# Numerical Design and Optimisation of Self-Compacting High Early-Strength Cement-Based Mortars
## 3 Response Models
### T-Funnel Model  
Regarding the T-funnel data, attempts to fit a quadratic model to all points lead to a severe lack of fit, potentially due to strong non-linearities. F15 exhibits an abnormal residual which compromises the fit; the following modelling steps exclude this point (see Figure 3).  
The inverse transformation was applied to the T-funnel response, as found in previous works [3,14]. Examining the magnitude of the effects, it can be perceived that Vs/Vm (factor X\({}_{4}\)) and Vw/Vp (factor X\({}_{3}\)) are dominant compared to Sp/p (factor X\({}_{2}\)), Vw/Vc  
Figure 3: T-funnel results: F15 excluded (arrow).  
Figure 2: (**a**) Normal probability plot of the externally studentised residuals for the D-flow fitted model, and (**b**) plot of the externally studentised residuals versus D-flow predicted values.  
(factor X\({}_{1}\)), and Vfs/Vs (factor X\({}_{5}\)), as obtained in the D-flow response (see Table 7). All five factors are significant, as well as three of the interactions and the other three s-order terms. The predicted R\({}^{2}\) of 0.9827 is in reasonable agreement with the adjusted R\({}^{2}\) of 0.9897, i.e., the difference is negligible.  
The estimated coefficients of the regression model for the T-funnel (in coded values) are presented in the second column of Table 8. Additionally, Table 8 presents standard errors and confidence intervals. To check the regression assumptions on the residuals, a normal probability plot was examined in Figure 4a and other plots such as the one in Figure 4b. No outliers were detected, although F15 has been considered a leverage point. In the author's view, this point does not considerably compromise the fitted model's quality.  
\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 Significance of & Model & 0.0640 & 11 & 0.0058 & 394.23 & \textless{}0.0001 \\ Regression & Residual & 0.0005 & 34 & 0 & & \\ Total & 0.0645 & 45 & & & \\ \hline Lack of Fit & Lack of Fit & 0.0005 & 27 & 0 & 3.14 & 0.0607 \\  & Pure Error & 0 & 7 & \(5.457\times 10^{-6}\) & & \\ \hline  & Vw/Vc & **0.0041** & 1 & 0.0039 & 305.22 & \textless{}0.0001 \\  & Sp/p & 0.0010 & 1 & 0.0011 & 82.2 & \textless{}0.0001 \\  & Vw/Vp & **0.0251** & 1 & 0.0243 & 1895.62 & \textless{}0.0001 \\  & Vs/Vm & **0.0317** & 1 & 0.0271 & 2113.06 & \textless{}0.0001 \\ Partial significance of & Vfs/Vs & 0.0001 & 1 & 0.0001 & 6.9 & 0.0131 \\ each predictor & (Vw/Vc) \(\times\) (Sp/p) & 0.0001 & 1 & 0.0001 & 11.61 & 0.0018 \\ variable & (Vw/Vc) \(\times\) (Vs/Vm) & 0.0004 & 1 & 0.0003 & 23.66 & \textless{}0.0001 \\  & (Vw/Vp) \(\times\) (Vs/Vm) & 0.0011 & 1 & 0.0009 & 73.86 & \textless{}0.0001 \\  & (Vw/Vc)\({}^{2}\) & 0.0001 & 1 & 0.0001 & 6.79 & 0.0138 \\  & (Vw/Vp)\({}^{2}\) & 0.0001 & 1 & 0.0001 & 4.84 & 0.0351 \\  & (Vfs/Vs)\({}^{2}\) & 0.0001 & 1 & 0.0001 & 8.39 & 0.0067 \\ \hline \end{tabular}
\end{table}
Table 7: ANOVA results for the T-funnel response.  
Figure 4: (**a**) Normal probability plot of the externally studentised residuals for the T-funnel fitted model, and (**b**) plot of the externally studentised residuals versus T-funnel fitted values.