## 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
A plot of the externally studentized residuals versus the predicted values was used to assess the assumption of homoscedasticity (equal variance). In other words, linear regression assumes that the variance in the residual versus predicted value plot is constant, with a symmetric shape. Therefore, a symmetrically shaped distribution of errors around the zero in the scatter plot indicates homoscedasticity, while asymmetric shapes in the scatter plot are heteroscedastic. The asymmetric shapes in the scatter plot indicate that the linear model is not suitable. Figure 12 shows that the predicted values versus residual values are distributed around zero in a symmetric shape. Thus, the linear model was suitable to predict the accumulated strain of the warm mix asphalt. To check the assumption of independence in the linear regression, a plot of the externally studentized residuals versus the run number is worthwhile. The residuals versus run number plot is used to detect autocorrelation between disturbances. Autocorrelation occurs when the residuals have a pattern where they remain positive or negative. As shown in Figure 13, there was no autocorrelation among the disturbances, since the residuals randomly transitioned between positive and negative values, which indicates that the dependent variable of accumulated strain was not independent. Figure 11: Plot of the normal probability of the linear model. Figure 12: Plot of the externally residuals versus predicted values of the linear model. The plot of the actual versus predicted values is influential to understand the linear relationship between the actual and the predicted values. Also, the assumption of linearity was checked through a plot of the relationship between the actual and the predicted values. As observed in Figure 14, there is a strong agreement between the actual and predicted values, since the data of the actual values and predicted values are close to the 1:1 line. The strong agreement indicates the capability of the linear model to predict the accumulated strain. The regression coefficient of the variance inflation factor (VIF) was utilized to check the assumption of multicollinearity in the models. The VIF is determined based on the variance of the predicted model divided by the variance of every factor. Multicollinearity occurs when the independent variables are highly correlated to each other, and thus a large change may occur in the model if one of the factors is removed or changed. To ensure that the multicollinearity assumption is met, the VIF value should not exceed 5. The variance inflation factor for the factors of the linear model was 1, which implies that the multicollinearity assumption is met. To introduce a better understanding of the effect of the applied compressive stress and production temperatures on the accumulated strain of the warm mix asphalt, Figure 15 is worthwhile. As indicated in Figure 15, the applied compressive stress and production temperatures have a great influence on the accumulated strain of the warm mix asphalt. As the applied load increases, the accumulated strain significantly increase. Also, the higher the production temperature, the better the permanent deformation resistance. Figure 14: Plot of the actual versus predicted values of the linear model. Figure 13: Plot of the externally residuals versus run number of the linear model.