## 4 Developing Prediction Models for the Performance of the Warm Mix Asphalt
### Developing a Prediction Model for the Accumated Strain of the Warm Mix Asphalt  
Table 7 displays the data input to DESIGN EXPERIM. In Table 7, the factors represent the compressive stress and the production temperatures applied, while the response was the observed values of the accumulated strain at the applied loads of 100 kPa and 200 kPa. The fit summary of the models is summarized in Table 8. As reported in Table 8, a linear model is suggested to predict the accumulated strain of the warm mix asphalt. This is because the sequential \(p\)-value is significant, while the sequential \(p\)-value of the other models was not significant. Also, the lack of fit of the linear model is insignificant because the \(p\)-value is greater than 0.05. This, in turn, suggests that the model suits the data. The high coefficient of determination of the linear model implies that there is a very strong relationship between the compressive stress, production temperatures, and accumulated strain applied. In addition, the less difference between the adjusted R\({}^{2}\) and predicted R\({}^{2}\), which was less than 0.2, refers to the high agreement between the factors and the response. Figure 10: Plot of the interaction between the factors and the response of the 2 FI model. Figure 9: Plot of actual versus predicted values of the 2 FI model. Table 9 shows the ANOVA for the linear model. As seen in Table 9, the significant \(p\)-value of the factors implies that the applied compressive stress and production temperatures had a considerable effect on the accumulated strain of the warm mix asphalt. The linear model is expressed in Equation (2). The assumptions of the linear model should be investigated to examine its adequacy. Figure 11 illustrates the assumptions of the normality of the linear regression. The plot of normal probability displays the shape of the error distribution. The straight distribution of the errors indicates that linear regression fits the data well, whereas the other shapes such as right skew, left skew, S, or long and short tails, imply that the error distribution is not normal, and thus that linear regression is not suitable. As seen in Figure 11, the residuals are distributed normally, since the points are close to the line of the theoretical residual. In addition, the Anderson-Darling test was carried out to check the assumption of normality. The \(p\)-value of the Anderson-Darling test was insignificant (\(p\)-value = 0.3505 > 0.05), which indicates that the errors were normally distributed. \begin{table}
\begin{tabular}{c c c} \hline
**Compressive Stress (kPa)** & **Production Temperature (\({}^{\circ}\)C)** & **Accumulated Strain (\%)** \\ \hline  
\hline \end{tabular}
\end{table} 
Table 7: Input data of the accumulated strain model. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Source** & **Sequential \(p\)-Value** & **Lack of Fit \(p\)-Value** & **R2** & **Adjusted R2** & **Predicted R2** & **Performance** \\ \hline Linear & \textless{}0.0001 & 0.2923 & 0.9820 & 0.9796 & 0.9740 & Suggested \\
2 FI & 0.1337 & 0.4513 & 0.9848 & 0.9815 & 0.9749 & \\ Quadratic & 0.4662 & 0.3082 & 0.9854 & 0.9809 & 0.9722 & Aliased \\ \hline \end{tabular}
\end{table} 
Table 8: Fit summary of the models. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Source** & **Sum of Squares** & **df** & **Mean Square** & **F-Value** & \(p\)**-Value** & **Performance** \\ \hline Model & 0.2620 & 2 & 0.1310 & 409.13 & \textless{}0.0001 & Significant \\ A-Compressive stress (kPa) & 0.2404 & 1 & 0.2404 & 750.57 & \textless{}0.0001 & Significant \\ B-Production temperature (\({}^{\circ}\)C) & 0.0217 & 1 & 0.0217 & 67.69 & \textless{}0.0001 & Significant \\ Residual & 0.0048 & 15 & 0.0003 & & & \\ Lack of Fit & 0.0012 & 3 & 0.0004 & 1.39 & 0.2923 & Insignificant \\ Pure Error & 0.0036 & 12 & 0.0003 & & & \\ Cor Total & 0.2668 & 17 & & & & \\ \hline \end{tabular}
\end{table} 
Table 9: ANOVA for the linear model.