# Numerical Design and Optimisation of Self-Compacting High Early-Strength Cement-Based Mortars
## 3 Response Models
### Compressive Strength: Rc,24 h Model  
Table 11 presents the ANOVA results for Rc,24 h. Examining the magnitude of the effects, it can be perceived that Vw/Vc (factor X\({}_{1}\)), Vw/Vp (factor X\({}_{3}\)), and Vfs/Vs (factor X\({}_{5}\)) are dominant compared to Sp/p (factor X\({}_{2}\)). The reduced quadratic model is significant (\(p<0.0001\)) and there is no lack of fit. The analysis of variance confirms the previous interpretation: Vw/Vc, Vw/Vp, and Vfs/Vs are significant terms of the model, i.e., the main effects Vw/Vc, Vw/Vp, and Vfs/Vs, the second order effects (Vs/Vm) \(\times\) (Vfs/Vs), (Vw/Vc)\({}^{2}\), and (Vw/Vp)\({}^{2}\), are those that most significantly influence the response Rc,24 h. In addition, the predicted R\({}^{2}\) of 0.9326 is reasonably close to the adjusted R\({}^{2}\). \begin{table}
\begin{tabular}{c c c c c} \hline
**Factor** & **Coefficient Estimate** & **Standard Error** & **95\% CI Low** & **95\% CI High** \\ \hline Intercept & 11.43 & 0.0893 & 11.25 & 11.61 \\ Vw/Vc & \(-\)0.4053 & 0.0724 & \(-\)0.5513 & \(-\)0.2592 \\ Vw/Vp & \(-\)0.261 & 0.0724 & \(-\)0.4071 & \(-\)0.115 \\ Vs/Vm & 0.2608 & 0.0724 & 0.1147 & 0.4068 \\ (Vw/Vp) \(\times\) (Vs/Vm) & 0.1826 & 0.0842 & 0.0127 & 0.3525 \\ (Vs/Vm)\({}^{2}\) & \(-\)0.2834 & 0.0631 & \(-\)0.4108 & \(-\)0.156 \\ \hline \end{tabular}
\end{table} 
Table 10: F24 h model coefficients in terms of coded values. Figure 5: (**a**) Normal probability plot of the externally studentised residuals for the F,24 h fitted model, and (**b**) plot of the externally studentised residuals versus F,24 h fitted values. The estimated values of the coefficients of the regression model that will be used to predict Rc,24 h in the coded values are presented in the second column of Table 12. Additionally, Table 12 shows the standard errors and confidence intervals. Figure 6a illustrates the normal probability plot of the residuals of the observed compressive strength values. It is reasonable to consider that there are no normality issues in the data. An example plot of the residuals against the predicted values is presented in Figure 6b. There is no evidence of assumption violation regarding the regression model fit. The subsequent diagnostics include the identification of possible outliers in the regression or observations that could be leverage points. Points with a Cook's distance higher than 1.0 or high leverage were not found (see Figure 7). \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Test for** & **Source** & **Sum of Squares** & **Degrees of Freedom** & **Mean of Square** & **F-Value** & \(p\)**-Value** \\ \hline \multirow{2}{*}{Significance of Regression} & Model & 1850.32 & 10 & 185.03 & 119.74 & \textless{}0.0001 \\  & Residual & 60.26 & 39 & 1.35 & & \\  & Cor Total & 1910.58 & 49 & & & \\ \hline \multirow{2}{*}{Lack of Fit} & Lack of Fit & 51.87 & 32 & 1.62 & 1.35 & 0.3599 \\  & Pure Error & 8.40 & 7 & 1.20 & & \\  & V\(\mathrm{w}\)/\(\mathrm{Vc}\) & **1650.75** & 1 & 1650.75 & 879.84 & \textless{}0.0001 \\  & Sp/p & 5.37 & 1 & 5.37 & 3.48 & 0.698 \\  & V\(\mathrm{w}\)/\(\mathrm{Vp}\) & **61.35** & 1 & 61.35 & 39.70 & \textless{}0.0001 \\ \multirow{2}{*}{Partial significance of each predictor variable} & V/\(\mathrm{V}\)/\(\mathrm{V}\) & 3.05 & 1 & 3.05 & 1.98 & 0.1677 \\  & V\(\mathrm{f}\)/\(\mathrm{V}\)s & **76.06** & 1 & 76.06 & 49.22 & \textless{}0.0001 \\  & (V\(\mathrm{w}\)/\(\mathrm{Vc}\)) \(\times\) (V\(\mathrm{w}\)/\(\mathrm{Vp}\)) & 6.08 & 1 & 6.08 & 3.93 & 0.0545 \\  & (V\(\mathrm{w}\)/\(\mathrm{Vc}\)) \(\times\) (V\(\mathrm{f}\)s/\(\mathrm{V}\)s) & 7.09 & 1 & 7.09 & 4.59 & 0.0385 \\  & (V\(\mathrm{w}\)/\(\mathrm{V}\)\(\mathrm{f}\)s/\(\mathrm{V}\)s) & 7.79 & 1 & 7.79 & 5.04 & 0.0305 \\  & (V\(\mathrm{w}\)/\(\mathrm{Vc}\))\({}^{2}\) & 18.43 & 1 & 18.43 & 11.93 & 0.0013 \\  & (V\(\mathrm{w}\)/\(\mathrm{Vp}\))\({}^{2}\) & 17.44 & 1 & 17.44 & 11.29 & 0.0018 \\ \hline \end{tabular} The three most significant parameters are typed in **bold**, and the most significant term is also **undefined**.  
\end{table}