Result and Discussion  
### Fitting the Models  
In this study, the lower and upper values for the factors chosen were set at +alpha (+ \(\alpha\) = 1.682) and \(-\)alpha (- \(\alpha\) = 1.682) respectively and therefore all the factor levels were selected within the desired limits. The experimental and predicted values for all the three different responses (DPPH, FRAP, and BCB) under various combinations of extraction conditions are given in Table 2. The results showed that the antioxidant activities of the red pitaya peels for the DPPH, FRAP, and BCB assays ranged from 62.12 to 77.45%, 6.05 to 7.38 mM Fe\({}^{2+}\)/g dry weight, and 78.18 to 93.47%, respectively, for the samples treated under extraction conditions as specified in Table 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Std. \({}^{\textbf{a}}\) Order**} & \multicolumn{2}{c}{**DPPH \({}^{\textbf{b}}\)**} & \multicolumn{2}{c}{**FRAP \({}^{\textbf{c}}\)**} & \multicolumn{2}{c}{**BCB \({}^{\textbf{d}}\)**} \\ \cline{2-7}  & **Exp. \({}^{\textbf{e}}\)** & **Pred. \({}^{\textbf{f}}\)** & **Exp. \({}^{\textbf{e}}\)** & **Pred. \({}^{\textbf{f}}\)** & **Exp. \({}^{\textbf{e}}\)** & **Pred. \({}^{\textbf{f}}\)** \\ \hline
1 & 62.12 & 62.07 & 6.13 & 6.14 & 82.79 & 82.86 \\
2 & 69.98 & 69.90 & 6.51 & 6.51 & 79.03 & 79.11 \\
3 & 68.01 & 67.96 & 6.05 & 6.05 & 78.18 & 78.38 \\
4 & 67.63 & 67.53 & 6.62 & 6.63 & 81.47 & 81.49 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The experimental and predicted values for the responses of the 1,1-Diphenyll-2-picrylhydrazyl (DPPH), ferric ion reducing antioxidant power (FRAP), and beta-carotene bleaching (BCB) assays under different extraction conditions. In order to fit the response function and experimental data to the second-order polynomial, the linearity and quadratic effect of the independent variables, their interactions, and their regression coefficients on the response variables were evaluated from variance analysis ANOVA as given in Table 3. The coefficient of determination, \(R^{2}\), for the regressed model predicting the antioxidant activities for the DPPH, FRAP, and BCB assays were 0.9992, 0.9993, and 0.9994, respectively, suggesting a good fit as the values were close to 1. Thus, the predicted second-order polynomial models defined the real behavior of the system very well. The absence of lack of fit (\(p>0.05\)) for all the responses also strengthened the accuracy of the models. The predicted model of the DPPH, FRAP, and BCB assays were obtained from the following second-order polynomial equations:  
\[\mathrm{Y_{DPPH}}=76.17-1.02\mathrm{A}+1.23\mathrm{B}+1.48\mathrm{C}-2.06 \mathrm{AB}-2.87\mathrm{AC}+0.35\mathrm{BC}-1.47\mathrm{A}^{2}-3.70\mathrm{B}^{ 2}-2.66\mathrm{C}^{2} \tag{7}\]  
\[\mathrm{Y_{FRAP}}=7.35+0.10\mathrm{A}-0.025\mathrm{B}+0.059\mathrm{C}+0.051 \mathrm{AB}-0.14\mathrm{AC}-0.032\mathrm{BC}-0.32\mathrm{A}^{2}-0.30\mathrm{B} ^{2}-0.35\mathrm{C}^{2} \tag{8}\]  
\[\mathrm{Y_{BCB}}=92.31+1.58\mathrm{A}+1.52\mathrm{B}+2.22\mathrm{C}+1.72 \mathrm{AB}+1.74\mathrm{AC}+2.04\mathrm{BC}-3.47\mathrm{A}^{2}-3.25\mathrm{B} ^{2}-2.90\mathrm{C}^{2} \tag{9}\]  
The adjusted \(R^{2}\) for the responses of the DPPH, FRAP, and BCB assays were 0.9986, 0.9986, and 0.9989, respectively, and these values were also relatively close to their corresponding \(R^{2}\) value. In addition, high values of adjusted \(R^{2}\) also advocated for the significance of the model for all the responses. The coefficient of variation for the DPPH, FRAP, and BCB assays were 0.26, 0.26, and 0.22. These small values of CV also revealed that the experimental results obtained were accurate and reliable with a high reproducibility.