## 4 Results and Discussion: Statistical Approach
### Multiple-Regression Analysis
#### 4.3.2 Model Development  
Based on all the aspects discussed above, a linear multiple-regression model was proposed in which the UPV was expressed as a linear combination of all the mechanical properties, as shown in Equation (4) (\(\mathit{UPV}_{S}\) is the standardized UPV value; \(\mathit{CS}_{S}\) is the standardized compressive strength; \(\mathit{ME}_{S}\) is the standardized modulus of elasticity; \(\mathit{STS}_{S}\) is the standardized splitting tensile strength; and \(\mathit{FS}_{S}\) is the standardized flexural strength). In this model, which presented a high accuracy (coefficient \(\mathrm{R}^{2}\) of 89.84%), it was found that, taking into account the sign, the sum of the coefficients of the mechanical properties was around 1. Thus, it could be stated that the use of standardized properties means that the percentage part of the UPV value that each mechanical property represents can be determined. In line with the above, the following two aspects stand out: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & \(\mathit{a}\) & \(\mathbf{R}^{2}\) **(\%)** \\ \hline Compressive strength & 0.933 & 87.52 \\ Modulus of elasticity & 0.769 & 80.43 \\ Splitting tensile strength & 0.730 & 53.86 \\ Flexural strength & 0.787 & 62.26 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Simple linear-regression model between standardized properties. * On the one hand, two mechanical properties had a negative sign, i.e., a negative contribution to the UPV value. One of these properties was the modulus of elasticity, which offset the value of compressive strength. The other one was flexural strength, which counteracted the value of splitting tensile strength. As a result, it can be affirmed that the UPV value is a weighting of the mean of the properties related to compressive behavior, compressive strength, and modulus of elasticity, and of the mean of the properties related to bending-tensile behavior, which are splitting tensile strength and flexural strength. * On the other hand, the weight of the tensile strength and flexural strength, properties related to bending-tensile behavior, had a negligible weight compared to the tensile strength. Figure 7: Standardized properties of SCC mixes: (**a**) compressive strength; (**b**) modulus of elasticity; (**c**) splitting tensile strength; (**d**) flexural strength; (**e**) UPV. properties related to compressive behavior; results that are in line with the correlations and the accuracy of the simple-regression models presented in previous sections. \[UPV_{S}=1.269\times CS_{S}-0.330\times ME_{S}+0.012\times STS_{S}-0.053\times FS _{S} \tag{4}\]  
The problem of the negative signs, which prevents a precise weighting of the UPV readings as a function of the mechanical properties, was addressed through a linear multiple-regression model. The model depended on the arithmetic mean of the properties related to compressive behavior (compressive strength and modulus of elasticity) and the mean of the properties related to bending-tensile behavior (splitting tensile strength and flexural strength). In Equation (5), it can be seen that the mean compressive behavior explained approximately 75% of the UPV value while the mean bending-tensile behavior represented approximately 25%. A similar result to the one in a previous study on the hammer rebound index [46], achieving an increase in the weight of the properties related to bending-tensile behavior compared to Equation (4). However, the coefficient R\({}^{2}\) of this model was only 78.69%, which meant a notable loss of accuracy compared to Equation (4). It is clear that trying to obtain an expression with a simple physical application involving all the mechanical properties implied a low estimation accuracy. \[UPV_{S}=0.725\times\left(\frac{CS_{S}+ME_{S}}{2}\right)+0.240\times\left(\frac {STS_{S}+FS_{S}}{2}\right) \tag{5}\]  
In view of the above discussion, it became clear that UPV was only slightly related to the properties linked to bending-tensile behavior.