Modeling dynamics in household car ownership over life courses: a latent class competing risks model

The number of studies examining the influence of life course events on car possession and change is relatively small. Many more studies in transportation research have analysed household car ownership as a function of socio-demographic variables such as occupation and household size in a cross-sectional manner. Potoglou and Kanaroglou (2008) examined the influence of family structure, socio-economic characteristics, and accessibility at the place of residence on car ownership using the MNL model. Oakil et al. (2016) applied logistic regression analysis to explore the effects of urbanization level, household composition, income, and employment on car ownership. In addition to the socio-economic characteristics, the built environment have been found to influence on car ownership. Guo (2013) investigated the impact of residential parking supply on private car ownership using the logit model. This study pointed out that parking supply significantly influences household car ownership decisions. When on-street parking becomes more available, a household tends to own more cars. Huang et al. (2017) investigated the effects of the metro transit surrounding residential on car ownership using cross-sectional data. Results show that metro is negatively associated with car ownership. Ding et al. (2018) found that built environment related variables have significant impacts on car ownership using an integrated multinomial logit and structural equation model. They showed that people living in areas with smaller block size and higher residential and employment densities are more likely to own a car.

Relative to the abundance of cross-sectional studies, the analysis of the effects of child-birth is considerably more modest. Life event may change the travel needs of household and therefore may trigger households to purchase a car. For example, the birth of a child may increase the need for a (second) car in the household. Verhoeven et al. (2005) were the first to apply Bayesian networks to examine the influence of life events on car ownership decisions. Beige and Axhausen (2012) found that moving out of the parents’ house leads to changes in car ownership. The probability of changes in car availability increases, when a new person is born in the household. Oakil et al. (2014) investigated change in household car ownership in response to other life events. Using a logistic regression model, Clark et al. (2016) analysed the impact of job change, baby birth, and marriage on the change in household car ownership. The estimation found that childbirth and job change are associated with an increasing number of cars. Adopting a more complicated and flexible model, Wang et al. (2018) found further evidence of this relationship. The results showed that the increase in household size, change in workplace/study location, and residential relocation trigger car acquisition.

In the models related to the dynamics of car ownership which consider the household car transactions over time (Chen and Niemeier 2005; Yamamoto 2008; Rashidi et al. 2011; Oakil et al. 2014, 2018), hazard models have been predominantly used. For example, Jong (1996) applied a hazard model focusing on the replacing of cars, modelling the time elapsed between two vehicle transactions. In addition, different baseline hazard distributions such as the exponential and Weibull distribution were compared. These models allow estimating interval times between successive events, but the type of car transaction is not considered because hazard models can only deal with the situation that duration exit concerns a single event.

To overcome this limitation, competing risks models may be applied. These models are capable of representing both the timing and the type of transaction simultaneously. In travel behaviour research, the application of competing risks models is rare. Leszczyc and Timmermans (2002) compared different competing risks models accounting for different conditions to predict the timing and duration of shopping activities. Auld et al. (2012) applied the model to predict activity generation. In the context of car ownership, Yamamoto et al. (2004) are one of the few researchers who applied a competing risks model to analyse actual and intended holding duration of a car. In their model, replacing, disposing, and purchasing were considered as the different event types. Mohammadian and Rashidi (2007) estimated a competing risks hazard model that considered the influence of socio-demographics and life course events.

In addition to the types of car ownership events, the problem of heterogeneity is another issue of concern in hazard modelling. To accommodate the heterogeneity which cannot be explained by covariates, researchers have included a random term to the hazard model (Baizán et al. 2003; Oakil et al. 2014) or estimated latent classes (Van Den Berg et al. 2012; Paleti et al. 2017). Bhat (1996) provided a unified methodological framework to estimate a proportional hazard duration model with a non-parametric baseline hazard distribution and a non-parametric unobserved heterogeneity distribution. Han and Hausman (1990) accounted for heterogeneity by adding a random term to the hazard model in their flexible parametric proportional hazards model. Most of these models performed better than the standard hazard model. However, the method of incorporating a random effect is difficult to interpret and is over-dependent on the assumed distribution of the random term. In addition, the distribution cannot reveal the source of the heterogeneity, which could be used in the prediction or simulation of the dynamic car ownership process.

As to the competing risks models, many researchers introduced heterogeneity by including a random component which follows a specific distribution (Yamamoto et al. 2004; Mohammadian and Rashidi 2007; Yamamoto 2008; Anastasopoulos et al. 2012; Hasan et al. 2013). However, usually the random term is identically distributed and independent of the observed covariates. Two issues deserve further research. First, these models cannot capture the relationship between unobserved heterogeneity in the reaction to life course events and the observed socio-demographics. Such information may be useful to simulate or predict people’s car ownership decision. Second, the results obtained from the random effect approach tend to be very sensitive to the choice of the mixture distribution (Heckman et al. 1984). A questionable assumption about the distribution of the random effect will lead to bias.

The latent class model, which assumes a discrete distribution of heterogeneity across classes, could specify the relationship between socio-demographic variables and latent classes. Moreover, it is easier to interpret. Latent class models assume the distribution of heterogeneity takes the form of discrete clusters. Bhat et al. (2004) extended the hazard-based formulation by including a latent segmentation scheme to classify individuals into regular and erratic. Lee and Timmermans (2007) developed a latent class accelerated hazard model for activity episode durations. In this model, the end of the activity is the only type of exit event. Kim et al. (2017) estimated a latent class hazard duration model to analyse different charging behaviour and classified the electric vehicle users into regular and random users by treating charging regularity as a latent variable. These studies proposed latent class models for common survival analysis with one exit event type. The latent class idea has been rarely applied to extend the competing risks model.

Competing risks occur when more than one type of exit event is possible. In car ownership decisions, competing risks are present because the purchasing, replacing and disposing of a car could be the endpoint of car ownership duration. When the primary outcome is purchasing a car, replacing and disposing a car serve as the competing events.

There are two conventional methods to assess the effects of covariates when competing risks are present: the cause-specific models and sub-distribution models (Putter et al. 2007). The first approach models the effects of covariates on cause-specific hazard functions and treat other type of risks as censored. The second type of models handles the effect on the cumulative incidence function (CIF) of each risk. Both approaches are valid and the choice of most appropriate approach has been discussed at length. The consensus (Koller et al. 2012; Haller et al. 2012; Wolbers et al. 2014a) is that the sub-distribution hazard model is more appropriate for estimating incidences or prediction in the presence of competing risks. However, the cause-specific hazard model is better than the sub-distribution models when the aim is to investigate the triggers of exit events (Andersen et al. 2012; Wolbers et al. 2014a, b). Because the current paper is concerned with the effect of covariates on different types of car ownership decisions, the cause-specific model is chosen. The types of cause include disposing, replacing, and purchasing a car. The cause of the end of a period is one of the m distinct types indexed by \( j \in \{ 1,2, \ldots ,m\} \). The hazard of cause j at time t can be formulated as Eq. (1). Correspondingly, the probability density function can be expressed as Eq. (2).where \( \lambda_{r} (t) \) is the hazard, \( f_{j} (t) \) is the probability density function. \( S_{{}} (t) \) is the survival function which represents the probability that an individual remains in the state until time t. T indicates the time of failure, J indicates the type of cause events.

Define \( \varLambda_{j} (t) \) as the cumulative hazard function of cause j at time t. Define \( S_{j} (t) \) as the survival function of event j at time t, which means the probability of state change caused by event j does not happen until time t. The survival function of cause j at time t can be formulated as:

As the competing risks are independent with each other, the joint survival function S becomes

The likelihood of a specific type of risk treats all other types of risks as censored. Thus, the competing risks are treated as censored data in the cause-specific models. Imagine a sample with N observations, which are subject to m risks. d_ij is the indicator that subject i has exit with cause j. z is the covariates for subject i. The likelihood function L can then be written as:

Following the studies discussed previously, life course events may change the decision of car transactions and terminate car holding duration. In this latent class competing risks model, life course event variables, and the attributes of vehicles including engine type and whether a first hand or second-hand vehicle which could increase or decrease the hazard of car transactions, are directly incorporated as the covariates of the competing risks. The life course events considered in the model consist of the change in household composition, house location and occupation. To take into account the possible lag and lead effects, household events happening in 1 year before or after the car ownership change were assumed to affect the hazard of car transaction.

In addition, a number of variables with respect to households’ socio-demographics are used to define the class membership. In general, households with different socio-demographics may have different reactions to life course events. The variables that have been incorporated to predict class membership consists of the characteristics of occupation and household composition when a car transaction event happens.

Let \( \lambda_{j}^{k} (t) \) represent the cause-specific baseline hazard function of risk j, for each subject in class k. The integrated baseline hazard function is denoted by \( \varLambda_{j}^{k} (t) \). Let \( \beta_{r}^{k} \) represent the effect of covariate z on type of cause j, for each subject in class k. The hazard function of j occurring at time t for each subject in class k, can be formulated as

Then, the conditional likelihood given that individual i with covariates vector z belongs to latent class k for the events j has the following form:

The specification of the competing risks model requires a decision regarding the form of the density function. In modelling car ownership duration, existing work (Yamamoto et al. 2004) has shown that the Weibull distribution is the best when compared with other common distributions for the baseline hazard functions of car transactions. In the current paper, the exponential distribution and the Weibull distribution are tested as the baseline hazard to get the best-fit model. The hazard of event j, \( \lambda_{r}^{k} (t) \), happened to the household that belongs to segment k without considering other covariates can be written as (in case of a Weibull baseline function)and (in case of an exponential baseline function),K represents the number of segmentations in all observations. \( \alpha_{j}^{k} \) is the size parameter and \( \gamma_{j}^{k} \) is the shape parameter of the Weibull distribution in the baseline hazard function for event j happened to the household that belongs to class k. The exponential baseline function in Eq. (8) is the special case of the Weibull baseline function when \( \gamma_{j}^{k} = 1 \).

Conventionally, the probability w_k is conditional on the covariates of households. A household i, given a set of socio-demographic variables \( {\mathbf{u}}_{i} \), is assumed to have a probability \( w_{ik} ({\mathbf{u}}_{i} ) \) that belongs to class k. The \( {\varvec{\uptheta}}_{k} \) refers to the parameters of the socio-demographic variables. An intercept is included in the \( {\varvec{\uptheta}}_{k} \). In our study, we represent the class membership using a logit-form function of socio-demographic variables. The probability that a household belongs to class k, w_k, can be presented by

Given Eqs. (7) and (10), the likelihood function for individual i, \( L_{i} \) is given by

As a consequence, the log-likelihood function of the latent class model is

To estimate the parameters of each competing risks in each class, the maximum likelihood estimation method is used.

Table 2 represents the sample distribution of the current socio-demographics and car ownership. It shows that the different age groups are more or less equally represented in the sample, which is similar to population statistics. The percentage for the group younger than 40 years of age is 26.1%; for the group between 41 and 60, it is 35.9%; for the group older than 61 years of age, it is 38.1%. Regarding the age of the youngest child, the percentage of households who have a child younger than 6 years old is 11.4%; for the groups between 7 and 12 years old, older than 12 years and no child, the percentages are 7.0%, 13.9%, and 67.6%, respectively. In addition, the majority of respondents (72.0%) is married or living together with a partner. The frequency distribution of house ownership levels suggests that the majority of respondents (60.7%) own a house. As for the number of cars, 11.9% of the respondents have no car. The percentage of households with 1 car is 60.3%, while households have more than 1 car represent less than one-third of the sample. As for the education, the education levels in Netherlands are categorized into two levels. Primary school, pre-vocational secondary education, general secondary education, and secondary vocational education are classified as low education level. Higher general and preparatory academic education, higher professional education, bachelor and master degree and higher are classified as high education level. The results show that the percentage of people who received a low education (52.3%) is slightly greater than the percentage of people who received a high education (47.7%).

In addition to respondents’ current socio-demographics, the car ownership changes that respondents experienced were recorded. The socio-demographics at the time of car ownership change were also recorded. The frequency distributions of the selected socio-demographic characteristics associated with the car transactions observation are listed in Table 3. As shown, the age of the respondents when car ownership changed was categorized into three categories. The majority of observations (53.5%) were made when respondents were between 24 and 40 years old. The percentages for 41–60 years and older than 61 years of age are respectively 32.8% and 13.7%.

More than 86% of the respondents have no children or a child older than 12 years old when a vehicle transaction was observed, while households with a child under 6 years represent 9.3% of the observations. When buying or selling a car, the percentage couples (56.4%) is slightly higher than the percentage singles (43.2%). Similarly, at that time, 53.3% owned a house. The frequency distribution of education levels suggests that the majority of the respondents (53.6%) did not hold a high education degree when they bought/sold a car.

According to the estimated parameters, households in each class are most sensitive to baby birth when purchasing a car. The positive parameters of baby birth for the two classes mean that people are more likely to purchase a car when a baby is born. It suggests that baby birth generates a shift in travel demand such as escorting the child to a day care centre and other locations. In addition, the comfort and convenience of a car are important when a baby is present in the household. This is in line with the findings from previous studies (Oakil et al. 2014; Clark et al. 2016). The parameter of getting married is significant for class 1 but not for class 2, indicating that households in class 1 have a higher probability to purchase a car than those in class 2 when they get married. For the change of house location, the effects are similar for the households in different classes in the sense that car ownership increases when they move, indicating that people prefer to purchase a car when they move. The parameters for change of job in the two classes are both positive, which means that people prefer to purchase a car when they change their job. For car attributes, once the households own an electric car, they have a lower probability to purchase a new car. Owning a brand-new car has a similar effect for both classes. The baseline hazard shows a considerable difference between the baseline hazards of the two classes. The baseline hazard for class 2 is higher than that of class 1. Class 2 has a higher probability to purchase a car than class 1 without consideration of other covariates.

According to the sensitivity analysis of each types of life events (see Fig. 2, (a1), (b1), (c1), and (d1), the impact of baby birth on car purchasing is greater than that of other events for people in class 1, while the change of job has the biggest impact for people in class 2.

Through comparatively analysing the difference between two classes, we found that households are more sensitive to baby birth and marriage in class 1, and to a change of job and house relocation in class 2. Combined with the characteristics of the two classes, the effects of life course events on car purchasing indicates that young people tend to attach a greater weight to household composition when they decide on car ownership. In addition, households without a car are more likely to belong to class 1, which implies that people purchase the first car mostly because of changing household composition. For a mid-aged and aged household, people attach a higher weight to the change of job and house relocation. A change of job or house relocation implies a different commuting distance and public transport availability. A car purchasing decision seems to be an adaptation to this changing context to improve convenience.

Regarding car disposing, baby birth has a negative (− 1.47 for class 1 and − 0.48 for class 2) and significant effect on the probability of selling a car. In contrast to the parameter of the baby birth event, parameters of marriage, residential relocation, and change of job are all positive, which indicates that people have a higher probability to sell a car when these events happen. Specifically, for people in class 1, the probability of selling a car increases when they get married. For people in class 2, marriage has no significant influence on the decision to sell a car. Moving house increases the probability to sell a car for people belonging to class 2, while it does not have a significant effect on the people belonging to class 1. The probability of selling a car increases when people change job, regardless of which class a household belongs to. For the attributes of a car, when people own an electric vehicle, they are less likely to sell a car. In addition, whether their current car is a brand-new car has no significant influence on car selling decisions. Households in class 2 are more likely to sell their car as the γ of baseline hazard is higher than for households in class 1. In general, the difference between the two latent classes in selling a car is similar to the purchasing a car.

According to the sensitivity analysis of each types of life events (see Fig. 2, (a2), (b2), (c2), and (d2)), the probability of car disposal decreases sharply in the case of baby birth for people in class 1, while the probability of car disposal increases most obviously in the case of job change for people in class 2. Households in class 1 are more sensitive to changes in household composition while households in class 2 are more sensitive to changes in house location.

In the case of car replacement, households in class 1 are more likely to replace a car while people in class 2 are less likely to replace a car. People who get married are more likely to replace a car in both classes. Residential relocation has a significant influence on the decision of replacing a car for people belonging to class 2, while it has no significant influence for people belonging to class 2. Change of job will trigger replacing a car for both classes. Similar to the previous two types of car ownership change, households in class 1 are more sensitive to marriage and baby birth than house relocation and change of job. In contrast, households in class 2 are more sensitive to change of job and house location. The baseline hazard is higher for class 2 than class 1, indicating that households in class 2 are more likely to replace their car.