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Exploring the nonlinear effects of ridesharing on public transit usage: A cas e study of San Diego
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Zhaolin Zhanga, Guocong Zhaib*, Kun Xieb, Feng Xiaoa
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a School of Business Administration, Faculty of Business Administration, Southwestern University of
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Finance and Economics, Chengdu, 610031, China
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b Department of Civil & Environmental Engineering, Old Dominion University (ODU), 129C Kaufman
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Hall, Norfolk, VA, 23529, USA
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To cite: Zhang, Z., Zhai, G., Xie, K., Xiao, F. (2022) Exploring the nonlinear effects of ridesharing on
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public transit usage: A case study of San Diego. Journal of Transport Geography.
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Abstract
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The emergence of ridesharing services might complement or substitute public transit systems, leading to
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intricate relationships between the two services. However, limited studies focused on the nonlinear effects
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of ridesharing use frequency on public transit usage. Therefore, this paper investigated such nonlinear
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effects using the hierarchical negative binomial generalized additive model (HNBGAM), with the latest
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publicly available National Household Travel Survey (NHTS) dataset. The negative binomial and
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hierarchical negative binomial generalized linear models were also developed for comparison with the
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HNBGAM. The NHTS data involved travel information of 928 ridesharing users within 98 census tracts in
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San Diego. Two-level hierarchy (individual and census tract level) was constructed in the HNBGAM. In
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addition, the smooth function of the HNBGAM could help identify the nonlinear effects of ridesharing use
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frequencies on public transit usage. Socio-demographic factors (age, gender, race, household size, etc.) and
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built environment factors (e.g., population density, worker density, percentage of rental houses, and house
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unit density) were also considered in the modeling process. The findings revealed a negligible impact on
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public transit usage for occasional ridesharing use (from one to eleven times per month), a complementary
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effect for regular ridesharing use (from eleven to thirty-two times per month), and a substitution effect for
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active ridesharing use (more than thirty-two times per month). Understanding such nonlinear relationships
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could help policymakers make more informed decisions to avoid the over-substitution of public transit
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usage and better complement the public transport system.
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Keywords: Spatial dependence, Multilevel, Generalized additive models, Nonlinear effects, Ridesharing
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use frequency
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1. Introduction
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The ridesharing
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services provided by transportation network companies (TNCs) such as Uber, Lyft, and
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Didi Chuxing have reformed the mobility patterns of ridesharing users. More specifically, ridesharing
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services render the gaps between collective and individual mobility options. The high-efficient matching
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platform and car-based movements of ridesharing services provided travelers with more convenient and
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economic mobility alternatives than public transit and traditional taxi services, respectively (Ya ng e t a l .,
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2022). Ridesharing services could satisfy travel demands to some degree for under-developed areas or the
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mid-night periods where limited public transit resources are available at those places or periods. Besides
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more convenient characteristics, travelers might also share trips of similar origins or destinations via the
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ridesharing platform, thus reducing travel fares. Ridesharing services can also mitigate the unexpected
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supply disruptions and demand surges, especially for the temporal disruption of metro stations (Hoffmann
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et al., 2016).
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Prior studies have concluded that ridesharing services could compensate for public transit usage by tackling
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the first-and-last mile issues and thus improving the accessibility of the public transit (Hall et al., 2018; Yan
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For more details to distinguish ridesharing from other service modes (e.g., ride-sourcing, ride-hailing, ride-splitting, and
carsharing), please refer to (Chen et al., 2021b; Lyu et a l., 2021 ).
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et al., 2019). However, ridesharing services might also substitute public transit usage due to the competition
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between the two alternatives (Graehler et al., 2019; Schaller, 2021). Hence, mixed conclusions about the
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effects of ridesharing services on public transit are reached. We postulate that a simple dummy variable (0:
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without ridesharing services and 1: with ridesharing services) in prior studies was too mechanical to profile
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the ridesharing use frequency, inducing potentially mixed conclusions (Ghaffar et al., 2020; Hu et al., 2018).
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Ghaffar et al. (2020) found significant nonlinear relationships between ridesharing usage and public transit
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supply at the census tract level. Low public transit density is related to very few ridesharing trips, while
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middle-level public transit density is associated with the most ridesharing trips. On the one hand, if the
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nonlinear associations that indeed exist are ignored, the linearity assumption will lead to biased estimates
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for the coefficients, which hinder understanding the role of studied factors in influencing travel behavior
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(Wu et al., 2019). On the other hand, identifying such nonlinear associations could provide a new
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perspective to examine the impact of ridesharing usage on public transit usage, i.e., the complement (or
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substitution) effect is only shown within certain thresholds of ridesharing use frequency. For example, Ding
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et al. (2021) found that land use mix is positively associated with choosing transit for commuting when the
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entropy index is smaller than 0.60, while a negative correlation is observed after the threshold. Thus,
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identifying such nonlinear (threshold) associations has important implications for planning practice.
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Additionally, previous research has shown that different age groups have different tendencies to use public
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transit (Deka and Fei, 2019; Grimsrud and El-Geneidy, 2013), which prompts us to investigate whether
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there is a nonlinear association between age and public transit usage. This association also has practical
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implications, as it can help policymakers more accurately identify the propensity of different age groups to
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use public transit, which facilitates the development of differentiated and effective policies to incentivize
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public transit usage. Further, travel patterns of travelers within the same census tract or traffic analysis zone
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are usually correlated or spatially dependent (Ding and Cao, 2019). Commonly observed independently and
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identically distributed (i.i.d.) assumption would be violated without considering the correlations within the
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same census tract (Hong et al., 2014). Moreover, many studies have demonstrated that the effects of built
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environment variables may vary from region to region, requiring multilevel models to capture such spatially
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varying effects (Ding et al., 2021; Li et al., 2020a; Ma et al., 2018). More appropriate approaches should
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be implemented to identify such hierarchical structures.
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Because previous research that jointly identified the nonlinear effects and spatial dependence is relatively
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limited, this study focuses on the nonlinear effects of ridesharing use frequency on public transit usage by
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the hierarchical negative binomial generalized additive model (HNBGAM), using the latest national
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household travel survey (NHTS) dataset in San Diego. For comparison, the modeling results of the negative
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binomial and hierarchical negative binomial generalized linear models have also been estimated. The
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development of the HNBGAM would contribute to literature for more robust investigations on nonlinear
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effects. More specifically, the hierarchical components assume a random intercept of the HNBGAM varied
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among different census tracts. Additionally, the generalized additive components of the HNBGAM use
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smooth functions to profile nonlinear relationships between ridesharing use frequency and public transit
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usage. This study also explores the nonlinear effects of respondent age on public transit usage. Such
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explorations above can provide a reference to classify the contributing covariates and reveal the nonlinear
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effects. Therefore, this study can help policymakers make more informed decisions to avoid the over-
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substitution of public transit usage and better complement the public transit systems.
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The remainders are constructed as follows. Section 2 provides an overview of factors influencing public
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transit usage and corresponding modeling approaches. Section 3 describes the national household survey
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data of San Diego, following the descriptions of the relative methodologies. Section 5 presents and
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discusses the modeling results. The last section concludes the findings and provides corresponding
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suggestions.
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2. Literature review
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This section mainly summarizes the impacts of ridesharing services on public transit, it also briefly
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summarizes other factors that affect public transit usage. The corresponding modeling approaches are also
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reviewed to explore the implications of ridesharing services.
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2.1. Overview of factors influencing public transit usage
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We f ir st r ev ie w th e rel ate d lit er at ure abou t the eff ec ts o f ri de sh ar ing serv ic es on p ub li c tr an si t. The impacts
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of ridesharing services on public transit are summarized into three types: complement, substitution, and
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mixed effect, thus reaching a mixed conclusion in prior studies. Figure 1 shows that 32% of the 22 previous
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studies supported complement and substitution effects, respectively
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. However, more research (36%)
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argued that the relationship between the two alternatives is not a black-and-white issue.
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On the one hand, ridesharing services might complement public transit and improve the accessibility of
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public transit (Yan et a l. , 2 0 19 ). Yan e t a l . ( 20 1 9) concluded that integrating ridesharing services and public
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transit would sufficiently address the first-and last-mile issues and improve the accessibility of public transit.
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More specifically, mode shares of public transit would increase by 13% if the out-of-vehicle time of the
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integrated alternative were reduced to 3 minutes or less. Thus, ridesharing services might have a
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complementary effect on public transit. Similarly, Zgheib et al. (2020) indicated that the transit ridership
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would increase by 2% by introducing ridesharing as a feeder service. A possible synergy effect between
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ridesharing services and transit would increase transit mode shares from 33.53% to 36.89%, with a half
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reduction in total travel fares. For more details about the complementary impacts of ridesharing services on
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public transit, please refer to the corresponding studies by (Contreras and Paz, 2018; Grahn et al., 2020;
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Hall et al., 2018; Smith, 2016; Stiglic et al., 2018; Zhang and Zhang, 2018).
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Figure 1. The impacts of ridesharing on public transit usage in previous literature (N=22)
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On the other hand, ridesharing services would be more substitutive than complementary to public transit.
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For instance, ridesharing services might suppress travelers’ willingness to use public transit, resulting in a
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6% decrease in bus ridership and a 3% reduction in light rail ridership, by residential surveys in seven major
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cities in America (Clewlow and Mishra, 2017). In addition, ridesharing services would be more substitutive
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with public transit in rural areas where limited public transit coverage and service quality might shift more
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Although some literature only confirms one single effect (complement or substitution), these studies do not definitively deny the
existence of the other effect. However, we still retain these references to provide a more comprehensive literature review.
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trips from public transit into ridesharing services (Schwieterman, 2019). A more significant substitution
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effect than the complement effect for public transit has also been observed in other studies (Graehler et al.,
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2019; Schaller, 2018; Schaller, 2021; Tirachini and del Río, 2019; Wa rd e t a l. , 20 21).
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Some studies investigated the complement and substitution effects of ridesharing services on public transit
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simultaneously. Kong et al. (2020) sufficiently described the spatial-temporal effects of ridesharing services
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on public transit in Chengdu, China. For example, the substitution effect on public transit would be higher
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during daylight (8:00–18:00) than during other periods. In addition, the substitution effect of ridesharing
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services on public transit would be more evident in city centers and areas adjacent to metro stations. In
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contrast, the complementary impact of ridesharing services would exist in suburban areas short of public
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transit coverage. In particular, the complement and substitution relationships between ridesharing services
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and public transit mainly depended on the fleet size of ridesharing services based on the user-equilibrium
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model (Ke et al., 2021). Specifically, passengers are more inclined to use ridesharing services if the fleet
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size is large enough to meet requirements. However, some studies have revealed that the bidirectional effect
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of ridesharing services on public transit may exist (Babar and Burtch, 2020; Deka and Fei, 2019; Jin et al.,
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2018; Rayle et al., 2014; Yo u ng e t a l ., 2 02 0 ). Further, ridesharing services might also have no significant
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substitution or complementary impact on public transit because of a small market share (Habib, 2019;
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Malalgoda and Lim, 2019; Yo un g a n d F ar b er, 20 1 9).
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In general, there are a variety of factors that affect public transit usage, including external and internal
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factors. External factors are usually those that are not directly related to the transit system and its managers
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(such as population and employment levels), while internal factors are those that can be controlled by transit
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managers, such as fares and service levels (Taylor a nd F ink, 20 03). Here we only focus on the study related
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to external factors, specifically socio-demographic and built-environment factors. Because these factors are
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closely related to our research. Previous literature has revealed the association of several socio-demographic
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factors with transit usage, such as education, age, ethnicity, gender, driver status, income, and the number
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of household vehicles (Deka and Fei, 2019; Ding et al., 2021; Momeni and Antipova, 2022). For example,
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Deka and Fei (2019) used the 2017 NHTS data to identify how the variables (e.g., income, gender, driver
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status, and the number of household vehicles) affect transit usage. Additionally, a host of literature confirms
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the impact of built environment factors on transit usage (Deka and Fei, 2019; Ding et al., 2021; Li et al.,
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2020a; Li et al., 2020b; Ma et al., 2018). Li et al. (2020a) found that population density, worker density,
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and land use diversity are highly significant in cultivating transit ridership. Ding et al. (2021) explored the
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relationship between built environment attributes and transit commuting in Nanjing. The results revealed
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that some built environment attributes (such as floor area ratio, intersection density, bus stop density, bus
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network density, etc.) are significantly associated with the likelihood of using transit for commuting. Other
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studies have also confirmed the impact of some built environment characteristics on the use of public transit,
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e.g., population size, house unit density, worker density, the share of rented dwellin gs (Deka and Fei, 2019);
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road density, bike lane density, parking spaces, subway, and bus accessibility (Qian and Ukkusuri, 2015);
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residential building density, place of employment density, bus stop density, commercial establishment
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density, road density (Ma et al., 2018).
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2.2. Overview of the corresponding modeling approaches
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In this subsection, we will review the modeling approaches related to the implications of ridesharing
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services on transportation systems, the nonlinear relationship explorations, and the hierarchical structure in
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transportation research. A summa ry of the research met hods in the p revi ous liter atur e is show n in Table 1 .
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Discrete choice models such as the binary logit and mixed logit models were commonly used in travel
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behavior studies, especially for travel behaviors related to ridesharing services (Gehrke et al., 2019; Habib,
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2019; Henao and Marshall, 2019; Ya n e t al . , 2 01 9; Zgheib et al., 2020). Take Habib (2019) as an example.
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A m ulti nomial lo git m odel was used to in vest igate the comp etit ion b etwe en ri desh aring ser vices and othe r
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alternatives. In addition, Zgheib et al. (2020) developed a mixed logit to investigate the impacts of
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ridesharing services on transit.
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Ordinary least square (OLS) regression models were also developed to explore the impacts of ridesharing
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(Bekka et al., 2020; Contreras and Paz, 2018; Grahn et al., 2020; Zou and Cirillo, 2021), especially
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considering spatial dependence or random-effects (Graehler et al., 2019; Kong et al., 2020). Previous
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studies also used count data models such as Poisson, negative binomial, and their extensions to investigate
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the impacts of ridesharing services on transportation systems (Deka and Fei, 2019; Jiao et al., 2020; Kong
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et al., 2020; Zhang and Zhang, 2018). For instance, the negative binomial models were developed to
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investigate the impacts of shared mobility on residents' travel demands (Jiao et al., 2020).
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Tabl e 1. Summary of research method s in the literatu re
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Studies
Methods
Nonlinear effects
Spatial dependence
(Clewlow and Mishra, 2017;
Henao and Marshall, 2019;
Rayle et al., 2014)
Descriptive analysis
-
-
(Chen et al., 2021a; Sabouri
et al., 2020a; Shao et al.,
2022; Tu et al., 2021)
Machine learning
approaches
P
-
(Graehler et al., 2019)
OLS with random
effects
-
P
(Kong et al., 2020)
OLS with spatial
dependence
-
P
(Sabouri et al., 2020b)
Multilevel OLS
model
-
P
(Bekka et al., 2020; Contreras
and Paz, 2018; Grahn et al.,
2020; Zou and Cirillo, 2021)
OLS
-
-
(Wang et a l. , 20 18)
Multilevel discrete
choice model
-
P
(Tirachini and del Río, 2019;
Yo un g e t a l ., 2 0 20 )
Discrete choice
model (ordinal logit
model)
-
-
(Gehrke et al., 2019; Habib,
2019; Malalgoda and Lim,
2019; Yan et a l ., 20 1 9;
Zgheib et al., 2020)
Discrete choice
model (mixed logit
regression model)
-
-
(Deka and Fei, 2019; Jiao et
al., 2020; Zhang and Zhang,
2018)
Count data model
-
-
Our study
HNBGAM
P
P
Note: Pmeans “yes”, - means “no”.
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However, few prior studies above investigated the nonlinear impacts of ridesharing use frequency on
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transportation systems. Two commonly used approaches for nonlinear analyses are machine learning
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approaches and generalized additive models (GAMs) in transportation research. Machine learning
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approaches such as random forest, gradient boosted decision trees, Bayesian neural networks, and support
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vector machines have been widely applied to explore the nonlinear effects, i.e., nonlinear effects of built
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environment on intermodal transit trips (Chen et al., 2021a), nonlinear effects of land use and motorcycles
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on car ownership (Shao et al., 2022), and nonlinear effects of the built environment on ride-splitting (Tu et
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al., 2021). Despite the ability to sufficiently identify and present the nonlinear relationships, some
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shortcomings of the machine learning approaches still hinder the applications in nonlinear relationship
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analyses. For example, the machine learning approach could not identify whether the covariates were
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statistically significant in the nonlinear relationship studies. In addition, machine learning may lead to
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overfitting issues in exploring the nonlinear relationship between the built environment and travel behavior
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(Ding et al., 2021). Fortunately, the GAMs could tackle the potential problems listed above to some degree
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and identify the nonlinear relationships with flexible additive components simultaneously (Hastie, 2017;
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T.J. Hastie, 199 0; Wood, 2 017 ). For instance, Hu et al. (2018) developed an extensive GAM to understand
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how factors affected station-based car-sharing use frequency. Dhulipala and Patil (2020) identified factors
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influencing agricultural freight production and explored the nonlinear relationship by GAM. Additionally,
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some other studies used GAMs to explore the relationships between built environment factors and travel
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behaviors (Park et al., 2020; Wali et al ., 2 02 1; Yan g e t a l. , 2 02 0 ).
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Hierarchical models were developed to identify multilevel structures, thus avoiding biased estimate (Zhai
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et al., 2020). Only limited literature has considered the spatial structure when studying the impacts of
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ridesharing services (Sabouri et al., 2020a; Sabouri et al., 2020b). For instance, Sabouri et al. (2020b)
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developed a multilevel model to address how the built environment affects the demand for ridesharing
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services. (Sabouri et al., 2020a) used a multilevel Poisson and a machine learning method to understand the
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effect of ridesharing services on vehicle ownership based on the latest national household travel survey data.
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Moreover, hierarchical models have also been widely used in crash safety analyses (Xie et al., 2014; Zhai
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et al., 2020).
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Concisely, the nonlinear relationships between ridesharing use frequency and public transit usage should
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be further explored. In addition, the GAMs integrated with the hierarchical structure might be a helpful
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approach to sufficiently identifying and describing the potential nonlinear relationships. Therefore, this
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study developed the HNBGAMs to understand the nonlinear effects of ridesharing on public transit usage.
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3. Data
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3.1. Data source
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The 2017 NHTS conducted by the Federal Highway Administration (FHWA) could provide the use
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frequencies of ridesharing and public transit for each respondent (FHWA, 2019). The NHTS data also
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contains socio-demographic factors (age, gender, race, household size, etc.) and built environment factors
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(e.g., population density, worker density, percentage of rental houses, and house unit density). Additionally,
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we discriminated the two-level hierarchical structure in the data as shown in Figure 2: individual and census
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tract levels. Unfortunately, the publicly available NHTS dataset does not include census tract identifiers for
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privacy reasons. Identifying the actual census tract index might be reasonable by linking the built
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environment factors in the NHTS and the American Community Survey (ACS) datasets. However, the built
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environment factors exist some approximation errors at the census tract level in the NHTS dataset for
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privacy reasons, thus being unable to directly compare the relative factors between the two datasets.
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Otherwise, we could get the actual census tract identifiers without protecting the privacy of respondents.
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Take popul ation de nsity as an examp le. Th e po pulation d ensity i n th e NH TS would be coded as 17,000
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persons per square mile if the actual population density were more than 10,000 but less than 24,999 persons
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per square mile at the census tract level. Therefore, the unique census tract identifier was derived only if
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involving the same values of four built environment factors (i.e., population density, worker density,
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percentage of rental houses, and house unit density at the census tract level).
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Because of the potential heterogeneity (e.g., transportation infrastructures and traffic regulations) among
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different cities, we only took San Diego (Figure 3), a typical car-oriented area simultaneously owning well-
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developed public transit systems, as an example. In addition, as one of several areas early activating the
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ridesharing market in America, San Diego has also offered sufficient potential to reveal the impacts of
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ridesharing services on public transit usage in the well-developed periods of ridesharing services.
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Figure 2. Two-level hierarchical structure (individuals and census tracts)
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Figure 3. San Diego regional household travel survey area
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3.2. Data description
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After removing observations without key information, the National Household Travel Survey (NHTS)
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dataset involved 928 ridesharing users in San Diego, which accounted for 18.95% of regular travelers in
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the area. The regular travelers refer to all respondents surveyed by NHTS in San Diego. Table 2 summarizes
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the characteristics of ridesharing users and regular travelers in San Diego. The t-test is performed to assess
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whether the socio-demographic factors of ridesharing users are significantly different from those of regular
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travelers.
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The average public transit usage for ridesharing users and regular travelers is 1.74 times and 1.02 times per
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month, respectively. The potential complementary effects of ridesharing services on public transit might
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explain the significant difference (Stiglic et al., 2018). Intuitively, the mean value of ridesharing use
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frequency of ridesharing users (4.36) is much higher than that of regular travelers (0.83) because the regular
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travelers involve ridesharing users and non-ridesharing users.
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Other covariates involved socio-demographic and built environment factors, which were selected based on
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previous literature. Due to the limited data availability, only some of the potential independent variables
15
are considered. For socio-demographic factors, the average age of ridesharing users is 41.68, much younger
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than that of regular travelers. This result is consistent with Ya n g et al . ( 2 02 2 ). A significant difference in the
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ratio of white America is founded between ridesharing users (79%) and regular travelers (75%). 49% of
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ridesharing users are female. Ridesharing users tend to have a higher education background than regular
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travelers. The driver's license ownership ratio for ridesharing users and regular travelers is 96% and 91%,
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respectively. Regarding household children, the average number of household children for ridesharing users
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is lower than that of regular travelers. Consistently, ridesharing users tend to have a smaller household size
22
than regular travelers. Of ridesharing users, 30% have an annual household income of fewer than 75,000
23
USD, whereas the ratio for regular travelers is 26%. In terms of household vehicle ownership, ridesharing
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users tend to have a lower household vehicle ownership than regular travelers. The reason may be that
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people who do not have a private vehicle in the household are more likely to use ride-sourcing services
26
(Cramer and Krueger, 2016). For built environment factors, the average population density and work
27
density of ridesharing users are much higher than those of regular travelers. The ratio of rental house density
28
and the average house unit density of ridesharing users are also higher than those of regular travelers.
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Tabl e 2. Descriptive analysis for ride sharing users (N = 928) and regu lar travelers (N = 4,898)
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Vari a bl es
Definition
Ridesharing users
Regular travelers
p-
value
Mean
S.D.
Mean
S.D.
Dependent variable
Public transit
usage
Number of transit trips in
the past month
1.74
5.11
1.02
4.03
< 0.01
Key factors
Ridesharing use
frequency
Number of ridesharing trips
in the past month
4.36
4.50
0.83
2.60
< 0.01
Socio-demographic factors
Respondent age
Age (year)
41.68
14.87
51.48
18.48
< 0.01
Respondent
gender
Female = 1, Male = 0
0.49
0.50
0.51
0.50
0.26
Highly educated
respondent
Having bachelor’s degree or
higher one = 1, others = 0
0.73
0.44
0.55
0.50
< 0.01
Driver license
Having a driver license = 1,
having no driver license =0
0.96
0.20
0.91
0.28
< 0.01
Number of
household
children
Number of people (age <18)
in household
0.37
0.77
0.44
0.85
0.02
Household size
Number of people in the
2.34
1.14
2.56
1.03
< 0.01
9
household
Low-level
household
income
Annual household income
of fewer than 75,000 USD =
1, others = 0
0.30
0.46
0.26
0.44
0.01
Household
vehicle
ownership
Number of household
vehicles
2.10
1.14
2.21
1.20
0.01
Built environment factors
Population
density
Number of residents per
10,000 population per
square miles
0.89
0.62
0.74
0.56
< 0.01
Measured but excluded factors
Respondent race+
White America = 1, others
=0
0.79
0.41
0.75
0.43
0.01
Percentage of
rental houses++
Ratio of rental house units
and total house units
0.46
0.22
0.38
0.21
< 0.01
Work er de ns it y
Number of employed
residents per 10,000
population per square miles
0.33
0.16
0.29
0.16
< 0.01
House unit
density
Number of house unit per
10,000 house units per
square miles
0.46
0.47
0.33
0.35
< 0.01
Samples
Sample weight of respondents
542.87
670.09
569.41
702.6
2
-----
Notes:
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1. + shows the insignificance at the 0.05 level for the NBGLM and HNBGLM.
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2. ++ means the insignificance at the 0.05 level for the HNBGLM.
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4. Methodology
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This study adopted the HNBGAM to explore the hierarchical structure (spatial dependence) and nonlinear
6
effects of ridesharing on public transit usage. In particular, the negative binomial generalized linear model
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(NBGLM) and hierarchical negative binomial model (HNBGLM) were also developed as a comparison.
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(1) NBGLM
10
The following functional forms are commonly used in NBGLM:
11
ln#(𝜃!)= 𝛽"+ *𝛽#
$
#%& 𝑋#!
(
1
)
where
𝑦!
denotes the corresponding observed number of public transit usage for ridesharing user i,
𝜃!
is the
12
expectation of public transit usage
𝑦!
,
#𝑖 = 1,2,…,𝑛.
𝑋#!
represents the
𝑝'(
explanatory variable for
13
ridesharing user i,
##𝛽"
and
#𝛽##
represent the corresponding intercept and the coefficient of the
14
𝑝'(#
explanatory variable, respectively.
15
16
The public transit usage
𝑦!
follows a Poisson-gamma (negative binomial) distribution. The probability of
17
public transit usage
𝑦!
could be formulated as
18
NB(𝑦!|𝜙,𝜃!) ≡ Pr#(𝑌!= 𝑦!|𝜙,𝜃!)=Γ(𝑦!+𝜙)
Γ(𝑦!+𝜙)Γ(𝜙)𝑦!>𝜃!
𝜃!+𝜙?)!>𝜙
𝜃!+𝜙?*
(
2
)
where
#𝜙
denotes the inverse dispersion parameter of negative binomial distribution. The log-likelihood
19
10
function (with weights
𝑤!
) is given by
1
𝐿𝐿
(
𝛽
)
=
*
𝑤!
+
!%&
{
−𝜃!+𝑦!ln
(
𝜃!
)
−ln(𝑦!!)
}
(
3
)
(2) HNBGLM
2
Level 1 model:
3
lnG𝜃!,H = 𝛽", +*𝛽#,
$
#%& 𝑋#!,
(
4
)
4
Level 2 model:
5
𝛽", = 𝛾"" + *𝛾"-
.
-%& 𝑍-, +𝜀",
(
5
)
where
𝜃!,
is the expectation of response variable
𝑦!,
, the subscript
𝑖
represents the
𝑖
-th individual while
𝑗
6
represents the
𝑗
-th census tract. For level 1 transit usage contributing factors,
𝑋#!,
donates the individual-
7
level variables such as the number of ridesharing trips for individual
𝑖
in census tract
𝑗
.
#𝛽#,
donates the
8
regression coefficient associated with
𝑋#!,
, and
𝜀",
is the random effect at the census tract level where
9
𝜀",∼
N (0,
𝜎/
0
). For level 2 transit usage contributing factors,
𝑍-,
donates the census tract-level variables
10
such as population density in census tract
𝑗
,
#𝛾"-
is a regression coefficient associated with
𝑍-,
, and
𝛽",
is
11
the intercept, which varies among census tracts.
12
13
(3) HNBGAM
14
Level 1 model:
15
logG𝜃!,H = 𝛽", +𝑓&G𝑅𝑆!,H +𝑓0G𝑅𝐴!,H +*𝛽#,
$
#%& 𝑋#!,
(
6
)
16
Level 2 model:
17
𝛽", = 𝛾"" + *𝛾"-
.
-%& 𝑍-, +𝜀",
(
7
)
where
𝑅𝑆!,
and
𝑅𝐴!,
are ridesharing use frequency and age for ridesharing user i in census tract
𝑗
,
18
respectively. The difference between HNBGLM and HNBGAM is that
𝑅𝑆!,
and
𝑅𝐴!,#
have been smoothed
19
by smooth functions
𝑓&
(
.
) and
𝑓0
(
.
), respectively. The introduced smooth functions are aimed to reflect a
20
more complex relationship between explanatory variables and
𝑌!,
. The smoothers called thin plate
21
regression splines are applied here for versatility (Woo d, 2 00 3, 2017). Hence, the smooth functions can be
22
written as linear combinations of basic functions that do not depend on the dependent variable, convenient
23
for prediction and estimation.
24
25
Unlike the NBGLM and HNBGLM, the HNBGAM could be estimated by the penalized maximum
26
likelihood. The penalized maximum likelihood function could be given by
27
𝐿𝐿(𝛽)= **𝑤!,
1
,%&
+"
!%& Y−𝜃!, +𝑦!,lnG𝜃!, H− ln(𝑦!, !)Z −1
2𝜆&\Y𝑓&
22G𝑤!,𝑅𝑆!,HZ0d𝑥
−1
2𝜆0\Y𝑓0
22(𝑤!,𝑅𝐴!,)Z0d𝑥
(8)
where
𝑛,
represents the number of individuals in the
𝑗
-th census tract, m represents the number of census
28
11
tracts.
1
2
The penalized maximum likelihood function in equation (8) includes the standard maximum likelihood
3
component and the penalized component. The terms
𝜆&
and
𝜆0
denote the smoothing parameters that
4
control the smoothness of the model. For more details about the penalized maximum likelihood function,
5
please refer to (T.J. Hastie, 199 0).
6
7
Finally, the NBGLM and HNBGLM are developed by the glmmTMB package of the programing language
8
R (Brooks et al., 2017); the HNBGAM is estimated by the gamm4 package of the programing language R
9
(Wood et al ., 2 017 ). Insignificant variables (p-value > 0.05) were excluded for the three models. For
10
example, the percentage of rental houses is insignificant at the 0.05 level for the HNBGLM, thus being
11
removed in the modeling process. Then, the multicollinear relationships among independent variables are
12
verified by the VIF test in Table 3. In general, multicollinear relationships would be acceptable if the VIF
13
values are lower than five. For the NBGLM, the worker density and house unit density are removed for the
14
high correlations with the other built environment factors.
15
16
Tabl e 3 Results of the VIF test
17
Independent variables
VIF
Ridesharing use frequency
1.09
Respondent age
1.27
Respondent race
1.27
Respondent gender
1.04
Highly educated respondent
1.38
Driver license
1.32
Number of household children
2.82
Household size
3.91
Low-level annual household income
1.51
Household vehicle ownership
1.95
Population density
1.73
Percentage of rental houses
1.75
18
5. Results and discussions
19
This section summarized the impacts of ridesharing use frequency on public transit usage. Table 4 presents
20
the estimated parameters for the three modeling approaches, involving the NBGLM, HNBHLM, and
21
HNBGAM. A low er AIC value for the HNBGAM in dicates a better predict ion performance than the other
22
two models because of the hierarchical structure and nonlinear effects. Significant random parameters of
23
the HNBGAM in this study supported the investigations above. In particular, the large values of the
24
effective degree of freedom (EDF) for ridesharing use frequency and respondent age indicate more complex
25
nonlinear effects rather than simple linear relationships with public transit usage. The significant EDF
26
values also validate the existence of the nonlinear effects, corresponding with the motivations of this study.
27
28
The modeling results revealed that a one-time increase in ridesharing use frequency would be associated
29
with a 3.05% increase and a 4.08% increase in public transit usage for the NBGLM and HNBGLM,
30
respectively. Of course, the fixed effect of ridesharing use frequency in the HNBGAM also suggested an
31
increasing trend in public transit usage with the increase of ridesharing use frequency. However, the random
32
effects of the smooth functions in the HNBGAM gave us different insights into the nonlinear relationships.
33
Figure 4 suggested a negligible impact of ridesharing use frequency on public transit usage for occasional
34
ridesharing use (from one to eleven times per month). In addition, complementary effects of ridesharing on
35
public transit were also shown for regular ridesharing use (from eleven to thirty-two times per month). The
36
complementary effects reached the peak point (32.12 times increase) at about twenty-two times ridesharing
37
use per month. In contrast, substitution effects of ridesharing use on public transit were observed for active
38
12
ridesharing use (more than thirty-two times per month).
1
2
In addition, respondent age was positively associated with public transit usage. NBGLM and HNBGLM
3
revealed the same 1.01% increase in public transit usage with a one-year rise for respondent age. However,
4
the nonlinear effects of respondent age on public transit for the HNBGAM were described in Figure 5.
5
Yo un g (aged below 30) and old respondents (aged above 55) would increase public transit usage by at most
6
49.18% and 200.42%, respectively. Middle-aged respondents (aged 30 to 55) would be negatively
7
associated with public transit usage, presenting a “v” curve shape. More specifically, middle-aged
8
respondents would reduce public transit usage by 22.89% at the highest negative point of 45 years old.
9
10
As for other socio-demographic factors, white Americans were positively associated with public transit
11
usage only for the HNBGAM, with a 29.69% increase in public transit usage. Such positive correlations
12
were consistent with previous studies by (Meredith-Karam et al., 2021; Yo u ng and Farber, 2019). In
13
addition, female respondents would use public transit less frequently than males, about 12.19% (NBGLM),
14
25.17% (HNBGLM), and 37.50% (HNBGAM), respectively. This is likely because women travelers make
15
more household-serving activities such as grocery shopping, which results in complicated travel patterns
16
that require more link trips. Thus, public transit was less attractive due to women's complicated trips for
17
household chores (Patterson et al., 2005). Highly educated respondents would be more likely to increase
18
public transit usage by 49.18% (NBGLM), 55.27% (HNBGLM), and 136.32% (HNBGAM). In contrast,
19
respondents with driver's licenses would reduce public transit usage by 78.13% (NBGLM), 71.35%
20
(HNBGLM), and 74.08% (HNBGAM). In addition, an increase of one child was related to a 42.88%
21
(NBGLM), 51.32% (HNBGLM), and 6% (HNBGAM) decrease in public transit usage, respectively. The
22
potential reason could be that households with children usually prefer to drive to meet more complex travel
23
patterns, including trips to work, school, and leisure activities (Brown et al., 2016; Hensher and Reyes,
24
2000). One more household member increase would increase public transit usage by 52.20% (NBGLM),
25
60.00% (HNBGLM), and 66.53% (HNBGAM). Further, a lower-level annual household income (less than
26
75,000 USD) would increase public transit usage by 34.30%% (NBGLM), 42.31% (HNBGLM), and 23.66%
27
(HNBGAM). Private vehicle ownership was negatively correlated with the public transit ridership
28
(Manville et al., 2018; Paulley et al., 2006), where owning one more household vehicle would reduce public
29
transit usage by 10.42% (NBGLM), 6.76% (HNBGLM), and 27.39% (NBGAM).
30
31
Built environment factors have been widely considered in transportation practices (Deka and Fei, 2019;
32
Ding and Cao, 2019; Lee et al., 2014; Sabouri et al., 2020b). For population density, one thousand more
33
residents increase per square mile would increase transit use frequency by 1.92% (NBGLM), 23.87%
34
(HNBGLM), and 7.68% (HNBGAM). Consistent with (Melia et al., 2018; Zhao, 2013), urbanization
35
contributed to more public transit usage where young adults in denser areas and larger settlements are more
36
likely to commute by public transit. In addition, a one-percent increase in rental houses would increase
37
transit use frequency by 136.32% for the NBGLM and 115.98% for the NBGAM.
38
39
Tabl e 4. Results of NBGLM, HNBGLM and HNBGAM
40
Variables
NBGLM
HNBGLM
HNBGAM
Coef.
S.E.
Coef.
S.E.
Coef.
S.E.
Fixed effect
Intercepts
0.24***
0.02
-4.03***
0.72
-0.39***
0.05
Key factor
Ridesharing use frequency
0.03***
<0.01
0.04***
<0.01
0.41***
0.13
Socio-demographic factors
Respondent age
0.01***
<0.01
0.01***
<0.01
0.13***
0.05
13
Respondent race
-
-
-
-
0.26***
0.02
Respondent gender
-0.13***
<0.01
-0.29***
0.01
-0.47***
0.01
Highly educated respondent
0.40***
0.01
0.44***
0.01
0.86***
0.01
Driver license
-1.52***
0.01
-1.25***
0.01
-1.35***
0.03
Number of household children
-0.56***
<0.01
-0.72***
0.01
-1.02***
0.01
Household size
0.42***
<0.01
0.47***
<0.01
0.51***
0.01
Low-level annual household
income
0.42***
0.01
0.55***
0.01
0.27***
0.02
Household vehicle ownership
-0.11***
<0.01
-0.07***
<0.01
-0.32***
0.01
Built environment factors
Population density
0.19***
<0.01
2.14***
0.67
0.74***
0.01
Percentage of rental houses
0.86***
0.02
-
-
0.77***
0.05
Random effects
sd (intercepts)
-
-
17.21***
4.15
0.73***
0.01
sds (ridesharing use frequency)
-
-
-
-
6.50***
0.13
sds (respondent age)
-
-
-
-
0.43***
0.06
Smooth terms
EDF
p-value
f(ridesharing use frequency)
-
-
-
-
6.00**
< 0.01
f(respondent age)
-
-
-
-
8.00**
< 0.01
Statistical performance
AIC
1,420,985
1,309,228
1,207,484
1
Notes:
2
1. Significance codes: * p<0.05, ** p<0.01, *** p<0.001.
3
2. sd (intercepts) shows the standard deviation of the coefficient for the intercept across census tracts.
4
3. sds (Z) describes the standard deviation of the coefficients forming the smoothing spline for the
5
variable Z’s fixed effect.
6
7
8
14
1
Figure 4. Nonlinear relationships between ridesharing and public transit usage
2
3
Figure 5. Nonlinear relationships between respondent age and public transit usage
4
5
6. Conclusions
6
This study explores the nonlinear effects between ridesharing use frequency and public transit usage. The
7
varied impacts of ridesharing on public transit usage across different ridesharing use frequencies are
8
validated by the effective smooth terms in the HNBGAM. The hierarchical structures between ridesharing
9
users and census tracts were also considered to understand the spatial dependence among ridesharing users
10
within the same census tract. The significant random intercept supports the existence of spatial dependence
11
15
in the HNBGAM. Compared with the NBGLM and HNBGLM, the development of the HNBGAM
1
simultaneously identifies the nonlinear effects and spatial dependence with a lower AIC value. In general,
2
a lower AIC value means a better model fit. Therefore, this study mainly used the HNBGAM to identify
3
the nonlinear effects of ridesharing use frequency on public transit using the 2017 NHTS data in San Diego.
4
This paper could help policymakers better understand the nonlinear effects of ridesharing services on the
5
public transit system, especially avoiding the over-used utilization of rideshare services. Such over-used or
6
redundant ridesharing services might hugely substitute public transit usage, increasing energy consumption
7
and environmental emissions.
8
9
Specifically, occasional ridesharing use (less than eleven times per month) had a negligible impact on public
10
transit usage. In addition, ridesharing services would complement public transit for regular ridesharing use
11
(more than eleven times per month but less than thirty-two times per month) and increase public transit
12
usage by at most 32.12 times. The possible reason could be attributed to the regular transit commuters in
13
San Diego, where ridesharing services could help tackle the first-and-last-mile issues of public transit and
14
thus improve the accessibility of the public transit (Contreras and Paz, 2018; Grahn et al., 2020; Hall et al.,
15
2018; Smith, 2016; Zhang and Zhang, 2018). However, active ridesharing use (more than thirty-two times
16
per month) would reduce public transit usage because more comfortable and convenient mobility services
17
provided by ridesharing would be more likely to replace public transit (Ke et al., 2021; Kong et al., 2020).
18
Overall, our findings show that the relationships between ridesharing and public transit usage were related
19
to ridesharing use frequency. The conclusions would emphasize the nonlinear effects between utilizations
20
of the two alternatives in the decision-making process. Regular ridesharing might improve public transit by
21
assisting passengers in their first-and-last-mile travel, whereas active ridesharing use would be associated
22
with more car-based trips. However, the ridesharing services tend to substitute public transit severely under
23
the context of the pandemic because of public fears of COVID-19 (Osorio et al., 2022; Qi et al., 2021).
24
During the peri-COVID period, how to recover public transit ridership and reduce reliance on car-based
25
modes would be critical challenges for government agencies and transportation practitioners. First, the
26
optimal schedule planning for public transit to set a reasonable headway is essential to reduce passenger
27
transfer time and missed connections. The holding strategies for public transit (such as schedule-based and
28
headway-based) should be applied more in practice according to the well-documented effect on improving
29
punctuality (Van de Kaa, 2010; Wa ng an d S un , 20 20 ). Second, demand-responsive feeder transit systems
30
such as demand-responsive connectors and customized buses would improve the accessibility of public
31
transit services like regular ridesharing use addressing the first-and-last mile issues (Ya n e t a l. , 2 0 19 ; Yan g
32
et al., 2021). Third, the number of transit transfers and travel time are key factors affecting the travel demand
33
of public transit services (Guo and Wilson, 2007). A locally well-designed public transit and on-demand
34
feeder transit systems would reduce the number of transfers, thus increasing the utilities of public transit
35
services. Improving the reliability and efficiency of travel time for public transit could also boost choice
36
preferences for public transit services (Ya ng a nd C he rr y, 2 01 7 ). The transit priority signal designs would
37
be helpful solutions to increase the reliability and efficiency of public transit services, mitigating the impacts
38
of other vehicles on transit vehicles.
39
40
Further, some built environment variables also played an essential role in public transit usage. Higher
41
population density and percentage of rental houses were positively associated with public transit usage,
42
consistent with previous research (Melia et al., 2018). Population density could proxy for local accessibility
43
to goods and mobility services (Holz-Rau et al., 2014). Traffic analysis zones with higher population density
44
and percentage of rental houses were correlated with more commuting travel demands and intensive public
45
transit configurations (e.g., number of public transit stations, transit lines, and infrastructure maintenance),
46
attracting more transit trips. On the one hand, planners might develop appropriate transportation
47
configurations such as transit station density and route layout to satisfy the commuting travel demand. Wel l -
48
designed transportation network helps increase coverage while reducing duplicate routes. Such measures
49
could attract more public transit trips shifted from ridesharing services, especially for ridesharing users with
50
low vehicle ownership. Transit network redesign is becoming one of the biggest trends in transit planning
51
16
right now, which is underway in many cities, such as Austin, Baltimore, Columbus, Dallas, Denver, Houston,
1
Indianapolis, and Jacksonville (Lee and Miller, 2018; Ziedan et al., 2021). On the other hand, “Mobility as
2
a Service (MaaS)” platform is proposed to coordinate multiple transportation modes into one integrated
3
platform. Travelers are expected to receive more convenient, efficient, and cost-effective “door-to-door”
4
transportation services. Currently, the MaaS platform is mainly based on public transit systems, such as
5
UbiGo (Sweden), NaviGoGo (Scotland), Jelbi (Germany), Whim (Finland), Beeline (Singapore). Actively
6
promoting the MaaS platform will help to enhance the attractiveness of transit services, especially in
7
densely populated metropolitan areas that face serious sustainability challenges such as traffic pollution and
8
congestion.
9
10
This study was also subject to some limitations. First, only limited built environment factors were applicable
11
to identify the heterogeneity among different cities. Commonly used built environment factors such as bus
12
stop numbers and road density are unavailable for this NHTS data. Therefore, only San Diego was
13
considered to understand the impacts of ridesharing on public transit usage despite a relatively small sample
14
size. Second, despite of slight difference between the derived and actual census tract identifiers, spatial
15
dependence at the census tract level indicates that the coincidence of values similarity for built environment
16
factors is associated with the location similarity (Anselin, 1988). One derived census tract identifier might
17
contain many actual census tract identifiers spatially closed. Of course, it would be a better way to get the
18
actual census tract identifiers with the agreement of the government agencies and avoid approximation
19
errors of built environment factors at the census tract level. Alternatively, the approach to deriving census
20
tract identifiers would be meaningful if the actual census tract identifiers are unavailable due to different
21
reasons. Such an approach to deriving census tract identifiers like model-based clustering algorithms in
22
previous studies (Ashik et al., 2022; Kumar et al., 2020; Liao and Scheuer, 2022; Su et al., 2022) might
23
capture spatial dependence and somewhat alleviate approximation errors of the built environment factors.
24
Third, besides ridesharing use frequency and respondent age, more smooth terms for other control variables
25
would be included to systematically examine the nonlinear effects on other alternatives in the future. This
26
paper could not evaluate the temporal trend of the ridesharing usage's impact on public transit due to data
27
unavailability. However, the NHTS survey duration was relatively short, and the temporal variations could
28
be neglected in this paper.
29
30
Acknowledgments
31
The contents of this paper present the views of the authors who are responsible for the facts and accuracy
32
of the data presented herein. The contents of the paper do not reflect the official views or policies of the
33
agencies. The work was partially funded by Transportation Informatics Lab, Department of Civil &
34
Environmental Engineering at Old Dominion University (ODU). The work described in this paper was also
35
supported by the National Natural Science Foundation of China (71861167001) and the National Science
36
Fund for Distinguished Young Scholars (72025104).
37
38
Declarations of interest
39
The authors reported no potential conflict of interest.
40
41
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