## 3 Results
### Predictive Models  
Results of the three logistic regressions models indicated that there was a significant association between kinematic parameters of the PTW vehicle and the likelihood of loss of control (Model-1: \(\chi^{2}(3)\) = 62.256; \(p<0.001\), Nagelkerke R\({}^{2}\) = 0.812; Model-2: \(\chi^{2}(2)\) = 72.090; \(p<0.001\), Nagelkerke R\({}^{2}\) = 0.914; Model-3: \(\chi^{2}(2)\) = 80.827; \(p<0.001\), Nagelkerke R\({}^{2}\) = 1.00). Table 1 includes the test of significance for each of the coefficients in the logistic regression models. As we can see, the two predictors are not significant in Model 3. The expressions of the predictive models developed at this stage are shown in Equations (6)-(8). The probability of loss of control event (\(P\)) predicted by the models is computed following Equation (2). \[Y(Model\_1)=-6.3+(-36.9)\ Gyro\ Norm\ RMS\Big{(}\frac{nd}{s}\Big{)}+(20.6)\ Gyro\ Z\ Max\Big{(}\frac{nd}{s}\Big{)} \tag{6}\]  
\[\hskip 14.226378 pt+(2.8)\ Lat\ Accel\ Max\Big{(}\frac{nd}{s}\Big{)}\]  
\[Y(Model\_2)=-5.7+(-29.3)\ Roll\ Acc\ RMS\Bigg{(}\frac{rad}{s^{2}}\Bigg{)}+(6.9) \ Roll\ Acc\ Max\Bigg{(}\frac{rad}{s^{2}}\Bigg{)} \tag{7}\]  
\[Y\Big{(}Model\_3\Big{)}=-1147.9\ +(203.5)\ Roll\ Acc\ ratio\Bigg{(}\frac{peak}{rms}\Bigg{)}+(24.6)\ Yaw\ Acc\ Max\ \Bigg{(}\frac{rad}{s^{2}}\Bigg{)}\,. \tag{8}\]  
In Model-1 and Model-2, the negative coefficient for Gyro Norm RMS and Roll Acc RMS indicates that for a fixed value of the rest of predictors, a braking trial with higher Gyro Norm RMS (in Model-1) and trial with a higher Roll Acc RMS (in Model-2) is less likely to have a loss of control event than a lower value. This is due to the fact that RMS are linked to the skill level of the riders, and expert riders can handle control under high demanding dynamic behavior that could cause a loss of control for less skill riders. In average RMS values of Gyro for expert riders are higher than for less skilled riders. As an example, the average Gyro Norm RMS for expert riders was 0.23 rad/s (SD = 0.11) compared to the average of 0.12 rad/s for intermediate riders (SD = 0.05). The increase of the rest of predictors is positively related to a higher probability of loss of control. The large coefficients of Model-3 together with the high \(p\)-values presented are signs of risk of collinearity and overfitting. The presence of collinearity does not affect the reliability of the prediction model, but does affect the assessment of the actual contribution of each variable as predictor. Lasso regularization of Model-3 using 10-fold cross-validation was done to limit the magnitude of the estimated coefficients and to reduce their variability. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(\beta\)**Coeff.** & **Std. Error** & **Wald** & \(p\)**-Values** \\ \hline
**MODEL-1** & & & & \\ \hline Gyro Norm RMS & \(-36.9\) & 19.2 & 3.619 & 0.057 \\ Lat Accel Max & 2.8 & 1.1 & 6.420 & 0.011 \\ Gyro-Z\_Max & 20.6 & 7.9 & 6.595 & 0.010 \\ Intercept & \(-6.3\) & 2.152 & 9.110 & 0.003 \\ \hline
**MODEL-2** & & & & \\ \hline Roll Acc Max & 6.9 & 4.4 & 2.710 & 0.100 \\ Roll Acc RMS & \(-29.3\) & 19.0 & 2.613 & 0.106 \\ Intercept & \(-5.7\) & 3.3 & 3.287 & 0.070 \\ \hline
**MODEL-3** & & & & \\ \hline Roll Acc peak-to-RMS & 203.5 & 3356.0 & 0.004 & 0.952 \\ Yaw Acc Max & 24.6 & 504.8 & 0.002 & 0.961 \\ Intercept & \(-1147.9\) & 19010.4 & 0.004 & 0.952 \\ \hline \end{tabular}
\end{table} 
Table 1: Logistic Regression Analysis of 165 braking trials for loss of control cases classification. Figure 6 shows cross-validated deviance of various levels of regularization. As \(\lambda\) increases (moving from right to left on the \(x\)-axis), the coefficients shrink towards zero. The green and blue circle indicate the value of \(\lambda\) (lambda) at the minimum cross-validated deviance and the point with minimum cross-validated deviance plus one standard deviation.