# Numerical Design and Optimisation of Self-Compacting High Early-Strength Cement-Based Mortars
## 3 Response Models
### Flexural Strength: F,24 h Model  
No leverage points were found for the F,24 h model and the Cook's distance was below 60%. Table 9 presents the ANOVA results for the F,24 h model. Examining the magnitude of the effects, it can be perceived that Vw/Vc (factor X\({}_{1}\)), Vw/Vp (factor X\({}_{3}\)), and Vs/Vm (factor X\({}_{4}\)) are dominant compared to Sp/p (factor X\({}_{2}\)) and Vfs/Vs (factor X\({}_{5}\)). The analysis of variance confirms the previous interpretation. The quadratic model is significant. In this case, Vw/Vc, Vw/Vp, and Vs/Vm are significant terms of the model, i.e., the main effects Vw/Vc, Vw/Vp, and Vs/Vm, and the second order effects (Vw/Vp) \(\times\) (Vs/Vm) and (Vs/Vm)\({}^{2}\) are those that most significantly influence the F,24 h response. The predicted R\({}^{2}\) of 0.5671 is in reasonable agreement with the adjusted R\({}^{2}\) of 0.6216 (i.e., the difference is less than 10%). Although R\({}^{2}\) did not reach a value close to 1, the model was considered useful for navigating the design space.  
The estimated values of the regression model coefficients in the coded values are presented in the second column of Table 10. Additionally, Table 10 presents standard errors and confidence intervals. Figure 5a presents the normal probability plot of the residuals of the observed values of flexural strength. The externally studentised residuals are presented in Figure 5b and show no strong evidence of violating the white noise assumptions of the remaining error in the regression model. It is reasonable to consider that there are no normality issues in the data.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Factor** & **Coefficient Estimate** & **Standard Error** & **95\% CI Low** & **95\% CI High** \\ \hline Intercept & 0.0688 & 0.0011 & 0.0667 & 0.071 \\ Vw/Vc & **0.0115** & 0.0007 & 0.0101 & 0.0129 \\ Sp/p & 0.005 & 0.0006 & 0.0038 & 0.0062 \\ Vw/Vp & **0.0248** & 0.0006 & 0.0236 & 0.026 \\ Vs/Vm & **-0.0302** & 0.0007 & \(-\)0.0315 & \(-\)0.0289 \\ Vfs/Vs & \(-\)0.0016 & 0.0006 & \(-\)0.0028 & \(-\)0.0004 \\ (Vw/Vc) \(\times\) (Sp/p) & \(-\)0.002 & 0.0007 & \(-\)0.0035 & \(-\)0.0006 \\ (Vw/Vc) \(\times\) (Vs/Vm) & \(-\)0.0037 & 0.0007 & \(-\)0.0051 & \(-\)0.0022 \\ (Vw/Vp) \(\times\) (Vs/Vm) & \(-\)0.0061 & 0.0007 & \(-\)0.0076 & \(-\)0.0047 \\ (Vw/Vc)\({}^{2}\) & \(-\)0.0017 & 0.0007 & \(-\)0.0031 & \(-\)0.0003 \\ (Vw/Vp)\({}^{2}\) & \(0.0011\) & 0.0005 & 0.0001 & 0.0021 \\ (Vfs/Vs)\({}^{2}\) & \(-\)0.0014 & 0.0005 & \(-\)0.0024 & \(-\)0.0003 \\ \hline \hline \end{tabular} The three most significant parameters are typed **bold** and the most significant term is also **underlined**.  
\end{table}
Table 8: T-funnel model coefficients (coded values).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Test for** & **Source** & **Sum of Squares** & **Degrees of Freedom** & **Mean of Square** & **F-Value** & **\(p\)-Value** \\ \hline \multirow{2}{*}{Significance of Regression} & Model & 18.65 & 5 & 3.73 & 16.44 & \textless{}0.0008 \\  & Residual & 9.53 & 42 & 0.2269 & & \\  & Total & 28.18 & 47 & & & \\ \hline \multirow{2}{*}{Lack of Fit} & Lack of Fit & 7.88 & 35 & 0.2252 & 0.9574 & 0.5812 \\  & Pure Error & 1.65 & 7 & 0.2352 & & \\ \hline \multirow{2}{*}{Partial significance of each predictor variable} & Vw/Vc & **7.11** & 1 & 7.11 & 31.36 & \textless{}0.0001 \\  & Vw/Vp & **2.95** & 1 & 2.95 & 13.01 & 0.0008 \\ \multirow{2}{*}{The three most significant parameters are typed in **bold**, and the most significant term is also **underlined**.  
\end{table}
Table 9: ANOVA results for the F,24 h response.