## 3 Results
### Outcomes of the Multiple Linear Regression Model  
Eight multiple linear regression models were developed for acceleration and deceleration behaviors for four different density levels. It should be noted that the acceleration and deceleration were modeled separately to prevent the averaging effect of positive and negative values of the acceleration when they are modeled together. The predictor variables of the model were relative speed (DV), which was calculated as the speed of the leading pedestrian minus the speed of the pedestrian under consideration (i.e., \(v_{i+1}(t)-v_{i}(t)\)), spacing (DS), and gender, while the dependent variable was time-delayed instantaneous acceleration (\(a_{i}(t+\tau)\)). Males and females were coded as \"0\" and \"1\", respectively. To avoid multicollinearity, instantaneous speed (\(v_{i}(t)\)) was not used as a predictor variable because it had a significantly high correlation with relative speed and spacing. Outcomes of the linear regression models for acceleration and deceleration behaviors are described below. * Models for acceleration behavior  
Only the positive values of instantaneous acceleration (\(a_{i}(t+\tau)\)) were considered in these models, and four models were developed for the considered density levels. As shown in Table 1, the ANOVA results indicated that all four models for acceleration behavior were significant. The R-squared values were 0.428, 0.288, 0.243, and 0.189 for N = 14, N = 20, N = 24, and N = 30 cases, respectively. The highest R-squared value was found for N = 14 case (global density = 0.81 m\({}^{-1}\)). All the predictor variables were found to be significant for all the density levels except spacing, which was not significant for N = 24 cases (global density = 1.38 m\({}^{-1}\)). Gender was significant for all the density levels. The unstandardized coefficients for all the significant predictors are shown in Table 2. For all four density levels, relative speed and spacing had a positive association with instantaneous acceleration. That is, people tend to accelerate when the relative speed is increasing, or in other words, when the leading pedestrian's speed is higher. This is logical that an individual tends to accelerate to gain speed to reach the leading individual and to move ahead with the crowd when the individual's speed is lower than the leading pedestrian. The negative sign of the gender indicates that males are likely to accelerate significantly more than females for all the density levels. Spacing also had a positive association with instantaneous acceleration, which is logical that people tend to accelerate to gain speed when the spacing in front of them increases. Gender had the greatest significant effect on acceleration behaviors for the lowest density level, which decreased with increasing density levels. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**Sum of**} & \multirow{2}{*}{**df**} & \multicolumn{2}{c}{**Mean**} & \multirow{2}{*}{**F**} & \multirow{2}{*}{**Sig.**} \\  & & **Squares** & & & & \\ \hline \multirow{3}{*}{N = 14} & Regression & 26.229 & 3 & 8.743 & 541.478 & 0.000 \\  & Residual & 35.086 & 2173 & 0.016 & & \\ \cline{1-1}  & Total & 61.315 & 2176 & & & \\ \hline \multirow{3}{*}{N = 20} & Regression & 40.382 & 3 & 13.461 & 757.273 & 0.000 \\  & Residual & 99.719 & 5610 & 0.018 & & \\ \cline{1-1}  & Total & 140.100 & 5613 & & & \\ \hline \multirow{3}{*}{N = 24} & Regression & 66.333 & 3 & 22.111 & 1433.751 & 0.000 \\  & Residual & 207.021 & 13,424 & 0.015 & & \\ \cline{1-1}  & Total & 273.353 & 13,427 & & & \\ \hline \multirow{3}{*}{N = 30} & Regression & 64.867 & 3 & 21.622 & 1701.145 & 0.000 \\ \cline{1-1}  & Residual & 278.269 & 21,893 & 0.013 & & \\ \cline{1-1}  & Total & 343.136 & 21,896 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: ANOVA results for the multiple regression models for acceleration.