## 3 Materials and Methodology
### Development of a Modeling Framework to Estimate Traffic Volume
Tolerance ranges from 0 to 1, with values closer to 0 indicating a high degree of multicollinearity and values closer to 1 indicating a low degree of multicollinearity. In our model, the VIF values obtained--2.955 for SC, 2.867 for BC, and 1.614 for EC--are all well below the VIF threshold of 5, indicating negligible multicollinearity. Furthermore, the corresponding tolerance values--0.338 for SC, 0.349 for BC, and 0.619 for EC--support this conclusion, as they are consistent with low multicollinearity. While some fields consider VIF values above 5 problematic, the VIF values below 3 in our model suggest minimal multicollinearity, which is widely accepted in the literature as indicating negligible or moderate collinearity [54]. Moreover, by adopting a more conservative VIF threshold of 3, we ensure the robustness and stability of our regression model, minimizing the potential inflation of variance in coefficient estimates. These results align well with similar studies, such as those by [PERSON] et al. [55] and [PERSON] et al. [26], which reported the absence of multicollinearity among identified topological parameters. This consistency reinforces the reliability and robustness of the model parameters and assures the validity of our findings. To test the goodness-of-fit of the proposed model, we used the coefficient of determination (\(R^{2}\)) and the mean absolute percentage error (\(MAPE\)). Here, \(R^{2}\) is a statistical measure that represents the proportion of variance in the dependent variable explained by the independent variables in the regression model. It measures how well the regression line fits the observed data points. The \(MAPE\) is a measure of the accuracy of a prediction model. This is calculated as the mean of the absolute percentage errors between the predicted and actual values. According to [PERSON] [31], \(R^{2}\) and \(MAPE\) together provide a sound understanding of a model's predictability. This study initially utilized 80% of the randomly selected data for calibration (i.e., a random subset of the calibration data) and 20% for validation. Table 2 presents the goodness-of-fit assessment of the proposed model. In considering the model performance, the explanatory variables of SC, BC, and EC produced acceptable goodness-of-fit values (\(R^{2}>0.75\)) and a lower mean absolute percent error (\(MAPE<40\%\)). Hence, the model comprising SC, BC, and EC explanatory variables was considered an acceptable model with which to assess traffic volume changes under critical-link attacks. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Specifications**} & \multirow{2}{*}{\
\begin{tabular}{c} **Correlations—AADT and** \\ **Explanatory Variables** \\ \end{tabular} } & \multicolumn{2}{c}{**Presence of Multicollinearity**} & \multirow{2}{*}{
\begin{tabular}{c} **Coefficient Value** \\ **VIF** \\ \end{tabular} } \\ \cline{3-4}  & & **Tolerance** & & **VIF** \\ \hline b (SC) & 0.817 ** & 0.338 & 2.955 & 64.220, \(p<0.0001\) \\ c (BC) & 0.790 ** & 0.349 & 2.867 & 51.935, \(p<0.0001\) \\ d (EC) & 0.665 ** & 0.619 & 1.614 & 31.295, \(p<0.0001\) \\ a (Constant) & & & & 186.873, \(p<0.0001\) \\ \hline \hline \end{tabular}  
* Correlation is significant at the 0.01 level (2-tailed).  
\end{table} 
Table 1: Statistics and specifications of the proposed model. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{5}{c}{**Goodness-of-Fit Assessment**} \\ \hline \multicolumn{5}{c}{**Calibration (80\% of the Sample)**} & \multicolumn{5}{c}{**Validation (20\% of the Sample)**} \\ \hline
**N** & **Adjusted \(R^{2}\)** & **MAPE** & **N** & **Adjusted \(R^{2}\)** & **MAPE** \\ \hline
510 & 0.747 & 0.375 & 128 & 0.768 & 0.361 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Goodness-of-fit assessment of the proposed model.