## 3 Results and Discussion
### Analysis of Variance and Validation of the BBD  
Analysis of variance (ANOVA) is a method used to assess the suitability of the models employed (for MB dye removal) and to evaluate the significant parameters and validity of the MB dye removal findings. Table 5 presents the ANOVA statistics for the removal of MB dyes. The statistics in Table 5 display F-values that indicate the adequacy of the models applied in MB dye removal. The F-value for the MB dye removal was found to be 303.30 (\(p\)-value = 0.001). The R\({}^{2}\) and adjusted R\({}^{2}\) values for MB dye removal are 0.9974 and 0.9942, respectively. These findings demonstrate a good agreement between the empirical and calculated data [24]. In general, variables with a \(p\)-value less than 0.05 are considered statistically significant in terms of their effect on MB adsorption efficiency [25]. As a result, the model terms for MB dye removal, A, B, C, AB, AC, BC, A\({}^{2}\), B\({}^{2}\), and C\({}^{2}\), have been determined to be statistically significant. These terms are specified in Equation (4), including the outputs (MB dye removal) obtained after their correlations and the investigated variables in the form of a second-degree quadratic polynomial.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Sample** & **C** & **O** & **Fe** & **Ca** & **Al** & **K** \\ \hline PP & 40.3 & 58.4 & - & - & 0.03 & 1.1 \\ BCA & 82.9 & 15.9 & - & - & - & 1.12 \\ BCA-Mag/AB & 36.4 & 49.6 & 8.24 & 5.75 & - & - \\ BCA-Mag/AB+MB & 40.3 & 46.1 & 9.7 & 3.7 & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 4: EDX analysis results of samples.  
Figure 3: BBD plots of (**a**) normal plot of residuals; (**b**) predicted vs. actual values; (**c**) distributions of residuals in relation to the run numbers; and (**d**) [PERSON]’s distance plots.  
In addition, the BBD graphs assist in the validation of the applied BBD by examining the nature of the residuals' distribution, determining the level of relationship between the theoretical and experimental values of MB dye removal [26]. Figure 3a shows the normal probability plots of the residuals for the MB removal model. In these plots, the points closely align with a straight line, indicating that the points are uniformly and accurately distributed, highlighting that the residuals are independent [27].  
Figure 3b presents diagrams showing the relationship between the actual and theoretical values in MB dye removal, clearly indicating a strong relationship between the empirical outputs and the statistically expected values, which demonstrates that the models are statistically reliable. Figure 3c depicts the distributions of residuals in relation to the run numbers of the MB dyes. The points in Figure 3c are randomly scattered close to zero, indicating that the models are accurate and reliable. [PERSON]'s distance is another statistical method used to validate the applied models. The [PERSON]'s distance plots for MB removal are provided in Figure 3d. In the [PERSON]'s distance plot, points with a magnitude greater than one are considered outliers, questioning the reliability of the regression model. As a result, all points in Figure 3d are below one, with most being below 0.2 and many notable points seen as equal to 0; this suggests the validity of the applied models [28].