## 3 Results and Discussion
### Statistical Modeling
#### 3.3.2 Performance Measures  
The assessment of the model fitting was performed based on five measures: \(R^{2}\), adjusted \(R^{2}\), Akaike Information Criteria (AIC), and Bayesian Information Criteria (BIC). These measures are commonly adopted as performance measures in statistical modelling (see, e.g., (Kal  
From the results, the following was observed:  
* For both models, all experimental \(q_{u}\) values fall inside the 95% predicted intervals, showing a good agreement with the fitted models;
* The error between the experimental and the predicted \(q_{u}\) values is not large;
* Apparently, both models overestimate the \(q_{u}\) value in the right upper region shown in Figures 14 and 15. This region was not contemplated in a preliminary phase of the designed plan. The Root Mean Square Error (RMSE) measures the average deviation between a statistical model's predicted values and the actual values. In other words, RMSE quantifies how dispersed the model residuals are, revealing how tightly the observed data clusters are around the predicted values [32]. RMSE can be used to compare the effectiveness of models by testing their prediction capability on points that were not used in model fitting, namely validation points. The lower the value of RMSE, the closer the data points are to the regression curve, resulting in greater accuracy of the model. In our framework, considering all 10 data points of the validation phase, the RMSE is smaller for Model 1 (43.11 kPa) than for Model 2 (55.44 kPa). However, considering just the six validation runs that do not go far outside the sample space (i.e., points 22, 24, 28, 29, 30, and 31), the RMSE decreases considerably in both Model 1 (RMSE = 28.11 kPa) and Model 2 (RMSE = 30.37 kPa). This could indicate that the models can predict well near the region associated with the initially considered range (see Table 2). Notice that, according to Figure 16, in both models, there are three points that do not fall inside the initial boundaries of the plan (marked with squares) and are farther away \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Validation Phase**} \\ \hline
**ID** & \(\mathbf{t_{FA}}\) & \(\mathbf{t_{L}}\) & \(\mathbf{c_{SH}}\) **(mol/kg)** & \(q_{u}\) **(kPa)** \\ \hline
22 & 0.300 & 0.170 & 14.00 & 152.10 \\
23 & 0.400 & 0.170 & 11.76 & 228.25 \\
24 & 0.300 & 0.180 & 14.00 & 115.50 \\
25 & 0.300 & 0.180 & 16.00 & 131.22 \\
26 & 0.300 & 0.170 & 16.00 & 168.64 \\
27 & 0.300 & 0.180 & 18.00 & 138.87 \\
28 & 0.25 & 0.15 & 10 & 57.24 \\
29 & 0.30 & 0.16 & 9 & 78.45 \\
30 & 0.15 & 0.14 & 7 & 56.80 \\
31 & 0.20 & 0.15 & 6 & 59.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Validation phases of the model and their respective \(\mathrm{q_{u}}\). Figure 16: Experimental x predicted points according to (**a**) Model 1 and (**b**) Model 2. The numbers indicated next to the marker correspond to the ID number given in Table 5.  
from the diagonal line. These points are numbered 24, 25, and 27, and their liquid content (\(t_{L}\)) is the same and equal to 0.18. In these points, both models predict higher strengths when compared to experimental values. A possible explanation may be that, for the studied region (\(t_{L}\in 0.14\)-0.17), a linear behavior was found; meanwhile, beyond that initial range, the relation may be nonlinear. If these three points were excluded from the initial set of 10 runs, the RMSE values would decrease considerably in both models (respectively, equal to 19.05 kPa for Model 1 and 6.65 kPa for Model 2). Therefore, Model 2 is proposed to predict \(q_{u}\) inside the sample region. A further investigation is needed to refine the dependence of \(q_{u}\), for very high values of \(t_{EA}\) and \(c_{SH}\) or the \(c_{A}\). That was not the intention of the present experimental plan, due to limitations in the practical implementation of the mentioned mixtures. Figure 17 illustrates how fast the optimum \(q_{u}\) can be achieved by increasing the values of \(t_{EA}\), \(t_{L}\), or \(c_{SH}\).