# Numerical Design and Optimisation of Self-Compacting High Early-Strength Cement-Based Mortars
## 3 Response Models
### Significant Individual and Interaction Effects  
The final models (in terms of coded factors) are illustrated in Figure 8 regarding the relative impact of each factor by comparing the corresponding coefficient with the remaining (higher values of the estimated coefficient indicate the higher influence of the design variable in the response). The sign (positive or negative) of the estimated coefficient is marked on each bar's label. We recall that a positive coefficient means that the response (or transformed response) variable will increase if the given mixture parameter increases and vice-versa. \begin{table}
\begin{tabular}{c c c c c} \hline
**Factor** & **Coefficient Estimate** & **Standard Error** & **95\% CI Low** & **95\% CI High** \\ \hline Intercept & 60.65 & 0.2733 & 60.09 & 61.2 \\ Vw/Vc & \(-\)6.17 & 0.1889 & \(-\)6.56 & \(-\)5.79 \\ Sp/p & \(-\)0.3522 & 0.1889 & \(-\)0.7342 & 0.0299 \\ Vw/Vp & \(-\)1.19 & 0.1889 & \(-\)1.57 & \(-\)0.8081 \\ Vs/Vm & 0.2655 & 0.1889 & \(-\)0.1165 & 0.6476 \\ Vfs/Vs & \(-\)1.33 & 0.1889 & \(-\)1.71 & \(-\)0.9431 \\ (Vw/Vc) \(\times\) (Vw/Vp) & 0.4357 & 0.2197 & \(-\)0.0088 & 0.8802 \\ (Vw/Vc) \(\times\) (Vfs/Vs) & 0.4706 & 0.2197 & 0.0261 & 0.9151 \\ (Vs/Vm) \(\times\) (Vfs/Vs) & \(-\)0.4934 & 0.2197 & \(-\)0.9379 & \(-\)0.049 \\ (Vw/Vc)\({}^{2}\) & 0.5639 & 0.1633 & 0.2336 & 0.8941 \\ (Vw/Vp)\({}^{2}\) & 0.5486 & 0.1633 & 0.2183 & 0.8788 \\ \hline \end{tabular}
\end{table} 
Table 12: Rc,24 h model coefficients in terms of coded values. Figure 7: Rc,24 h: (**a**) [PERSON]’s distance and (**b**) leverage. The response surfaces corresponding to the two main effects for each response of the mixtures are illustrated in Figure 9. For the D-flow and the T-funnel variation, the most significant factor is the volume of sand concerning the mortar (Vs/Vm) and the content of water/fines factors (Vw/Vp). In addition, it was possible to confirm the great influence of the water/cement factor (Vw/Vc) on the flexural (F,24 h) and compressive strength (Rc,24 h) of the mortars. In the order of inversely proportional influence, there is the fine sand concerning medium sand (Vfs/Vs) and the content of water/fines factors (Vw/Vp). The other factors have a smaller influence on the responses, exhibiting smaller estimated coded coefficients. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model Terms** & **D-Flow (mm)** & **1/(T-Funnel) (s)** & **F,24 h (MPa)** & **Rc,24 h (MPa)** \\ \hline Independent & \(-\)2154.81 & \(-\)1.3582 & 5.400 & 293.77 \\ Vw/Vc & 5375.51 & 1.2092 & \(-\)4.529 & \(-\)257.75 \\ Sp/p & 15,060.05 & 13.4743 & - & \(-\)186.47 \\ Vw/Vp & 1141.35 & 1.1255 & \(-\)36.616 & \(-\)289.80 \\ Vs/Vm & \(-\)1375.32 & 1.4119 & 85.487 & 32.94 \\ Vfs/Vs & \(-\)365.99 & 0.0374 & - & \(-\)7.31 \\ (Vw/Vc) \(\times\) (Vw/Vp) & \(-\)1050.80 & - & - & 85.43 \\ (Vw/Vc) \(\times\) (Sp/p) & \(-\)17,105.24 & \(-\)12.0954 & - & - \\ (Vw/Vc) \(\times\) (Vs/Vm) & 723.96 & \(-\)0.8550 & - & - \\ (Vw/Vp) \(\times\) (Vs/Vm) & - & \(-\)2.2390 & 66.741 & - \\ (Vw/Vp) \(\times\) (Vfs/Vs) & 349.05 & - & - & 35.06 \\ (Sp/p) \(\times\) (Vfs/Vs) & 6673.75 & - & - & - \\ (Vs/Vm) \(\times\) (Vfs/Vs) & - & - & - & \(-\)68.53 \\ (Vw/Vc)\({}^{2}\) & \(-\)2581.92 & \(-\)0.2148 & - & 70.43 \\ (Vw/Vp)\({}^{2}\) & - & 0.3375 & - & 168.84 \\ (Vs/Vm)\({}^{2}\) & - & - & \(-\)123.018 & - \\ (Vfs/Vs)\({}^{2}\) & - & \(-\)0.0602 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Final equations in terms of actual factors: estimated coefficients. Figure 8: Influence of each factor on fresh state responses, in which (+) Positive effect in response models; (\(-\)) Negative effect in response model (see Table 13). The equations in terms of actual factors can be used to predict the response for the given levels of each factor (Table 13 and Figure 10). Uncertainty in the fitted models (confidence interval in Tables 6, 8, 10 and 12) is provided for a 95% confidence level.