## 3 Results
### Developing Master Curves for UPV and Dynamic Modulus  
In order to develop the maturity master curves for UPV and/or dynamic modulus the following steps were followed:  
1. Using the general logarithmic form, \(y=a\ ln(MI)+b\), the coefficients of \(ln(M\!I)\), \"\(a\)\" and/or \"\(a\)\", were set to a constant value (i.e., \(a\) = average of coefficient \"\(a\)\" and \"\(a\)\" of all mixture equations). New coefficients \"\(b\)\" and/or \"\(b\)\" for each mixture were then calculated. The resulting R\({}^{2}\) values are reported in Table 3, while the corresponding relationships for all the mixes are shown in Figure 2.
2. In the next step, one mixture was selected as the reference mixture. As can be seen from Figure 2, Mix 5 represents a typical mixture used in Maryland for infrastructure projects, identified as MD 7, with test results very close to the center of all mixtures. Therefore, Mix 5 was selected as the reference mixture. \begin{table}
\begin{tabular}{c c c c} \hline
**Mix** & **UPV vs. Temp-Time** & **E\({}_{d}\) (RTG) vs. Temp-Time** & **Compressive Strength vs. Temp-Time** \\ \hline
1 & 0.99 & 0.95 & 0.97 \\
2 & 0.95 & 0.97 & 0.95 \\
3 & 0.86 & 0.97 & NA * \\
4 & 0.96 & 0.96 & NA * \\
5 & 0.92 & 0.97 & 0.98 \\
6 & 0.96 & 0.96 & 0.99 \\
7 & 0.97 & 0.95 & 0.92 \\
8 & 0.92 & 0.97 & 0.90 \\
9 & 0.94 & 0.95 & 0.98 \\
10 & 0.99 & 0.99 & 0.99 \\
11 & 0.95 & 0.96 & 0.99 \\ \hline \end{tabular}  
* Limited data; \(p<0.05\) for all cases.  
\end{table} 
Table 2: R\({}^{2}\) for best fit logarithmic models between NDTs and compressive strength with temperature-time. 3. The vertical shift factors, which are the differences between the intercepts of the governing equations of any mixture and the reference mixture (Mix 5), were calculated. Examples are shown in Figure 2. The resulting master curves from shifting are shown in Figure 3 for both the dynamic modulus and UPV. 4. In the next step, the shift factors were related to concrete mixture properties. The characteristics of a concrete mix most related to the shift factor were identified based on Pearson's correlation analysis. Table 4 includes the shift factors of each concrete mixture along with the concrete properties and the hydration temperatures during curing. The Pearson's correlation coefficients are summarized in Table 5. The parameters with higher absolute values of Pearson's correlation coefficients were selected to be the best predictors of the shift factors. Linear regression multivariate analysis was then used to identify the relationship describing the shift factor, Equation (5): \[y=\beta_{0}+X_{1}\beta_{1}+X_{2}\beta_{2}+X_{3}\beta_{3}+\varepsilon\] (5) where \(y\) is the response variable, \(x\) is the independent variable (predictor), \(\beta\) is the unknown parameter, and \(\varepsilon\) is the residual. In a linear model, the parameters enter linearly. However, the predictors themselves do not have to be linear. Here, the objective is finding \(\overset{\rightarrow}{\beta}\) (i.e., \(\beta_{0}\), \(\beta_{1}\), \(\beta_{2}\), and \(\beta_{3}\)) in order to minimize the residual error (\(\varepsilon\)). Further details can be found in [PERSON] [36]. Bivariate analysis on the shift factors and each one of these selected variables indicated that the best relationships have polynomial. To describe the shift factors, polynomial relationships (second degree) for each shift factor (UPV or dynamic modulus) are presented in Tables 6 and 7. In selecting the best models, R-squared, \(p\)-value, residual standard error, and simplicity of the model are considered (i.e., highest R-square and \(p\)-value less than 0.05). Residual standard error (RSE), also known as the model sigma, is a variant of the RMSE (root-mean-squared error) attuned for the number of predictors in the model. The lower the RSE, the better the model. In practice, the difference between RMSE and RSE is very minor, especially for large multivariate data.