## 3 Methodology
### Model Specification  
This study analyzes travel mode choice within the utility maximization framework [39; 40], where the utility \(U_{j}\) of alternative \(j\) is composed of a deterministic part \(V_{j}\) and a random error term \(\varepsilon_{j}\):  
\[U_{j}=V_{j}+\varepsilon_{j}. \tag{1}\]  
The deterministic utility is specified as  
\[V_{j}=\beta_{0j}+\sum_{i}\beta_{ij}X_{ij}, \tag{2}\]  
where \(\beta_{0j}\) is the alternative-specific constant capturing intrinsic preferences, and \(X_{ij}\) are explanatory variables such as travel time or cost.  
Although machine learning methods (e.g., random forests, gradient boosting, neural networks) offer high predictive accuracy [41; 42], they lack behavioral interpretability and require large datasets [43]. Since this study focuses on explaining mode choice behavior, especially differences between daytime and nighttime travel, discrete choice models (DCMs) are preferred for their interpretability and suitability for moderate sample sizes. The multinomial logit (MNL) model, based on the assumption that the error terms \(\varepsilon_{j}\) follow an i.i.d. Gumbel distribution, defines choice probabilities as  
\[P_{j}=\frac{\exp(V_{j})}{\sum_{k\in C}\exp(V_{k})}, \tag{3}\]  
where \(C\) denotes the choice set.  
However, MNL imposes the Independence of Irrelevant Alternatives (IIA) assumption, which can be violated if alternatives share unobserved characteristics, leading to biased estimates. To relax IIA, the nested logit (NL) model groups alternatives into nests \(B_{m}\), allowing correlated unobserved utility components within nests [44]. The choice probability  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Attributes** & **Walking** & **Bicycle** & **Private Car** & **PT** & **RP** & **Taxi** \\ \hline Travel time (min) & 25, 35, 45, 55 & 9, 14, 19, 24 & 5, 8, 11, 14 & 6, 9, 12, 15 & 6, 9, 12, 15 & 5, 8, 11, 14 \\ Egress time (min) & - & - & - & 2, 4, 6 & 4, 6, 8 & 2, 4 & - \\ Degress time (min) & - & - & 1, 3, 5 & 2, 4, 6 & 2, 4, 6 & - \\ Waiting time—day (min) & - & - & - & 4, 8, 12 & 4, 8, 12 & - \\ Waiting time—night (min) & - & - & - & 15, 30, 45 & 15, 30, 45 & 2, 7, 12 \\ Transfer (within PT) & - & - & - & 0, 1 & - & - \\ Travel costs (EUR) & - & - & 1.5, 3, 4.5 & 0, 1.7, 3.2, & 0, 1, 3.5, 4.5, & 12, 18, 24 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Attribute levels in the stated choice experiment.  
decomposes into the product of the probability of choosing nest \(m\) and the conditional probability of choosing alternative \(j\) within that nest:  
\[P_{j}=P_{j|m}\times P_{m} \tag{4}\]  
with  
\[P_{j|m}=\frac{\exp\left(\frac{V_{j}}{\lambda_{m}}\right)}{\sum_{k\in B_{m}}\exp \left(\frac{V_{k}}{\lambda_{m}}\right)} \tag{5}\]  
and  
\[P_{m}=\frac{\exp(\lambda_{m}I_{m})}{\sum_{l=1}^{M}\exp(\lambda_{l}I_{l})} \tag{6}\]  
where \(\lambda_{m}\in(0,1]\) is the nest correlation parameter and  
\[I_{m}=\ln\sum_{k\in B_{m}}\exp\left(\frac{V_{mk}}{\lambda_{m}}\right) \tag{7}\]  
is the inclusive value summarizing nest attractiveness.  
Model parameters are estimated via maximum likelihood, implemented with the Apollo package in R [45]. This approach balances explanatory power with computational feasibility given the sample size and the study's behavioral focus.