## 2 Methodology
### Regression Modelling
#### 2.2.1 Multiple Logistic Regression  
We use a multiple logistic regression model to analyse the effect of trackwork, along with other explanatory variables, on the train running time delay increases (1)/decreases (2). Logistic regression is commonly used to study functional relationships between a categorical dependent variable and one or more independent variables [35; 36]. The response \begin{table}
\begin{tabular}{l l c c c c c} \hline \hline
**Regression** & **Train Running Time Delay** & **Obs.** & **Mean** & **St. Dev.** & **Min** & **Max** \\ \hline \multirow{4}{*}{Multiple logistic} & Increase \(\geq\)1 min & 27,182,634 & 0.22 & 0.42 & 0 & 1 \\  & Decrease \(\geq\)1 min & 27,182,634 & 0.45 & 0.50 & 0 & 1 \\  & Increase \(\geq\)5 min & 27,182,634 & 0.03 & 0.16 & 0 & 1 \\  & Increase \(\geq\)10 min & 27,182,634 & 0.01 & 0.11 & 0 & 1 \\ \hline \multirow{4}{*}{Negative binomial} & Increase \(\geq\)1 min & 406,563 & 15 & 44 & 0 & 2111 \\  & Decrease \(\geq\)1 min & 406,563 & 30 & 100 & 0 & 3625 \\  & Increase \(\geq\)5 min & 406,563 & 1.70 & 5.28 & 0 & 142 \\  & Increase \(\geq\)10 min & 406,563 & 0.74 & 2.91 & 0 & 86 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistical summary of the analysed response variables for logistic and negative binomial regression models. variable for the first model captures the presence and absence of train running time delay increase while passing an analysed track segment, coded as 1 and 0 accordingly. In the second model, the response variable reports the presence and absence of train delay decrease in the same circumstances coded as 1 and 0 accordingly. The multiple regression model predicts the train running time delay increase/decrease (\(\gamma\)) occurrence by the explanatory (\(x_{i}\)) variables described in Table 2. The summary of this model is presented in the equation:  
\[\text{logit}(Y)=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{ 4}x_{4}+\beta_{5}x_{5} \tag{1}\]  
where:  
* \(Y\) is the response variable capturing the presence or absence of the train running time delay increase (1 min) for the first model and of running time delay decrease (1 min) for the second model, given the predictor variables. The possible values are 0 or 1;
* \(X_{1},X_{2},\dots,X_{5}\) are the predictor variables in the model (trackwork, track type, train subtype, train enter status, and day time, respectively);
* \(\beta_{0}\) is the intercept term, and \(\beta_{1},\beta_{2},\dots,\beta_{5}\) are the coefficients for each predictor variable. The explanatory variable trackwork is a binary variable where 1 is assigned to cases where the train passage on the studied track segment overlaps with scheduled trackwork; otherwise, it is: 0. Track type, train type, train enter status, and night are categorical explanatory variables representing the track type, train subtype, whether the train is on time, early, or late, and whether the train operates at night, respectively. The time variable shows when the train passed the analysed line day (0) or night (1). [PERSON]'s chi-squared test was used to check the independence of qualitative variables entering the regression model. The results show that all tested variables were independent. The selection variables chosen for this model were made by testing several logistic models. For ease of interpretation, in line with multiple logistic regression coefficients, we computed the odds ratio (OR). OR is a measure of association between a given exposure in a logistic regression and an outcome \(Y\):  
\[\text{OR}=\frac{\text{odds}(Y=1)}{\text{odds}(Y=0)} \tag{2}\]  
The OR, therefore, indicates how much more likely the event is to happen given a particular exposure (in this case, trackwork) compared to its absence. An OR greater than 1 suggests a higher likelihood of the event when the exposure is present, whereas an OR less than 1 indicates a reduced likelihood.