## 4 Results and Discussion
### Case 2: \(E_{u\,50}\)  
A similar study was performed for \(\mathrm{E_{u\,50}}\). A full quadratic model was estimated by PLS regression after expanding the original data set with the quadratic and cross-product terms of the quadratic expansion. All variables were considered as continuous variables. The detailed explanations given in the previous section will not be repeated again. The model selection plot (Figure 8) sets the optimal model with five components, which can predict 89% of the model response. The response plot (Figure 9) shows that the model predicts quite well the predicted response versus the actual response in the leave-one-out cross-validation. The ANOVA test for the PLS model with five components is presented in Table 7. Once again, as the \(p\)-value is under 0.05, the model is significant [51]. Table 7 also shows the performance statistics for the PLS model. Statistically, the model with five components  
Figure 8: PLS model selection plot—\(\mathrm{E_{u\,50}}\). Figure 7: SEM image of a chemically stabilized soil sample additivated with MWCNTs (0.01%) dispersed in a solution of Amber 4001 (3%) for 7 curing days. has a quite high R\({}^{2}\) of 92% and R\({}^{2}\) (pred) of 89% (Table 7), which are good indicators of its fitting ability and predictive accuracy. The analysis of the standard coefficients of E\({}_{\mathrm{u}\,50}\) (Table 8) leads to similar conclusions as for q\({}_{\mathrm{u}\,\mathrm{max}}\). The MWCNT concentration (x3) and its quadratic relation (x3.x3) are the most important factors influencing the response (E\({}_{\mathrm{u}\,50}\)). While x3 has a positive effect on E\({}_{\mathrm{u}\,50}\), x3.x3 has a negative impact. It means that the CNT concentration has a non-linear behavior, i.e., there is an optimum value of CNT concentration that maximizes the response. As stated previously, the probability of agglomeration increases with the MWCNT concentration, resulting in a loss of MWCNT beneficial properties. On the other hand, the surfactant concentration (x1) and surfactant hydrodynamic diameter (x2) have a negative impact on E\({}_{\mathrm{u}\,50}\), but, once again, when combined (x1.x2), the impact becomes positive, meaning that these variables depend on each other. Standardization was conducted in a similar way as for q\({}_{\mathrm{u}\,\mathrm{max}}\). As it was previously discussed, an increase in the surfactant concentration or surfactant hydrodynamic diameter promotes particles' spacing (especially regarding cement particles), resulting in a less strong and stiff stabilized matrix. Again, when both effects are combined, the impact on E\({}_{\mathrm{u}\,50}\), the undrained Young's modulus, becomes positive, meaning that these two variables (surfactant concentration and surfactant hydrodynamic diameter) depend on each other, following what was observed previously for q\({}_{\mathrm{u}\,\mathrm{max}}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & & & \\ \hline
**Source** & **DF** & **SS** & **MS** & **F-test** & **P-test** \\ \hline Regression & 5 & 2906.39 & 581.278 & 71.46 & 0.000 \\ Residual & 33 & 268.42 & 8.134 & & \\ Error & 38 & 3174.81 & & & \\ \hline \multicolumn{5}{c}{**Performance Statistics for the PLS Model**} \\ \hline
**Components** & **X Variance** & **Er** & **R\({}^{2}\)** & **PRESS** & **R\({}^{2}\) (pred)** \\ \hline
1 & 0.28485 & 1596.57 & 0.497112 & 2098.69 & 0.338954 \\
2 & 0.60152 & 1292.49 & 0.592890 & 1607.12 & 0.493788 \\
3 & 0.69374 & 531.93 & 0.832454 & 633.92 & 0.800329 \\
4 & 0.99941 & 433.51 & 0.863453 & 539.30 & 0.830132 \\
5 & 1.00000 & 268.42 & 0.915454 & 362.65 & 0.885773 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Summary of the performance statistics for the PLS model—E\({}_{\mathrm{u}\,50}\). Figure 9: PLS response plot (5 components)—E\({}_{\mathrm{u}\,50}\). The PLS residual vs. leverage plot (Figure 10) shows at least two points in the border of leverage; thus, they can be considered as possible outliers, but the result continues to be significant.