# Effect of Human Hair Fibers on the Performance of Concrete Incorporating High Dosage of Silica Fume
## 4 Anova
### Optimization and Experimental Validation  
Multiple response optimization approaches are used to create the validation combination design. If the goal is to maximize reaction and mechanical strength, this answer gives four options. It is suggested that you select the option that meets your highest expectations. Experimental validation has been performed on the RSM equations. One technique to validate the model is to plot the residual normal plot, which is a straight line. As a result, if all points are almost parallel to the normal line, the predicted value will produce more accurate results than expected. The validity of the model and the capacity to select the appropriate extraction parameters based on the response outcomes are determined using normal residual plots. Table 8 shows the experiment and RSM results to summarize optimization mix results. The mean and standard deviation show that 2% HHFC and 15% SF, 21.5 MPa compressive strength, 2.94 MPa flexural strength, and 2.78 MPa split tensile strength can be achieved.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Source**} & **Sum of** & \multirow{2}{*}{**df**} & \multirow{2}{*}{**Mean Square**} & \multirow{2}{*}{**F-Value**} & \multirow{2}{*}{**\(p\)-Value**} & \multirow{2}{*}{**Significance of the Model**} \\ \cline{5-6}  & & & **Squares** & & & \\ \hline \multirow{5}{*}{Compressive Strength} & Model & 139.90 & 3 & 46.63 & 5.82 & 0.0124 & Significant \\ \cline{2-6}  & A-Cement & 130.41 & 1 & 130.41 & 16.28 & 0.0020 & \\ \cline{2-6}  & B-Human Hair Fiber & 5.40 & 1 & 5.40 & 0.6746 & 0.4289 & \\ \cline{2-6}  & C-Silica Fume & 5.01 & 1 & 5.01 & 0.6259 & 0.4456 & \\ \cline{2-6}  & Residual & 88.13 & 11 & 8.01 & & & \\ \cline{2-6}  & Cor Total & 228.03 & 14 & & & & \\ \hline \multirow{5}{*}{Split Tensile Strength} & Model & 6.24 & 3 & 2.08 & 5.59 & 0.0141 & Significant \\ \cline{2-6}  & A-Cement & 0.2982 & 1 & 0.2982 & 0.8013 & 0.3899 & \\ \cline{2-6}  & B-Human Hair Fiber & 3.40 & 1 & 3.40 & 9.15 & 0.0116 & \\ \cline{2-6}  & C-Silica Fume & 2.54 & 1 & 2.54 & 6.83 & 0.0241 & \\ \cline{2-6}  & Residual & 4.09 & 11 & 0.3722 & & & \\ \cline{2-6}  & Cor Total & 10.34 & 14 & & & & \\ \cline{2-6}  & Model & 6.24 & 3 & 2.08 & 5.59 & 0.0141 & Significant \\ \cline{2-6}  & A-Cement & 0.2982 & 1 & 0.2982 & 0.8013 & 0.3899 & \\ \hline \multirow{5}{*}{Flexural Strength} & Model & 0.8897 & 9 & 0.0989 & 7.50 & 0.0117 & Significant \\ \cline{2-6}  & A-Marble Powder & 0.6440 & 1 & 0.6440 & 48.89 & 0.0004 & \\ \cline{2-6}  & B-PP Fiber & 0.0179 & 1 & 0.0179 & 1.36 & 0.2880 & \\ \cline{2-6}  & C-Cement & 0.0005 & 1 & 0.0005 & 0.0394 & 0.8492 & \\ \cline{2-6}  & Residual & 0.0790 & 6 & 0.0132 & & & \\ \cline{2-6}  & Lack of Fit & 0.0789 & 5 & 0.0158 & 159.28 & 0.0601 & Significant \\ \cline{2-6}  & Pure Error & 0.0001 & 1 & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 6: ANOVA Analysis of the Response Models.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & **Compressive** & **Split Tensile** & **Flexural** & \multirow{2}{*}{**Source**} & \multirow{2}{*}{**Sequential**} \\  & **Strength** & **Strength** & **Strength** & & **\(p\)-Value** \\ \hline
**Std. Dev** & 2.83 & 0.1207 & 0.6101 & Linear & \textless{}0.0001 \\ \hline
**Mean** & 20.59 & 2.33 & 9.81 & Linear & \textless{}0.0001 \\ \hline
**C.V\%** & 13.74 & 5.18 & 6.22 & Linear & \textless{}0.0001 \\ \hline \(\mathbf{R}^{2}\) & 0.8852 & 0.8720 & 0.8674 & Linear & \textless{}0.0001 \\ \hline
**Adj. \(\mathbf{R}^{2}\)** & 0.8672 & 0.8520 & 0.8210 & Linear & \textless{}0.0001 \\ \hline
**Pred. \(\mathbf{R}^{2}\)** & 0.8642 & 0.8487 & 0.8143 & Linear & \textless{}0.0001 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Models\({}^{\prime}\) validation.