# Modeling Public Transportation Use Among Short-Term Rental Guests in Madrid
## 2 Materials and Methods
### Ordered Logistic Regression  
When the outcome variable \(Y_{i}\) takes on an ordinal set of J categories {1,2,..., j}, ordered logistic regression (also known as the proportional-odds model) provides a way to model the cumulative probabilities of these ordered responses as a function of covariates \(X_{i}\). A common example is a Likert-scale survey item (e.g., \"strongly disagree\" to \"strongly agree\" on a 5-point scale), and this regression has been used previously, for instance, to model the perception of safety and comfort [29]. In such settings, treating the responses as nominal (multinomial) ignores the ordering, while treating them as continuous (linear regression) imposes arbitrary interval assumptions. The ordered logistic model links these cumulative probabilities to a linear predictor via the logit link (Equation (1)). The ordered logistic models were implemented using the 'MASS' package (Version 7.3-65) [30] in R (Version 4.4.0) [31].  
\[log\bigg{(}\frac{P(Y_{i}\leq j|X_{i})}{1-P(Y_{i}\leq j|X_{i})}\bigg{)}=\alpha_{ j}-X_{i}^{T}\beta \tag{1}\]  
Here, \(\beta\) is the vector of the regression coefficients, common across all thresholds, \(X_{i}^{T}\) is the vector of independent variables, and \(\alpha_{1}<\alpha_{2}<\cdots<\alpha_{J-1}\) are the ordered intercepts (often called _cut points_ or _thresholds_).  
The ordered logistic model can also be expressed in the form of a latent-variable model, with \(Y_{i}^{*}\) being an unobserved dependent variable of the ith observation, following Equation (2).  
\[Y_{i}^{*}=X_{i}^{T}\beta+\varepsilon \tag{2}\]  
Here, \(\varepsilon\) is the error term, which follows a standard logistic regression function. The categories of the response can be associated with the latent variable following Equation (3), which is applied to the 3-level response variable of this study:  
\[Y_{i}=\begin{cases}0,&Y_{i}^{*}\leq\alpha_{1}\\ 1,&\alpha_{1}<Y_{i}^{*}\leq\alpha_{2}\\ 2,&\alpha_{2}<Y_{i}^{*}\end{cases} \tag{3}\]  
In interpreting ordered logistic regression results, each estimated coefficient reflects the change in the underlying propensity toward higher versus lower response categories: a positive coefficient implies that increasing the corresponding predictor shifts the distribution of  
Figure 3: Location of the accommodations (STRs) of the surveyed tourists.  
outcomes upward (i.e., toward higher-valued categories), while a negative coefficient shifts it downward. Because the model is formulated in terms of cumulative odds, exponentiating a coefficient yields an odds-ratio interpretation, a multiplicative change in the odds of being in a higher rather than a lower category for a one-unit increase in the predictor, holding all else constant. Importantly, under the proportional-odds assumption, this effect is assumed to be constant across all threshold splits of the response scale, allowing a single coefficient to summarize the influence of the predictor on the ordering of outcomes.  
The interpretation of the marginal effects is also important because they translate these odds-ratio results into changes in predicted probabilities for each outcome category, which are often more directly interpretable. They quantify how a one-unit change in a predictor increases or decreases the probability of each specific category and can be reported at the mean of covariates or averaged over the sample. By showing how probability mass shifts among categories, marginal effects provide a more intuitive view of the substantive impact compared to odds ratios alone.