## 3 Results and Discussion
### Experimental Correlation between UCS and \(\mathrm{v_{P}}\) in This Study
#### 3.2.2 Multiple Regression Analyses  
Results of the multiple regression analyses through ANOVA showed that there was no significant effect of the saturated density \(\rho_{\text{sat}}\) in the UCS results (\(p>0.05\)), considering both linear and quadratic models or derived from the interactions with the remaining independent variables. On the other hand, the linear effects and interaction of \(\
u_{\text{p}}\) and n contributed greatly to explaining the experimental model. Then, after eliminating nonsignificant coefficients, two models were generated to predict the UCS over the range of sampled experimental conditions presented in Equations (21) and (22):  
\[\text{UCS}=-185+55.8\
u_{\text{p}}-24.87\text{n}, \tag{21}\]  
\[\text{UCS}=2.0137\text{e}^{0.794\
u_{\text{p}}-0.40\text{ln}}. \tag{22}\]  
The ANOVA test of the models adjusted in Equations (21) and (22) was suitable for representing UCS under the range of sampled parameters. There was no lack of fit of the models, and the values of \(R^{2}\) (0.8228) for the multiple regression and \(R^{2}\) (0.8490) for the exponential model validated it. The ANOVA showed that the indicated factors influenced the UCS because of \(\text{F}_{\text{regression}}\) (23.21) and \(\text{F}_{\text{exponential}}\) (28.12) \(>\text{F}_{\text{Theoretical}}\) (\(\text{F}_{2,9}=3.8625\)) at a 95% confidence level. Additionally, the \(p\)-values of both models (Table 5) indicated that they were statistically significant, while the Mallows' \(\text{C}_{\text{p}}\) statistic indicated that the adjusted models were relatively accurate and did not show bias in the estimation of the true regression coefficients.  
Figure 3a shows that the UCS increased linearly for larger values of \(\
u_{\text{p}}\) and lower values of n. In addition, a lower UCS was attained for large levels of n and low values of \(\
u_{\text{p}}\). Figure 3b shows that there was an exponential decrease of the UCS when increasing porosity and an exponential increase of UCS when increasing \(\
u_{\text{p}}\). Therefore, considering the goodness parameters of both fits, the potential model Equation (22) best represented the uniaxial compressive strength under the studied conditions.  
The predicted Equation (22) and experimental values of UCS are compared in Figure 4. This figure shows that the observed versus predicted data usually followed the diagonal line (1:1). The normality test determination coefficients were very high (\(p=0.77\)), which indicates a good correlation.  
Figure 3: Multiple regression analysis response surface of UCS: (**a**) Equation (21) and (**b**) Equation (22).  
The levels of significance of the regression coefficients (after a transformation of the exponential with the multiple exponent model) were investigated by means of an \(F\)-test. The \(F\)-value (28.12) of the Equation (22) model and the corresponding significance level equal to zero (Table 5) suggest that at least one of the independent variables influenced the dependent variable. On the other hand, all significance levels of the model parameters were lower than \(\alpha\) = 0.05, so there were significant individual correlations between dependent and independent variables. Finally, the ANOVA test results suggest that the developed empirical correlation Equation (22) between UCS (dependent variable) and \(\mathrm{v_{P}}\) and \(\mathrm{n}\) (independent variables) was significant.  
The significance of the multiple regression model Equation (22) can be determined by examining the dispersion of the residuals of the predicted values. For such a purpose, the normal distribution (\(p>0.05\)) was checked. As can be seen in Figure 5, the cumulative probability of a normal distribution was compatible with the cumulative probability of the experimental distribution. Therefore, the hypothesis about the normal distribution of the results is correct.