## 5 Results and Discussion
### Results from Integrated Consideration and Footfall Model
#### 5.2.1 Omitted Variable Bias and Fit Loss Due to Linearity and Normality Assumptions  
The robustness of the proposed footfall model was assessed by comparing its performance against several benchmark models that did not account for both metro consideration and non-normality. The benchmark models included linear regression, log-linear, Poisson regression, and negative binomial regression models. As presented in Table 5, the results indicated that the proposed model, which incorporated both metro consideration and non-normality, outperformed the alternatives with a goodness-of-fit value of 0.83. In comparison, the linear regression model achieved a fit of 0.67, the log-linear model 0.82, the Poisson regression model 0.78, and the negative binomial regression model 0.63. These findings underscore the significance of considering both features to enhance the predictive accuracy and explanatory power of the footfall model. The residual Q-Q plots also reveal a substantial deviation from normality, leading to a lack of fit in the linear model (Figure 8). For the footfall model validation, the dataset was similarly split into 70% training and 30% testing subsets. The goodness-of-fit measure, R\({}^{2}\), was compared between the test model (calibrated on the test data) and the train model (calibrated on the train data) when both were applied to the test data. The absolute percentage difference in R\({}^{2}\) was 4.04%, with the test model achieving an R\({}^{2}\) of 0.82 and the train model achieving an R\({}^{2}\) of 0.79 on the test data. Based on these findings, the models were re-estimated using the full dataset to leverage all available information and are reported in this section. \begin{table}
\begin{tabular}{l l} \hline \hline
**Models** & **R\({}^{2}\)** \\ \hline Proposed model with both consideration & 0.83 \\ and non-normality \\ Linear regression model & 0.67 \\ Log–linear model & 0.82 \\ Poisson regression model & 0.78 \\ Negative binomial regression model & 0.63 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of alternate model structures for metro footfall model. Ignoring the non-linearity leads to omitted variable bias with regard to three key variables. Bus cost and personal mobility resource variables turned out to be significant in the non-linear models but were insignificant in the linear model (Table 6). In addition, the availability of bicycles for hire in metro stations for first- or last-mile connectivity shows false significance in the linear and non-linear model, which disregards consideration. Bus cost is insignificant in the non-linear model without consideration, whereas population density has no effect in the linear model. Thus, disregarding non-linearity and non-normality of footfall leads to neglecting the effect of competing modes and some socio-demographic variables, thereby reducing the overall goodness of fit. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Linear Model Without**} & \multicolumn{2}{c}{**Non-Linear Model Without**} & \multicolumn{2}{c}{**Non-Linear Model with**} \\  & \multicolumn{2}{c}{**Consideration Effect**} & \multicolumn{2}{c}{**Consideration Effect**} & \multicolumn{2}{c}{**Consideration Effect**} \\ \hline
**Variables** & **Coef** & **t-Stat** & **Coef** & **t-Stat** & **Coef** & **t-Stat** \\ \hline Intercept & 1975.00 & 7.15 & 32.49 & 12.83 & 23.41 & 14.79 \\ Before AM peak (\textless{}8 a.m.) & \(-\)144.40 & \(-\)5.95 & \(-\)2.78 & \(-\)12.47 & \(-\)2.80 & \(-\)12.57 \\ AM peak (8 a.m.\(-\)11 a.m.) & 209.50 & 7.33 & 1.84 & 7.03 & 1.83 & 6.97 \\ Afternoon Off-peak (11 a.m.\(-\)4 p.m.) & \(-\)27.20 & \(-\)1.12 & 0.23 & 1.02 & 0.21 & 0.96 \\ PM peak (4 p.m.\(-\)8 p.m.) & 221.20 & 8.36 & 2.06 & 8.49 & 2.04 & 8.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Model results of hourly footfall at metro stations. Figure 8: Q–Q plot of the residuals obtained from the proposed model and the benchmark linear model.