## 3 Results and Discussions
### Model Fit and Adequacy for the Stability of ACM Stabilized O/W-Type Nanoemulsion  
The precise stability parameters for D\({}_{(3,2)}\) obtained from the mixture design for determining the ACM stabilized O/W-type nanoemulsion formulation is presented in Table 1. The summary of the model statistics is shown in Table 2. The quadratic mixture model for D\({}_{(3,2)}\) was significant (F [235.71, 0.13] = 1805.99; \(p\)-value \(<\) 0.0001) in describing the component effects (ACM, oil, and water) on D\({}_{(3,2)}\) stability parameter (Table 2). There was only a 0.01% chance that a model F-value this large could occur due to noise. The model's lack of fit (F [0.17, 0.11] = 0.3104; \(p\) = 1.49) was not significant, and the predicted R-squared value was 0.9977 indicating a model with adequate goodness-of-fit (Table 2). All variables were fitted to a quadratic model, and the residual errors were calculated to determine the goodness of the model fit. All variables were fitted to a polynomial quadratic model, and the residual errors were calculated to determine the goodness of model fit. The normality of data was confirmed with the normality plot of the residuals adequacy tool. The straight line obtained from the plot indicated a normal distribution of residuals and showed that changes in transformation would not lead to any improvement of the analyzed data. Studies have shown that the Box-Cox plot, defined as \"the natural log of the sum of the squares of the residuals against lambda,\" indicates whether lambda should be transformed [8; 18]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Independent Variable**} & \multicolumn{2}{c}{**Dependent Variable**} \\ \hline
**Formulation/Run** & **A: ACM b (\%)** & **B: Oil b (\%)** & **C: Water b (\%)** & **D\({}_{(3,2)}\) b ** \\ \hline
1 & 1.0 & 9.6 & 89.4 & 31.9 \(\pm\) 0.6 a  \\
2 & 1.0 & 9.6 & 89.4 & 32.4 \(\pm\) 0.6 a  \\
3 & 1.0 & 3.0 & 96.0 & 10.2 \(\pm\) 0.3 a  \\
4 & 5.0 & 7.3 & 87.7 & 7.0 \(\pm\) 0.2 a  \\
5 & 1.0 & 5.7 & 93.3 & 23.2 \(\pm\) 0.6 a  \\
6 & 5.0 & 3.0 & 92.0 & 6.7 \(\pm\) 0.1 a  \\
7 & 5.0 & 3.0 & 92.0 & 6.6 \(\pm\) 0.1 a  \\
8 & 5.0 & 10 & 85.0 & 7.4 \(\pm\) 0.2 a  \\
9 & 2.3 & 10 & 87.7 & 15.4 \(\pm\) 0.4 a  \\
10 & 1.0 & 3.0 & 96.0 & 10.4 \(\pm\) 0.4 a  \\
11 & 2.9 & 3.1 & 93.9 & 15.9 \(\pm\) 0.5 a  \\
12 & 3.3 & 6.2 & 90.5 & 13.8 \(\pm\) 0.4 a  \\
13 & 5.0 & 10.0 & 85.0 & 7.4 \(\pm\) 0.2 a  \\
14 & 1.0 & 8.0 & 91.0 & 29.1 \(\pm\) 0.5 a  \\
15 & 5.0 & 4.6 & 90.4 & 7.0 \(\pm\) 0.2 a  \\
16 & 3.3 & 6.2 & 91.0 & 13.6 \(\pm\) 0.4 a  \\ \hline \hline \end{tabular}  
* Indicated values are reported as means \(\pm\) standard deviation (n = 5), b ACM—African catfish mucilage, oil—sunflower oil, and water—MilliQ water, and D\({}_{(3,2)}\)—Sauter mean diameter.  
\end{table} 
Table 1: Composition and observed responses from randomized runs for D-optimal mixture design. In this case, the statistical tool did not recommend any transformation for the D\({}_{(3,2)}\) response variables of ACM stabilzed O/W-type nanoemulsions. Hence, the model was found adequate to determine the effects of the three ingredients (ACM, oil, and water) on the precise stability parameters for particle size as a function of Sauter mean diameter D\({}_{(3,2)}\). Based on these values, the quadratic mixture model was found to be adequate in determining the effects of the three ingredients. The final equation in terms of actual components for D\({}_{(3,2)}\) is given by Equation (5). \[ \begin{array}{l}\gamma_{1}\left(\text{BS flux}\%\right)\;=\;-267.43\;*\;( \text{ACM})\;+\;6.48\;*\;(\text{Oil})\;+\;0.64\;*\;(\text{Water})\;+\;3.55\;*\;( \text{ACM}\;*\;\text{Oil})\;+\;2.86\\ \;*\;(\text{ACM}\;*\;\text{Water})\;-\;0.12\;*\;(\text{Oil}\;*\;\text{Water}) \end{array}  \tag{5}\]  
The R\({}^{2}\) value for the D\({}_{(3,2)}\) model was 0.9992 and the closeness of this value to 1 makes it represent a good fit.