## 3 Analysis and Results
### RSM Results for Single, Tandem, and Tridem Axles
#### 3.1.1 Single Axle with Dual Tires
The normal probability plot is particularly useful for assessing the distribution of residuals; a straight line on this plot indicates a favorable fit for linear regression. Any deviations from this line, such as right or left skewness, could suggest potential issues with the model's assumptions about error distribution [29; 30]. To assess homoscedasticity, the assumption that the variance of errors is constant across all levels of the independent variables, externally studentized residuals were plotted against predicted values. A scatter plot with symmetrically shaped errors around zero supports the assumption of homoscedasticity, whereas any asymmetric patterns would indicate heteroscedasticity [31]. Figure 3 demonstrates a symmetrical pattern around zero, confirming the model's suitability for accurately predicting the developed stresses. The residuals versus run number plot is instrumental in detecting autocorrelation among disturbances. The absence of autocorrelation is indicated by residuals transitioning randomly between positive and negative values, as observed in Figure 3. This randomness suggests that the stresses are well modeled by the independent variables [32]. However, the actual versus predicted values plot provides a visual representation of the model's predictive capability. Figure 3 shows a strong correlation between actual and predicted values, with data points closely aligning with the 1:1 line. This alignment underscores the model's accuracy in predicting developed stresses under various loading conditions, validating its utility in pavement design. Equation (1) describes the quadratic model. \[\sigma_{z}=10^{(2.31928-0.026869*A-0.072919*B+0.000267*A*B+0.000387*A^{2}+0.00067 6*B^{2})}*\left(\frac{\textit{tire load}}{4500}\right) \tag{1}\]  
where:  
\(\sigma_{z}\) = Vertical stress;  
A = spacing between tires in the Y direction;  
B = thickness of the pavement. \begin{table}
\begin{tabular}{c c c c c} \hline  & \multicolumn{4}{c}{**Fit Summary**} \\ \hline
**Source** & **Sequential \(p\)-Value** & **Adjusted R\({}^{2}\)** & **Predicted R\({}^{2}\)** & **Performance** \\ \hline Linear & \textless{}0.0001 & 0.9197 & 0.9047 & \\ \hline
2 FI & 0.4196 & 0.9188 & 0.892 & \\ \hline Quadratic & \textless{}0.0001 & 0.9903 & 0.9862 & Suggested \\ \hline Cubic & \textless{}0.0001 & 0.9973 & 0.9953 & Aliased \\ \hline \end{tabular}
\end{table} 
Table 2: Fit summary of the various models for single axle with dual tires. Figure 3: Plots analyzing the quadratic model of developed stress for single axle. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Source** & **Sum of Squares** & **df** & **Mean Square** & **F-Value** & **\(p\)-Value** & **Performance** \\ \hline Model & 11.19 & 5 & 2.24 & 651.13 & \textless{}0.0001 & significant \\ \hline A-A & 0 & 1 & 0 & 0.0035 & 0.9534 & \\ \hline B-B & 1.61 & 1 & 1.61 & 469.74 & \textless{}0.0001 & \\ \hline AB & 0.0192 & 1 & 0.0192 & 5.58 & 0.0256 & \\ \hline A\({}^{2}\) & 0.0026 & 1 & 0.0026 & 0.7674 & 0.3887 & \\ \hline B\({}^{2}\) & 0.7346 & 1 & 0.7346 & 213.75 & \textless{}0.0001 & \\ \hline Residual & 0.0928 & 27 & 0.0034 & & & \\ \hline Cor Total & 11.28 & 32 & & & \\ \hline \multicolumn{6}{c}{Fit statistics} \\ \hline Std. Dev. & 0.0586 & \multicolumn{2}{c}{R\({}^{2}\)} & 0.9918 \\ \hline Mean & 0.8933 & \multicolumn{2}{c}{Adjusted R\({}^{2}\)} & 0.9903 \\ \hline C.V. \% & 6.56 & \multicolumn{2}{c}{Predicted R\({}^{2}\)} & 0.9862 \\ \multicolumn{6}{c}{Adeq Precision} & 76.0539 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: ANOVA for quadratic model for single axle model.