## 3 Results and Discussion
### Regression Analysis  
The results were analyzed using the analysis of variance (ANOVA), and the statistical significance was checked using an F-test, as shown in Table 3. The \(F\)-value (13.3) of the regression model (\(p<0.01\)) showed a high significance, confirming the adequacy of the quadratic model. The high value of the determination coefficient \(R^{2}\) (0.9447) demonstrated that the model adequately represented a real relationship between the independent variables, which showed that 94% of the variation in the response could be explained by the model [25]. The \(p\)-value for the lack of fit (0.442) implied that the lack of fit was not significant compared to the pure error. Furthermore, the coefficient of variation (\(C.V.=4.86\%\)) indicated a high degree of precision and high reliability of the experimental results. The values of Prob \(>F\) were less than 0.05, indicating that the model terms were significant [26]. A regression analysis of the experimental design shown in Table 3 demonstrates that the linear model terms (\(X_{1}\), \(X_{2}\), and \(X_{3}\)) and interacting model terms (\(X_{1}X_{3}\)) were significant for the flavonoid yield.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**RUN** & \(\mathbf{X_{1}}\) & \(\mathbf{X_{2}}\) & \(\mathbf{X_{3}}\) & \(\mathbf{Y}\) \\  & **(_w/w \%_)** & **(_w/w \%_)** & **(mL g\({}^{-1}\))** & **(mg g\({}^{-1}\))** \\ \hline
1 & \(-1\) & 0 & \(-1\) & 2.37 \\
2 & 0 & \(-1\) & \(-1\) & 3.46 \\
3 & \(-1\) & 0 & 1 & 3.80 \\
4 & 0 & \(-1\) & 1 & 3.81 \\
5 & \(-1\) & 1 & 0 & 2.70 \\
6 & 0 & 1 & 1 & 3.59 \\
7 & 1 & 0 & 1 & 4.14 \\
8 & 1 & 0 & \(-1\) & 3.54 \\
9 & 1 & 1 & 0 & 3.62 \\
10 & 0 & 1 & \(-1\) & 2.68 \\
11 & 0 & 0 & 0 & 3.75 \\
12 & 0 & 0 & 0 & 3.48 \\
13 & 0 & 0 & 0 & 3.37 \\
14 & 1 & \(-1\) & 0 & 3.89 \\
15 & 0 & 0 & 0 & 3.38 \\
16 & \(-1\) & \(-1\) & 0 & 3.11 \\
17 & 0 & 0 & 0 & 3.38 \\ \hline \hline \end{tabular} Note: * \(p\)-value less than 0.05 indicates statistical significance; ** \(p\)-value less than 0.01 indicates statistically extreme significance.  
\end{table}
Table 2: Experimental design and results for response surface analysis.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **Sum of Squares** & **df** & **Mean Square** & \(F\) **Value** & \(p\)**-Value Prob \(>F\)** & **Significance** \\ \hline Model & 3.31 & 9 & 0.37 & 13.3 & 0.0013 & ** \\ \(X_{1}\) & 1.29 & 1 & 1.29 & 46.65 & 0.0002 & ** \\ \(X_{2}\) & 0.35 & 1 & 0.35 & 12.81 & 0.009 & ** \\ \(X_{3}\) & 1.35 & 1 & 1.35 & 49.01 & 0.0002 & ** \\ \(X_{1}X_{2}\) & \(4.70\times 10^{-3}\) & 1 & \(4.70\times 10^{-3}\) & 0.17 & 0.6925 & \\ \(X_{1}X_{3}\) & 0.17 & 1 & 0.17 & 6.21 & 0.0414 & * \\ \(X_{2}X_{3}\) & 0.076 & 1 & 0.076 & 2.74 & 0.1419 & \\ \(X_{1}{}^{2}\) & 4.01 \(\times 10^{-3}\) & 1 & \(4.01\times 10^{-3}\) & 0.15 & 0.7144 & \\ \(X_{2}{}^{2}\) & 0.051 & 1 & 0.051 & 1.85 & 0.2161 & \\ \(X_{3}{}^{2}\) & \(2.06\times 10^{-3}\) & 1 & \(2.06\times 10^{-3}\) & 0.075 & 0.7925 & \\ Residual & 0.19 & 7 & 0.028 & & & \\ Lack of Fit & 0.088 & 3 & 0.029 & 1.11 & 0.442 & Not significant \\ Pure Error & 0.11 & 4 & 0.026 & & & \\ Cor Total & 3.5 & 16 & & & \\ \(R^{2}\) & 0.9447 & & & & \\ \(C.V.\) & 4.86 & & & & \\ \hline \hline \end{tabular} Note: * \(p\)-value less than 0.05 indicates statistical significance; ** \(p\)-value less than 0.01 indicates statistically extreme significance.  
\end{table}
Table 3: Analysis of variance (ANOVA) for the quadratic model.