Analyzing car ownership in Quebec City: a comparison of traditional and latent class ordered and unordered models

Abstract

Private car ownership plays a vital role in the daily travel decisions of individuals and households. The topic is of great interest to policy makers given the growing focus on global climate change, public health, and sustainable development issues. Not surprisingly, it is one of the most researched transportation topics. The extant literature on car ownership models considers the influence of exogenous variables to remain the same across the entire population. However, it is possible that the influence of exogenous variable effects might vary across the population. To accommodate this potential population heterogeneity in the context of car ownership, the current paper proposes the application of latent class versions of ordered (ordered logit) and unordered response (multinomial logit) models. The models are estimated using the data from Quebec City, Canada. The latent class models offer superior data fit compared to their traditional counterparts while clearly highlighting the presence of segmentation in the population. The validation exercise using the model estimation results further illustrates the strength of these models for examining car ownership decisions. Moreover, the latent class unordered response models perform slightly better than the latent class ordered response models for the metropolitan region examined.

Appendices

Appendix 1: estimation results of the traditional models

See Tables 5, 6, 7, 8.

Table 5 Traditional ordered logit model (OL) estimates with transit accessibility interactions (N = 5218)

Table 6 Latent segmentation based ordered logit model (OL) estimates with only transit accessibility as the segmentation variable (N = 5218)

Table 7 Traditional multinomial logit model (MNL) estimates with transit accessibility interactions (N = 5218)

Table 8 Latent segmentation based multinomial logit model (OL) estimates with only transit accessibility as segmentation variable (N = 5218)

Appendix 2: mathematical formulation of latent class models

Let us consider S homogenous segments of households (the optimal number of S is to be determined). We need to determine how to assign the households probabilistically to the segments for the segmentation model. The utility for assigning a household q (1, 2, …, Q) to segment s is defined as:

$$U_{qs}^{*} = \beta_{s}^{\prime } z_{q} + \xi_{qs}$$

(1)

$z_{q}$ is a (M × 1) column vector of attributes that influences the propensity of belonging to segment s, $\beta_{s}^{\prime }$ is a corresponding (M × 1) column vector of coefficients and $\xi_{qs}$ is an idiosyncratic random error term assumed to be identically and independently Type 1 Extreme Value distributed across households q and segment s. Then the probability that household q belongs to segment s is given as:

$$P_{qs } = \frac{{\exp (\beta_{s}^{\prime } z_{q} )}}{{\mathop \sum \nolimits_{s} \exp (\beta_{s}^{\prime } z_{q} )}}$$

(2)

Within the latent segmentation approach, the probability of household q choosing auto ownership level k is given as:

$$P_{q} \left( k \right) = \mathop \sum \limits_{s = 1}^{S} (P_{q} \left( k \right) | s )(P_{qs} )$$

(3)

where $P_{q} \left( k \right) |s$ represents the probability of household q choosing auto ownership level k within the segment s. Note that the choice construct of car ownership considered to compute $P_{q} \left( k \right) |s$ may be either the ordered or unordered response mechanism.

Now, if we consider the car ownership levels of households (k) to be ordered,

$$y_{qs}^{*} = \alpha_{s}^{\prime } x_{q} + \varepsilon_{qs} ,\,\;y_{q} = k,\quad {\text{if}}\,\,\psi_{{s_{k - 1} }} < y_{qs}^{*} < \,\psi_{{s_{k} }}$$

(4)

where $y_{qs}^{*}$ is the latent propensity of household q conditional on q belonging to segment s. $y_{qs}^{*}$ is mapped to the ownership level $y_{q}$ by the $\psi$ thresholds ($\psi_{{s_{0} }} = - \infty$ and $\psi_{{s_{k} }} \, = \,\infty$) in the usual ordered-response fashion. $x_{q}$ is a (L × 1) column vector of attributes that influences the propensity associated with car ownership. $\alpha$ is a corresponding (L × 1) column vector of coefficients and $\varepsilon_{qs}$ is an idiosyncratic random error term assumed to be identically and independently standard logistic distributed across households q. The probability that household q chooses car ownership level k is given by:

$$P_{q} \left( k \right)|s = \varLambda \left( {\psi_{{s_{k} }} - \alpha_{s}^{\prime } x_{q} } \right) - \varLambda \left( {\psi_{{s_{k - 1} }} - \alpha_{s}^{\prime } x_{q} } \right)$$

(5)

where $\varLambda (.)$ represents the standard logistic cumulative distribution function (cdf).

If we consider the car ownership levels (k) to be unordered, we employ the usual random utility based multinomial logit (MNL) structure. Equation (6) represents the utility $U_{qk}$ that household q associates with car ownership level k if that household belongs to segment s

$$U_{qk} | s = \alpha_{s}^{\prime } x_{q} + \varepsilon_{qk}$$

(6)

$x_{q}$ is a (L × 1) column vector of attributes that influences the propensity associated with car ownership. α is a corresponding (L × 1)-column vector of coefficients and $\varepsilon_{qk}$ is an idiosyncratic random error term assumed to be identically and independently generalized extreme value (GEV) distributed across households q. Then the probability that household q chooses car ownership level k is given as:

$$P_{q} \left( k \right) | s = \frac{{\exp (\alpha_{s}^{\prime } x_{q} )}}{{\mathop \sum \nolimits_{k} \exp (\alpha_{s}^{\prime } x_{q} )}}$$

(7)

The log-likelihood function for the entire dataset with appropriate $P_{q} (k)|s$ for ordered and unordered regimes is provided below:

$$L = \sum\limits_{q = 1}^{Q} {\log \left( {P_{q} \left( {k_{q}^{*} } \right)} \right)} ,$$

(8)

where k _q * represents the ownership level chosen by household q.

Appendix 3: estimation results of the traditional models

See Tables 9, 10.

Table 9 Traditional ordered logit model (OL) estimates with all variables (N = 5218)

Table 10 Traditional multinomial logit model (MNL) estimates with all variables (N = 5218)

Appendix 4: elasticity effects

The exogenous variable coefficients do not directly provide the magnitude of impacts of variables on the probability of car ownership levels. For better understanding the impacts of exogenous factors, we compute the relevant elasticities for changes in selected variables. The calculation results are presented in Table 11. For the analysis, we selected three socio-demographic variables (number of employed adults, number of children and number of transit pass holders) and two land use attributes (transit accessibility and residential density). Note that the elasticity effects were computed for the OL, LSOL II, MNL and LSMNL II models.

Table 11 Elasticity effects of important variables

The results illustrate that both full-time working adults and part-time working adults increase household car ownership levels. However, as expected full-time working adults had greater impact on increasing vehicle ownership levels (2 or more) compared to the part-time working adults. The impact of change in number of children demonstrates the likelihood of vehicle fleet size reduction with similar impacts in magnitude in all the models. The reduction in fleet size observed in the elasticity analysis, while counterintuitive, is consistent with the coefficients of that variable in the models and is similar across all models; in particular, with respect to the large percentage increase in zero-car households it should be kept in mind that the base proportion of those households is not very large (10 %). It might be useful to investigate this result further in future analysis.

Increase in number of transit pass holders resulted in a decrease in car ownership levels. The decreasing effect was more pronounced for 3 or more car ownership level. We can also see from the table that increase in transit accessibility and residential density reduces the probability of household’s owning 2 or more cars. However, between the two attributes, residential density has a greater impact on car ownership levels than transit accessibility. The computation exercise provides an illustration of the applicability of the proposed framework for policy analysis.