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Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice
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International Journal of Transportation Science and Technology journal homepage: www.elsevier.com/locate/ijtst 5 6
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Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice
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Xiaofeng Pan a,⇑, Zhi Zuo b
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a Department of the Built Environment, Urban Planning and Transportation Group, Eindhoven University of Technology, PO Box 513, 5600MB Eindhoven, the Netherlands b School of Architectural & Civil Engineering, Xinjiang University, 830047 Urumqi, PR China
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a r t i c l e
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Article history: Received 14 June 2019 Received in revised form 20 December 2019 Accepted 27 December 2019 Available online xxxx Keywords: Utility-difference threshold Bus service choice Stated preference experiment Heterogeneity
a b s t r a c t This paper provides an empirical case study involving the concept of utility-difference threshold in the context of bus service choice, which aims to explore the role of utilitydifference threshold in choice behavior. The underlying assumption is that consumers are unable to recognize a small utility difference between two alternatives, which may be either because they only have imperfect knowledge and limited perception ability or because they just simplify the choice task to save process efforts. Based on this assumption, a concept of utility-difference threshold is introduced to the model specification within which two alternatives are treated as the same. In this study, it is assumed that the utility-difference threshold could be a constant, a random term following a certain distribution or a function of some exogenous determinants, such as consumers’ sociodemographic characteristics. To test these assumptions, a stated choice experiment based on the rule of orthogonality was designed, in which two hypothetical kinds of bus service composing a choice set. A survey based on this design was carried out, in which 1784 valid observations were collected. The estimation results support the existence of utilitydifference threshold as well as its observed and unobserved heterogeneity. Ó 2020 Tongji University and Tongji University Press. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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1. Introduction
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It has been widely acknowledged that discrete choice modeling based on the rule of random utility maximization is a great tool to analyze consumer choice behavior, which traditionally postulates that when consumers decide to make a choice from a choice set, they first perceive utilities of alternatives based on observed (e.g., alternative-specific and contextual attributes) and unobserved (e.g., consumers’ mood when making decisions) factors and then choose the alternative with the highest perceived utility. Although such an assumption has become the dominant paradigm in consumer choice behavior analysis for many decades, studying how consumers really think when making a choice have never stopped. Therefore many relevant models based on various assumptions were proposed, among which some studies (e.g., Krishnan, 1977; Lioukas, 1984) argued that consumers may ignore or cannot recognize a small difference among alternatives – either because of their imperfect knowledge and limited perception ability or because they just want to simplify the choice task to save process
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Peer review under responsibility of Tongji University and Tongji University Press. ⇑ Corresponding author at: PO Box 513, VRT 8.23, 5600 MB Eindhoven, the Netherlands. E-mail address: [email protected] (X. Pan). https://doi.org/10.1016/j.ijtst.2019.12.001 2046-0430/Ó 2020 Tongji University and Tongji University Press. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: X. Pan and Z. Zuo, Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice, International Journal of Transportation Science and Technology, https://doi.org/10.1016/j.ijtst.2019.12.001
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efforts. In general, to take this assumption into consideration, a concept of ‘‘threshold” or ‘‘cut-off” or ‘‘boundary” (hereafter we refer to all of them as ‘‘threshold”) would be introduced which commonly indicates the insensitivity of consumers if the utility difference of two alternatives is smaller than the threshold. The threshold could be attribute-specific or alternative-specific. In the former, each attribute of alternatives could have a threshold and the overall utility of an alternative is the sum of each attribute-utility that is beyond the threshold (Cantillo and Ortúzar, 2005; Cantillo et al., 2006); in the latter, the overall utility of an alternative is derived based on the attribute levels of the alternative and two alternatives are treated as the same if the difference of their overall utilities is lower than a threshold (Cantillo et al., 2010). In addition, the threshold could be heterogeneous. On the one hand, the threshold could be various across population. In this sense, it is intuitive to assume that the threshold is random and also depends on consumers’ socio-demographic characteristics and some latent elements (e.g., knowledge level, perception ability and even personalities). On the other hand, the consumers may have different thresholds in terms of negative and positive utility differences, which leads to an asymmetric model framework. The consideration of threshold is important not only for academic researchers to interpret the choice behavior of consumers, but also for policy makers to propose and evaluate relevant policies. Policy makers are always interested in elasticities and willingness-to-pay associated with specific attributes, based on which to make policies to influence consumers’ behavior and ultimately generate economic benefits. The consideration of threshold tells that if the stimulus from a certain policy is too mild, consumers may not feel it at all, resulting in a totally failure of the policy. This could be a serious issue in stated preference experiment, a widely used approach for the analysis of willingness-to-pay and elasticities, in which the typical goal of experiment design is utility balance (Cantillo et al., 2010). Although the assumption of threshold has been proposed for decades, most studies involving the concept of threshold usually assume that consumers would generate utilities if the absolute levels of specific attributes of an alternative exceed the thresholds (e.g., Swait, 2001; Danielis and Marcucci, 2007; Deng et al., 2019). However, in this current study, the concept of threshold has a slight different meaning as it is assumed that consumers could only recognize the difference if the difference of utilities derived from (a certain attribute of) two alternatives exceeds the threshold. Until recently, there are some studies tried to make contributions to the concept of utility-difference threshold from the theoretical perspective or tried to take this concept into account to model consumers’ specific choice behavior. Obermeyer et al. (2015) proposed a flexible transformation functions which could easily capture the utility indifference and be applied in estimation software. Meanwhile, this study concluded that the value of travel time savings could be biased if the utility-difference threshold was ignored using a simulated data set. Bahamonde-Birke et al. (2017) extended the model specification in Cantillo et al. (2010), which focused on binary choice cases, into multiple choice cases by adding an extra random term to the utilities of alternatives that consumers perceived indifferent. Jang et al. (2018) introduced this assumption into the random regret-based models and also differentiated the potential effects of negative and positive indifferences. Wang et al. (2016) offered numerical experiments about a bi-modal equilibrium model in which utility indifference was taken into accounted. Specifically, this study focused on the effects of utility-difference thresholds on transit fare and frequency schemes. This current study is inspired by Cantillo et al. (2010) and provides an empirical case study in the context of bus service choice to explore the role of utility-difference threshold. To this end, a stated preference experiment design was carried out to collect data and respondents’ choices were modeled using the model with utility-difference threshold. Specifically, we assume not only the existence of utility-difference threshold in consumers’ choice behavior but also the existence of heterogeneity property of utility-difference threshold, which could be a random concept and a function of consumers’ sociodemographic characteristics. The remainder of the paper is structured as follows. Section 2 presents the details of the model framework proposed in Cantillo et al. (2010). Section 3 describes the details of the experiment design, data collection and data pre-process. Section 4 first compares the estimation results of the model with utility-difference threshold to those from the model with an quasiopt-out option, then shows the final estimation results, in which utility-difference threshold is assumed to be random and related to respondents’ socio-demographic characteristics, and finally discusses marginal change, the direct and cross elasticities associated with specific attributes. Section 5 finalizes the paper with some conclusions.
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2. Model with utility-difference threshold
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We consider a binary choice case. Assume a choice set contains two alternatives labeled as 1 and 2. Following the random utility theory, the utilities of these alternatives contain a systematic part derived from alternative-specific attributes and a random part indicating unobserved elements, respectively. Assume the systematic utility functions are linear-additive, then utilities of these two alternatives for a certain consumer n can be specified as the following, respectively:
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U nt1 ¼ V nt1 þ nt1 ¼ U nt2 ¼ V nt2 þ nt2 ¼
X
bx k k nt1k
X
bx k k nt2k
þ nt1
ð1Þ
þ nt2
ð2Þ
where U nt1 and U nt2 denote the overall utilities that the respondent n perceived for alternative 1 and 2 in choice task t, respectively; V nt1 and V nt2 denote the systematic utilities of the corresponding alternatives, respectively; nt1 and nt2 denote Please cite this article as: X. Pan and Z. Zuo, Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice, International Journal of Transportation Science and Technology, https://doi.org/10.1016/j.ijtst.2019.12.001
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the random utilities; xnt1k and xnt2k denote the kth attribute towards alternative 1 and 2 in choice task t, respectively; bk is the parameter of kth alternative-specific attribute with a generic value. Based on the conventional random utility maximization assumption of discrete choice models, the probability of choosing each alternative is:
Pnt1 ¼ PfU nt1 U nt2 g ¼ P fnt2 nt1 V nt1 V nt2 g
ð3Þ
Pnt2 ¼ PfU nt2 U nt1 g ¼ P fnt2 nt1 V nt1 V nt2 g
ð4Þ
However, with the assumption that consumers cannot recognize a small different between two alternatives, the concept of threshold is introduced – if the utility difference is lower than the threshold, then these two alternatives are treated as the same. Therefore, the probability of choosing each alternative is:
Pnt1 ¼ PfU nt1 U nt2 þ gg ¼ P fnt2 nt1 V nt1 ðV nt2 þ gÞg
ð5Þ
Pnt2 ¼ PfU nt2 U nt1 þ gg ¼ P fnt2 nt1 ðV nt1 þ gÞ V nt2 g
ð6Þ
If we assume the random terms in the utility functions of the two alternatives nt1 and nt2 distribute iid Gumbel, then the explicit expressions of the choice probabilities of these two alternatives could be given as the following, respectively:
Pnt1 ¼
exp ðV nt1 Þ exp ðV nt1 Þ þ exp ðV nt2 þ gÞ
ð7Þ
Pnt2 ¼
exp ðV nt2 Þ exp ðV nt1 þ gÞ þ exp ðV nt2 Þ
ð8Þ
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Therefore, the probability that the consumer n perceives no difference between these two alternatives, denoted as PnI , is specified as:
PntI ¼ PfjU nt1 U nt2 j gg ¼ 1 Pnt1 P nt2
ð9Þ
Here g in Eqs. (5)–(9) is the threshold parameter, which should be non-negative. If g ¼ 0, the model is reduced to the conventional logit model, which means consumers can recognize any difference between alternatives. Moreover, to capturing the heterogeneity property of utility-difference threshold, g could be a function of consumers’ socio-demographic characteristics and also randomly distributed, denoted as gn :
gn ¼ f ðznl ; eÞ
ð10Þ
where znl denotes the consumer n’s lth socio-demographic variable and e is a random term which follows a certain distribution. It is worthwhile noting that here a generic utility-difference threshold for negative and positive utility differences was applied, however, it is easily to expand it to an asymmetric thresholds case. Maximum likelihood method is applied for model estimation. Since the likelihood function involves a random distributed term, which needs integration and inevitably leads to a non-closed form, simulation should be applied (Train, 2009). Therefore, the simulated log-likelihood function for the model with utility-difference threshold could be give as the following:
SLL ¼
1 X XR ln Pnt1jenr yt1 Pnt2jenr yt2 PntIjenr ytI n r¼1 R
ð11Þ
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where SLL denotes the simulated log-likelihood, R denotes the number of random draws, enr is a draw from a certain distribution which is respondent-specific but stays constant in different choice tasks for a single respondent to reflect the fact that the choices from a single respondent are likely not independent, and yt1 , yt1 and ytI are dummy variables indicating whether the consumer n chooses the corresponding alternative in choice task t.
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3. Data collection
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The data set we used in this study is from a stated preference experiment about bus service satisfaction. In detail, respondents were required to evaluate a new bus service comparing the original one – both services were hypothetic and generated from the experiment design. A 5-degree Likert scale (‘‘much worse”, ‘‘worse”, ‘‘almost the same”, ‘‘better” and ‘‘much better”) was used. Indeed, the experiment design was originally designed for an ordered choice study, however, given the 5-degree Likert scale used in the experiment, it still can be applied in the current study after a data transformation. Specifically, those reporting ‘‘much worse” and ‘‘worse” could be seen as preferring the original bus service; those reporting ‘‘better” and ‘‘much better” could be seen as preferring the new service; and those reporting ‘‘almost the same” does not need any transformation. As the respondents were clearly reminded the exact meaning of each scale, we believe this data transformation would not create estimation bias. The attributes of bus services used in the experiment including their definitions and corresponding levels are shown in Table 1. Specifically, we separated the whole travel time into several parts: in-vehicle time, wait time, walk time and transfer time. Another attribute called ‘‘standing duration” is adopted indicating whether seat is available and if not how long he/she
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Please cite this article as: X. Pan and Z. Zuo, Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice, International Journal of Transportation Science and Technology, https://doi.org/10.1016/j.ijtst.2019.12.001
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Table 1 Attribute and corresponding levels of bus services. Attributes
Definitions
Levels
In-vehicle time Wait time Walk time Standing duration Ticket price Transfer time
time in a bus (min) time waiting for a bus, excluding the transfer time (min) time walking out of a bus, excluding the transfer time (min) ratio of standing time over in-vehicle time in a bus price of the ticket (¥) time for transfer (min)
35, 45, 55, 65 2, 6, 10, 14 2, 6, 10, 14 0, 0.33, 0.67, 1 0.5, 1, 1.5, 2 0, 3, 6, 9
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will stand on the bus averagely – this is an index which can measure seat availability. Since the standing time cannot exceed in-vehicle time, we used their ratio to represent standing duration. Finally, we use the software package Ngene (ChoiceMetrics, 2014) for an orthogonal fractional factorial design, based on which 32 profiles (i.e., choice sets) with 8 blocks were obtained. The survey was carried out in Dalian, China, in December 2016. 480 questionnaires were dispensed online as well as onsite. After deleting the observations with inconsistent and incomplete answers, 1784 observations from 446 respondents were deemed valid and used in the model estimation. The descriptive statistics of the sample in terms of their sociodemographic characteristics are shown in Table 2. From the summary of sample characteristics, we can see that the sample are almost uniform distributed in terms of gender, marital status. Meanwhile, most of them are mid-aged (accounting for 54.71%) and at least received a bachelor degree (accounting for 90.58%), and have low or middle income (accounting for 75.79%) and no driving experience (accounting for 43.05%). Moreover, 845 observations (accounting for 47.37%) choose the new bus service and 455 (accounting for 25.50%) choose the original one, and there are 484 observations (accounting for 27.13%) choose the ‘‘almost the same” option.
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4. Model estimation
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4.1. Model with utility-difference threshold vs. model with a quasi-opt-out option
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In many cases, respondents are provided an opt-out option, which indicates the dissatisfaction for any alternatives. Usually, in such cases, the utility of the opt-out option is treated as a constant. Although in this current study the ‘‘almost the same” option cannot be seen as an opt-out option, treating the utility of the ‘‘almost the same” option as a constant is still an
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Table 2 Descriptive statistics of the sample. Socio-demographic attribute
Level
Percentage
Gender
Male Female
45.07% 54.93%
Marital Status
Single Married
56.73% 43.27%
Age
20 21–30 31–40 41–50 51–65 66
13.68% 54.70% 22.20% 5.61% 2.91% 0.90%
Driving Experience (Year)
0 (0, 1] (1, 3] (3, 5] (5, 10] >10
43.05% 15.92% 19.06% 12.11% 6.95% 2.91%
Income (Yuan/Month)
3499 3500–6999 7000–9999 10,000–14,999 15,000–19,999 20,000
42.83% 32.96% 15.47% 6.05% 2.02% 0.67%
Education Level
High school or lower Bachelor Master/PhD
9.42% 76.23% 14.35%
Please cite this article as: X. Pan and Z. Zuo, Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice, International Journal of Transportation Science and Technology, https://doi.org/10.1016/j.ijtst.2019.12.001
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intuitive way for estimation – we label it as ‘‘quasi-opt-out” option. Therefore, in this sub-section, we compare the estimation results from the model with utility-difference threshold and the model with quasi-opt-out option. Both models were estimated based on maximum likelihood method using R package ‘‘maxLik” (Henningsen and Toomet, 2011). All attributes of the bus services entered the models continuously. The utility-difference threshold was assumed to be constant and none of the socio-demographic characteristics were taken into consideration. In addition, in the pretest results we found that the estimated values of taste parameters towards in-vehicle time, wait time, walk time and transfer time were too small, therefore we finally decided to shift their unit from ‘‘minute” to ‘‘hour”. The estimation results are presented in Table 3, which denotes that the utility-difference threshold is significant. Moreover, in terms of the taste parameters, results from both models show that all alternative-specific attributes are significant and have expected signs. However, the taste parameters from these two models are slightly different, especially for those with respect to wait time and walk time. Besides, various indexes of goodness of fit are computed, including AIC, BIC, adjusted Rho-squared. All of the indexes confirm that the model with utility-difference threshold outperforms the model with quasi-opt-out option.
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4.2. Heterogeneity of utility-difference threshold
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The estimation results in Table 3 already show the existence of utility-difference threshold. In this sub-section, we further assume that the threshold varies across population. Specifically, we assume that the threshold is random and also related to consumers’ socio-demographic characteristics. Given that the utility-difference threshold is non-negative, log-normal distribution was firstly applied to capture its randomness. However, the log-normal distribution has a long, thick tail, which may generate an implausibly large threshold value. Therefore, symmetric triangular distribution was also applied (hereafter we refer to these two models as LN-threshold model and TRI-threshold model, respectively). In detail, the threshold was specified as:
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gn ¼ exp gn þ e ; e Nð0; 1Þ
ð12Þ
gn ¼ gn þ gn e; e triangleð1; 1Þ where:
gn ¼ g0 þ
X l
ð13Þ
al znl
ð14Þ
Here znl denotes the consumer n’s lth socio-demographic characteristic and al denotes the impact of lth socio-demographic characteristic to the threshold and g0 is a constant. Therefore, if all socio-demographic characteristics are effects coded, the constant g0 represents the median for log-normal distribution and the median/mode/mean for symmetric triangular distribution. Halton sequence for the random utility-difference threshold was used for simulation. In addition, to investigate the influence of different numbers of Halton draws on estimation results, the model was estimated several times based on different
Table 3 Estimation results of models with utility-difference threshold and quasi-opt-out option.
Alternative-specific attributes In-vehicle time (h) Wait time (h) Walk time (h) Standing duration Ticket price (¥) Transfer time (h) Constant (new bus service) Constant (original bus service) Constant (quasi-opt-out-option) Utility-difference threshold
Model with utility-difference threshold
Model with quasi-opt-out option
coef.
std. err.
p-Value
coef.
std. err.
p-Value
2.5656 3.2485 2.4560 0.7720 0.3411 3.0519 0.5382
0.1875 0.4472 0.4415 0.0923 0.0603 0.5956 0.0477
0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000***
2.5801 2.5671 2.0559 0.7186 0.3882 2.9710 4.3767 3.6754
0.2118 0.5049 0.5081 0.1009 0.0695 0.6851 0.2566 0.2491
0.0000*** 0.0000*** 0.0001*** 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000***
0.7086
0.0294
# of estimated parameters (N) Sample size Initial log-likelihood Convergent log-likelihood AIC (AIC/N) BIC (BIC/N) Rho-squared Adjusted Rho-squared
0.0000*** 8 1784 1959.924 1685.341 3386.683 (423.335) 3430.576 (428.822) 0.1401 0.1360
8 1784 1959.924 1741.963 3499.926 (437.491) 3543.819 (442.977) 0.1112 0.1071
***p-value <0.01; **p-value <0.05; *p-value <0.1.
Please cite this article as: X. Pan and Z. Zuo, Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice, International Journal of Transportation Science and Technology, https://doi.org/10.1016/j.ijtst.2019.12.001
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numbers of draws, from 100 to 1000 with a step size 100. The final convergent log-likelihoods are shown in Fig. 1. Fig. 1 shows that the largest difference of final log-likelihood is less than 0.9 for the LN-threshold model and less than 0.4 for the TRI-threshold model, and the final convergent log-likelihoods for both models tend to be stable with the increasing of the number of Halton draws. Therefore, the estimation results with 1000 Halton draws for both models are reported in this sub-section. The model was estimated based on simulated maximum likelihood method using R package ‘‘maxLik” (Henningsen and Toomet, 2011). All attributes of the bus services entered the models as continuous values while the socio-demographic characteristics entering the models were effects coded. Table 4 shows the estimation results with 1000 Halton draws. The adjusted Rho-squared is 0.1378 and 0.1371 for LN-threshold and TRI-threshold models, respectively, which both indicate acceptable model performance (McFadden, 1979; Louviere et al., 2000). Since these two models are not nested, the adjusted likelihood ratio test (Ben-Akiva and Swait, 1986) was applied to compare their performance statistically. The probability that the TRI-threshold model being preferred is:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P q2ln q2tri > z / 2 z LLð0Þ þ ðK tri K ln Þ
ð15Þ
where q2ln and q2tri are adjusted Rho-squared for LN-threshold and TRI-threshold models, respectively; / is the standard normal cumulative density function; LLð0Þ is the initial log-likelihood; K tri and K ln are the numbers of parameters for the two models, respectively. Finally, it turns that the probability is around 0.0271. Together with AIC and BIC indexes, we conclude that the LN-threshold model is just slightly better than the TRI-threshold model. Therefore, in the following we present and discuss the results from both models. Another issue should be noted is that some literature (e.g., Hensher and Greene, 2003; Hensher et al., 2015) argued that although log-normal distribution can constrain the sign of a random parameter or component, unreasonable or biased results may be obtained due to its very long tail – in these cases, triangle distribution is usually suggested. In this study, the emphasis is not on the comparison between different distributions, but we think the readers should realize the difference as we will show in the following. In terms of the taste parameters, all alternative-specific attributes are significant and have expected signs. Results from the LN-threshold model has lower values than those from the TRI-threshold model. If we calculate the willingness-to-pay, we will find the TRI-threshold model would generate higher values of willingness-to-pay. Additionally, we included an alternative-specific constant in the utility function of the new bus service. This is principally not necessary for unlabeled choice context. However, considering the choice context in our experiment, which is not exactly an unlabeled choice originally, we believe that there might be bias if an alternative-specific constant is not taken into account. Besides, Hensher et al. (2015) also suggests to include this constant at the beginning even in unlabeled choice context and later removed it if it is found to be statistically insignificant. The estimation results from both models confirm a significant alternative-specific constant. In terms of the utility-difference threshold, the estimation results from both models confirm its observed and unobserved heterogeneity, which support our assumption that the utility-difference threshold varies across population. To present parsimonious models, the insignificant variables are removed. The 95% (average) interval of threshold ranges from 0.0082 to 14.0460 for LN-threshold model and from 0.1658 to 1.3174 for TRI-threshold model, which means a change of unit in terms of the attributes only has influence on a part of the population. Based on the results of the LN-threshold model, it tells that averagely one-minute change in in-vehicle time, wait time, walk time or transfer time would make the overall utility increased/decreased by 0.0526, 0.0612, 0.0470, or 0.0661, respectively, which implies only 16.33%, 18.37%, 14.91%, or 19.47% of the population could feel the change; a ¥1 change in ticket price would make the overall utility increased/decreased by 0.4709, implying 56.86% of the population could feel the change; a unit change in standing duration (e.g., from standing for the entire trip to sit for the entire trip or the opposite) would make the overall utility increased/decreased by 0.9157–69.94% of the population could feel the change. Based on the results of TRI-threshold model, it tells that averagely for
Fig. 1. Convergent log-likelihoods based on various numbers of Halton draws.
Please cite this article as: X. Pan and Z. Zuo, Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice, International Journal of Transportation Science and Technology, https://doi.org/10.1016/j.ijtst.2019.12.001
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X. Pan, Z. Zuo / International Journal of Transportation Science and Technology xxx (xxxx) xxx Table 4 Estimation results of the utility-difference threshold model with 1000 Halton draws. Model with log-normal dist.
Alternative-specific attributes In-vehicle time (h) Wait time (h) Walk time (h) Standing duration Ticket price (¥) Transfer time (h) Constant (new bus service) Utility-difference threshold Constant Std. deviation Education High school or below Master/PhD Bachelor
Model with triangular dist.
coef.
std. err.
p-Value
coef.
std. err.
p-Value
3.1533 3.6744 2.8207 0.9157 0.4709 3.9650 0.6801
0.3071 0.5574 0.5552 0.1228 0.0896 0.7971 0.0781
0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000***
2.6108 3.3143 2.4964 0.7872 0.3488 3.0909 0.5501
0.1914 0.4561 0.4505 0.0939 0.0615 0.6072 0.0488
0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000***
1.0812 1.8998
0.4180 0.7585
0.0097*** 0.0123**
0.7416
0.0440
0.0000***
0.3324 0.4170 0.0846
0.1920 0.1161
0.0299** 0.4663
0.1373 0.1793 0.0420
0.0659 0.0468
0.0065*** 0.3698
# of estimated parameters (N) Sample size Initial log-likelihood Convergent log-likelihood AIC (AIC/N) BIC (BIC/N) Rho-squared Adjusted Rho-squared
11 1784 1959.924 1678.819 3379.638 (307.2398) 3439.991 (312.7265) 0.1434 0.1378
10 1784 1959.924 1681.159 3382.317 (338.2317) 3437.184 (343.7184) 0.1422 0.1371
***p-value <0.01; **p-value <0.05; *p-value < 0.1.
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one-minute change in in-vehicle time, wait time, walk time, or transfer time, only 0.17%, 0.28%, 0.16%, or 0.24% of the population could feel the change; 11.06% of the population could feel a ¥1 change in ticket price; 55.96% of the population could feel a unit change in standing duration (e.g., from standing for the entire trip to sit for the entire trip or the opposite). Some conclusions can be made here. First of all, when calculate the willingness-to-pay, the utility-difference threshold must be taken into consideration – otherwise the results may be largely overestimated. Second, some attributes have limited improvement space, such as standing duration. Even the bus service in these attributes has got the best performance, there are still a part of population do not care about the changes. In this sense, policy makers should not pay much attention to such attributes. The estimation results also show that this heterogeneity property of utility-difference threshold is less related to consumers’ socio-demographic characteristics. According to our results, only consumers’ education level has significant (at the 95% probability level for the LN-threshold model and at the 99% probability level for the TRI-threshold model) influence on the effect of utility-difference threshold. In detail, consumers with a higher education level has a higher utility-difference threshold. A plausible reason is that consumers with a higher education level is more likely to simplify the choice tasks to save process efforts. Note that the sample used in this study cannot reflect the real distribution of population in Dalian. Although the estimated coefficients of the threshold may be biased from those for the whole population, this study already confirmed the observed and unobserved heterogeneity of utility-difference threshold based on the specific data set. Note the main objective of this study is to explore the role of utility-difference threshold rather than measure the effect of the threshold in population. Policy makers may be also interested in the elasticities associated to specific attributes. Therefore we also compute the average direct and cross elasticities, using the estimation results of LN-threshold and TRI-threshold models, when the performance of attributes of the new bus service are deteriorated by 1%. The results are presented in Table 5, which indicate that the loss of market share of the new bus service would not entirely goes to the original one – some of the sample cannot feel the changes therefore would in fact stick to the one that they preferred originally. In terms of elasticity with respect to a specific attributes, Table 5 indicates that deterioration in in-vehicle time is the severest, followed by ticket price. The elasticities based on different models are slightly different, which again confirms the importance of the assumption of the distribution of utility-difference threshold.
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5. Conclusions
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Due to various reasons, consumers may not recognize or does not care about a small difference among alternatives. The previous studies introduced the concept of threshold to indicate insensitivity of consumers if the difference of alternatives is smaller than the threshold. The threshold could be attribute-specific or alternative-specific. In addition, there may be heterogeneous threshold among population. This assumption is so importation that its ignorance may cause bias when calculate
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Table 5 Elasticities when attributes of the new bus services deteriorated by 1% (unit: %). New bus service
Original bus service
The almost same option
LN-threshold model In-vehicle time Wait time Walk time Standing duration Ticket price Transfer time
0.426 0.077 0.063 0.075 0.099 0.047
0.382 0.070 0.055 0.067 0.087 0.042
0.043 0.007 0.008 0.008 0.013 0.006
TRI-threshold model In-vehicle time Wait time Walk time Standing duration Ticket price Transfer time
0.430 0.085 0.068 0.078 0.090 0.046
0.368 0.074 0.056 0.067 0.073 0.038
0.062 0.011 0.011 0.011 0.017 0.008
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willingness-to-pay and elasticity associated with specific attributes. This study provides an empirical case study of utilitydifference threshold model proposed in Cantillo et al. (2010) in the context of bus service choice to explore the role of utilitydifference threshold. The stated preference experiment was applied to collect data and models were estimated using maximum likelihood method. The model with utility-difference threshold was compared with the one with quasi-opt-out option. Further, the utilitydifference threshold was assumed to be log-normally or symmetrically & triangularly distributed and also a function of consumers’ socio-demographic characteristics. Results provided acceptable goodness of fit and confirmed the existence and heterogeneity property of the utility-difference threshold. Moreover, results showed that the LN-threshold model was only slightly better than the TRI-threshold model while the estimated coefficients of the model were quite different both for taste parameters and utility-difference threshold parameters. Therefore, one should carefully determine the distribution of the threshold. In addition, results also showed that the utility-difference threshold was related to consumers’ education level that a higher education level would lead to a higher utility-difference threshold. We also analyzed consumers’ sensitivity to marginal change in terms of each alternative-specific attribute. Since the existence of utility-difference threshold, a marginal change in a certain attribute cannot trigger full awareness of the population. This sensitivity varies across attributes – to some attribute, due to its limited improvement space, even an alternative has already improved to its best performance, there are still a part of population choose to ignore this change. In addition, we also computed the direct and cross elasticities associated with all alternative-specific attributes, which also confirmed the importance of utility-difference threshold. This study also provides policy implications to the policy makers. It reminds the policy makers to make a single policy with a big change in some cases. For instance, if the transportation department is going to make a policy to release the road congestion, the results should be significant so that the most of the travelers could feel the improvements. However, in other cases it reminds the policy makers to not make a single policy with a big change. For instance, if the transit company is going to increase the ticker fare from ¥1 to ¥2, it is better to do it in several steps, such as increasing ¥0.5 in the first year and then increasing ¥0.5 in the second year, to avoid severe responses from travelers. There are still many things to do towards this topic. Here we just list a few. First of all, empirical applications in different choice context are always needed, both from stated and revealed preference data sets. Second, extension of the model to multiple choice context could make the model more attractive and universal. This extension may make the model choiceset dependent. In this sense, the random regret model (Chorus et al., 2008) and the relative utility model (Zhang et al., 2004) may could be served as references. A challenge on this topic is that a multiple choice case makes consumers’ choice behavior much more complicated as they have to compare each pair of alternatives in the choice set. Third, deep investigation of elements that could influence the utility-difference threshold is also needed. Our results show consumers’ sociodemographic characteristics are less related, which leads us to think about other elements, such as consumers’ knowledge level, perception ability, even personality and behavior habits. This may suffer from the difficulty of data collection but it seems promising.
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Conflict of Interest
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Acknowledgements and declarations
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The authors thank the anonymous reviewers for their comments, which help to improve this paper. On behalf of all authors, the corresponding author states that there is no conflict of interest.
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Contents lists available at ScienceDirect
International Journal of Transportation Science and Technology journal homepage: www.elsevier.com/locate/ijtst 5 6
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Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice
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Xiaofeng Pan a,⇑, Zhi Zuo b
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a Department of the Built Environment, Urban Planning and Transportation Group, Eindhoven University of Technology, PO Box 513, 5600MB Eindhoven, the Netherlands b School of Architectural & Civil Engineering, Xinjiang University, 830047 Urumqi, PR China
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a r t i c l e
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Article history: Received 14 June 2019 Received in revised form 20 December 2019 Accepted 27 December 2019 Available online xxxx Keywords: Utility-difference threshold Bus service choice Stated preference experiment Heterogeneity
a b s t r a c t This paper provides an empirical case study involving the concept of utility-difference threshold in the context of bus service choice, which aims to explore the role of utilitydifference threshold in choice behavior. The underlying assumption is that consumers are unable to recognize a small utility difference between two alternatives, which may be either because they only have imperfect knowledge and limited perception ability or because they just simplify the choice task to save process efforts. Based on this assumption, a concept of utility-difference threshold is introduced to the model specification within which two alternatives are treated as the same. In this study, it is assumed that the utility-difference threshold could be a constant, a random term following a certain distribution or a function of some exogenous determinants, such as consumers’ sociodemographic characteristics. To test these assumptions, a stated choice experiment based on the rule of orthogonality was designed, in which two hypothetical kinds of bus service composing a choice set. A survey based on this design was carried out, in which 1784 valid observations were collected. The estimation results support the existence of utilitydifference threshold as well as its observed and unobserved heterogeneity. Ó 2020 Tongji University and Tongji University Press. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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1. Introduction
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It has been widely acknowledged that discrete choice modeling based on the rule of random utility maximization is a great tool to analyze consumer choice behavior, which traditionally postulates that when consumers decide to make a choice from a choice set, they first perceive utilities of alternatives based on observed (e.g., alternative-specific and contextual attributes) and unobserved (e.g., consumers’ mood when making decisions) factors and then choose the alternative with the highest perceived utility. Although such an assumption has become the dominant paradigm in consumer choice behavior analysis for many decades, studying how consumers really think when making a choice have never stopped. Therefore many relevant models based on various assumptions were proposed, among which some studies (e.g., Krishnan, 1977; Lioukas, 1984) argued that consumers may ignore or cannot recognize a small difference among alternatives – either because of their imperfect knowledge and limited perception ability or because they just want to simplify the choice task to save process
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Peer review under responsibility of Tongji University and Tongji University Press. ⇑ Corresponding author at: PO Box 513, VRT 8.23, 5600 MB Eindhoven, the Netherlands. E-mail address: [email protected] (X. Pan). https://doi.org/10.1016/j.ijtst.2019.12.001 2046-0430/Ó 2020 Tongji University and Tongji University Press. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: X. Pan and Z. Zuo, Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice, International Journal of Transportation Science and Technology, https://doi.org/10.1016/j.ijtst.2019.12.001
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efforts. In general, to take this assumption into consideration, a concept of ‘‘threshold” or ‘‘cut-off” or ‘‘boundary” (hereafter we refer to all of them as ‘‘threshold”) would be introduced which commonly indicates the insensitivity of consumers if the utility difference of two alternatives is smaller than the threshold. The threshold could be attribute-specific or alternative-specific. In the former, each attribute of alternatives could have a threshold and the overall utility of an alternative is the sum of each attribute-utility that is beyond the threshold (Cantillo and Ortúzar, 2005; Cantillo et al., 2006); in the latter, the overall utility of an alternative is derived based on the attribute levels of the alternative and two alternatives are treated as the same if the difference of their overall utilities is lower than a threshold (Cantillo et al., 2010). In addition, the threshold could be heterogeneous. On the one hand, the threshold could be various across population. In this sense, it is intuitive to assume that the threshold is random and also depends on consumers’ socio-demographic characteristics and some latent elements (e.g., knowledge level, perception ability and even personalities). On the other hand, the consumers may have different thresholds in terms of negative and positive utility differences, which leads to an asymmetric model framework. The consideration of threshold is important not only for academic researchers to interpret the choice behavior of consumers, but also for policy makers to propose and evaluate relevant policies. Policy makers are always interested in elasticities and willingness-to-pay associated with specific attributes, based on which to make policies to influence consumers’ behavior and ultimately generate economic benefits. The consideration of threshold tells that if the stimulus from a certain policy is too mild, consumers may not feel it at all, resulting in a totally failure of the policy. This could be a serious issue in stated preference experiment, a widely used approach for the analysis of willingness-to-pay and elasticities, in which the typical goal of experiment design is utility balance (Cantillo et al., 2010). Although the assumption of threshold has been proposed for decades, most studies involving the concept of threshold usually assume that consumers would generate utilities if the absolute levels of specific attributes of an alternative exceed the thresholds (e.g., Swait, 2001; Danielis and Marcucci, 2007; Deng et al., 2019). However, in this current study, the concept of threshold has a slight different meaning as it is assumed that consumers could only recognize the difference if the difference of utilities derived from (a certain attribute of) two alternatives exceeds the threshold. Until recently, there are some studies tried to make contributions to the concept of utility-difference threshold from the theoretical perspective or tried to take this concept into account to model consumers’ specific choice behavior. Obermeyer et al. (2015) proposed a flexible transformation functions which could easily capture the utility indifference and be applied in estimation software. Meanwhile, this study concluded that the value of travel time savings could be biased if the utility-difference threshold was ignored using a simulated data set. Bahamonde-Birke et al. (2017) extended the model specification in Cantillo et al. (2010), which focused on binary choice cases, into multiple choice cases by adding an extra random term to the utilities of alternatives that consumers perceived indifferent. Jang et al. (2018) introduced this assumption into the random regret-based models and also differentiated the potential effects of negative and positive indifferences. Wang et al. (2016) offered numerical experiments about a bi-modal equilibrium model in which utility indifference was taken into accounted. Specifically, this study focused on the effects of utility-difference thresholds on transit fare and frequency schemes. This current study is inspired by Cantillo et al. (2010) and provides an empirical case study in the context of bus service choice to explore the role of utility-difference threshold. To this end, a stated preference experiment design was carried out to collect data and respondents’ choices were modeled using the model with utility-difference threshold. Specifically, we assume not only the existence of utility-difference threshold in consumers’ choice behavior but also the existence of heterogeneity property of utility-difference threshold, which could be a random concept and a function of consumers’ sociodemographic characteristics. The remainder of the paper is structured as follows. Section 2 presents the details of the model framework proposed in Cantillo et al. (2010). Section 3 describes the details of the experiment design, data collection and data pre-process. Section 4 first compares the estimation results of the model with utility-difference threshold to those from the model with an quasiopt-out option, then shows the final estimation results, in which utility-difference threshold is assumed to be random and related to respondents’ socio-demographic characteristics, and finally discusses marginal change, the direct and cross elasticities associated with specific attributes. Section 5 finalizes the paper with some conclusions.
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2. Model with utility-difference threshold
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We consider a binary choice case. Assume a choice set contains two alternatives labeled as 1 and 2. Following the random utility theory, the utilities of these alternatives contain a systematic part derived from alternative-specific attributes and a random part indicating unobserved elements, respectively. Assume the systematic utility functions are linear-additive, then utilities of these two alternatives for a certain consumer n can be specified as the following, respectively:
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U nt1 ¼ V nt1 þ nt1 ¼ U nt2 ¼ V nt2 þ nt2 ¼
X
bx k k nt1k
X
bx k k nt2k
þ nt1
ð1Þ
þ nt2
ð2Þ
where U nt1 and U nt2 denote the overall utilities that the respondent n perceived for alternative 1 and 2 in choice task t, respectively; V nt1 and V nt2 denote the systematic utilities of the corresponding alternatives, respectively; nt1 and nt2 denote Please cite this article as: X. Pan and Z. Zuo, Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice, International Journal of Transportation Science and Technology, https://doi.org/10.1016/j.ijtst.2019.12.001
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the random utilities; xnt1k and xnt2k denote the kth attribute towards alternative 1 and 2 in choice task t, respectively; bk is the parameter of kth alternative-specific attribute with a generic value. Based on the conventional random utility maximization assumption of discrete choice models, the probability of choosing each alternative is:
Pnt1 ¼ PfU nt1 U nt2 g ¼ P fnt2 nt1 V nt1 V nt2 g
ð3Þ
Pnt2 ¼ PfU nt2 U nt1 g ¼ P fnt2 nt1 V nt1 V nt2 g
ð4Þ
However, with the assumption that consumers cannot recognize a small different between two alternatives, the concept of threshold is introduced – if the utility difference is lower than the threshold, then these two alternatives are treated as the same. Therefore, the probability of choosing each alternative is:
Pnt1 ¼ PfU nt1 U nt2 þ gg ¼ P fnt2 nt1 V nt1 ðV nt2 þ gÞg
ð5Þ
Pnt2 ¼ PfU nt2 U nt1 þ gg ¼ P fnt2 nt1 ðV nt1 þ gÞ V nt2 g
ð6Þ
If we assume the random terms in the utility functions of the two alternatives nt1 and nt2 distribute iid Gumbel, then the explicit expressions of the choice probabilities of these two alternatives could be given as the following, respectively:
Pnt1 ¼
exp ðV nt1 Þ exp ðV nt1 Þ þ exp ðV nt2 þ gÞ
ð7Þ
Pnt2 ¼
exp ðV nt2 Þ exp ðV nt1 þ gÞ þ exp ðV nt2 Þ
ð8Þ
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3
Therefore, the probability that the consumer n perceives no difference between these two alternatives, denoted as PnI , is specified as:
PntI ¼ PfjU nt1 U nt2 j gg ¼ 1 Pnt1 P nt2
ð9Þ
Here g in Eqs. (5)–(9) is the threshold parameter, which should be non-negative. If g ¼ 0, the model is reduced to the conventional logit model, which means consumers can recognize any difference between alternatives. Moreover, to capturing the heterogeneity property of utility-difference threshold, g could be a function of consumers’ socio-demographic characteristics and also randomly distributed, denoted as gn :
gn ¼ f ðznl ; eÞ
ð10Þ
where znl denotes the consumer n’s lth socio-demographic variable and e is a random term which follows a certain distribution. It is worthwhile noting that here a generic utility-difference threshold for negative and positive utility differences was applied, however, it is easily to expand it to an asymmetric thresholds case. Maximum likelihood method is applied for model estimation. Since the likelihood function involves a random distributed term, which needs integration and inevitably leads to a non-closed form, simulation should be applied (Train, 2009). Therefore, the simulated log-likelihood function for the model with utility-difference threshold could be give as the following:
SLL ¼
1 X XR ln Pnt1jenr yt1 Pnt2jenr yt2 PntIjenr ytI n r¼1 R
ð11Þ
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where SLL denotes the simulated log-likelihood, R denotes the number of random draws, enr is a draw from a certain distribution which is respondent-specific but stays constant in different choice tasks for a single respondent to reflect the fact that the choices from a single respondent are likely not independent, and yt1 , yt1 and ytI are dummy variables indicating whether the consumer n chooses the corresponding alternative in choice task t.
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3. Data collection
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The data set we used in this study is from a stated preference experiment about bus service satisfaction. In detail, respondents were required to evaluate a new bus service comparing the original one – both services were hypothetic and generated from the experiment design. A 5-degree Likert scale (‘‘much worse”, ‘‘worse”, ‘‘almost the same”, ‘‘better” and ‘‘much better”) was used. Indeed, the experiment design was originally designed for an ordered choice study, however, given the 5-degree Likert scale used in the experiment, it still can be applied in the current study after a data transformation. Specifically, those reporting ‘‘much worse” and ‘‘worse” could be seen as preferring the original bus service; those reporting ‘‘better” and ‘‘much better” could be seen as preferring the new service; and those reporting ‘‘almost the same” does not need any transformation. As the respondents were clearly reminded the exact meaning of each scale, we believe this data transformation would not create estimation bias. The attributes of bus services used in the experiment including their definitions and corresponding levels are shown in Table 1. Specifically, we separated the whole travel time into several parts: in-vehicle time, wait time, walk time and transfer time. Another attribute called ‘‘standing duration” is adopted indicating whether seat is available and if not how long he/she
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Please cite this article as: X. Pan and Z. Zuo, Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice, International Journal of Transportation Science and Technology, https://doi.org/10.1016/j.ijtst.2019.12.001
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Table 1 Attribute and corresponding levels of bus services. Attributes
Definitions
Levels
In-vehicle time Wait time Walk time Standing duration Ticket price Transfer time
time in a bus (min) time waiting for a bus, excluding the transfer time (min) time walking out of a bus, excluding the transfer time (min) ratio of standing time over in-vehicle time in a bus price of the ticket (¥) time for transfer (min)
35, 45, 55, 65 2, 6, 10, 14 2, 6, 10, 14 0, 0.33, 0.67, 1 0.5, 1, 1.5, 2 0, 3, 6, 9
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will stand on the bus averagely – this is an index which can measure seat availability. Since the standing time cannot exceed in-vehicle time, we used their ratio to represent standing duration. Finally, we use the software package Ngene (ChoiceMetrics, 2014) for an orthogonal fractional factorial design, based on which 32 profiles (i.e., choice sets) with 8 blocks were obtained. The survey was carried out in Dalian, China, in December 2016. 480 questionnaires were dispensed online as well as onsite. After deleting the observations with inconsistent and incomplete answers, 1784 observations from 446 respondents were deemed valid and used in the model estimation. The descriptive statistics of the sample in terms of their sociodemographic characteristics are shown in Table 2. From the summary of sample characteristics, we can see that the sample are almost uniform distributed in terms of gender, marital status. Meanwhile, most of them are mid-aged (accounting for 54.71%) and at least received a bachelor degree (accounting for 90.58%), and have low or middle income (accounting for 75.79%) and no driving experience (accounting for 43.05%). Moreover, 845 observations (accounting for 47.37%) choose the new bus service and 455 (accounting for 25.50%) choose the original one, and there are 484 observations (accounting for 27.13%) choose the ‘‘almost the same” option.
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4. Model estimation
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4.1. Model with utility-difference threshold vs. model with a quasi-opt-out option
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In many cases, respondents are provided an opt-out option, which indicates the dissatisfaction for any alternatives. Usually, in such cases, the utility of the opt-out option is treated as a constant. Although in this current study the ‘‘almost the same” option cannot be seen as an opt-out option, treating the utility of the ‘‘almost the same” option as a constant is still an
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Table 2 Descriptive statistics of the sample. Socio-demographic attribute
Level
Percentage
Gender
Male Female
45.07% 54.93%
Marital Status
Single Married
56.73% 43.27%
Age
20 21–30 31–40 41–50 51–65 66
13.68% 54.70% 22.20% 5.61% 2.91% 0.90%
Driving Experience (Year)
0 (0, 1] (1, 3] (3, 5] (5, 10] >10
43.05% 15.92% 19.06% 12.11% 6.95% 2.91%
Income (Yuan/Month)
3499 3500–6999 7000–9999 10,000–14,999 15,000–19,999 20,000
42.83% 32.96% 15.47% 6.05% 2.02% 0.67%
Education Level
High school or lower Bachelor Master/PhD
9.42% 76.23% 14.35%
Please cite this article as: X. Pan and Z. Zuo, Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice, International Journal of Transportation Science and Technology, https://doi.org/10.1016/j.ijtst.2019.12.001
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intuitive way for estimation – we label it as ‘‘quasi-opt-out” option. Therefore, in this sub-section, we compare the estimation results from the model with utility-difference threshold and the model with quasi-opt-out option. Both models were estimated based on maximum likelihood method using R package ‘‘maxLik” (Henningsen and Toomet, 2011). All attributes of the bus services entered the models continuously. The utility-difference threshold was assumed to be constant and none of the socio-demographic characteristics were taken into consideration. In addition, in the pretest results we found that the estimated values of taste parameters towards in-vehicle time, wait time, walk time and transfer time were too small, therefore we finally decided to shift their unit from ‘‘minute” to ‘‘hour”. The estimation results are presented in Table 3, which denotes that the utility-difference threshold is significant. Moreover, in terms of the taste parameters, results from both models show that all alternative-specific attributes are significant and have expected signs. However, the taste parameters from these two models are slightly different, especially for those with respect to wait time and walk time. Besides, various indexes of goodness of fit are computed, including AIC, BIC, adjusted Rho-squared. All of the indexes confirm that the model with utility-difference threshold outperforms the model with quasi-opt-out option.
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4.2. Heterogeneity of utility-difference threshold
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The estimation results in Table 3 already show the existence of utility-difference threshold. In this sub-section, we further assume that the threshold varies across population. Specifically, we assume that the threshold is random and also related to consumers’ socio-demographic characteristics. Given that the utility-difference threshold is non-negative, log-normal distribution was firstly applied to capture its randomness. However, the log-normal distribution has a long, thick tail, which may generate an implausibly large threshold value. Therefore, symmetric triangular distribution was also applied (hereafter we refer to these two models as LN-threshold model and TRI-threshold model, respectively). In detail, the threshold was specified as:
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gn ¼ exp gn þ e ; e Nð0; 1Þ
ð12Þ
gn ¼ gn þ gn e; e triangleð1; 1Þ where:
gn ¼ g0 þ
X l
ð13Þ
al znl
ð14Þ
Here znl denotes the consumer n’s lth socio-demographic characteristic and al denotes the impact of lth socio-demographic characteristic to the threshold and g0 is a constant. Therefore, if all socio-demographic characteristics are effects coded, the constant g0 represents the median for log-normal distribution and the median/mode/mean for symmetric triangular distribution. Halton sequence for the random utility-difference threshold was used for simulation. In addition, to investigate the influence of different numbers of Halton draws on estimation results, the model was estimated several times based on different
Table 3 Estimation results of models with utility-difference threshold and quasi-opt-out option.
Alternative-specific attributes In-vehicle time (h) Wait time (h) Walk time (h) Standing duration Ticket price (¥) Transfer time (h) Constant (new bus service) Constant (original bus service) Constant (quasi-opt-out-option) Utility-difference threshold
Model with utility-difference threshold
Model with quasi-opt-out option
coef.
std. err.
p-Value
coef.
std. err.
p-Value
2.5656 3.2485 2.4560 0.7720 0.3411 3.0519 0.5382
0.1875 0.4472 0.4415 0.0923 0.0603 0.5956 0.0477
0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000***
2.5801 2.5671 2.0559 0.7186 0.3882 2.9710 4.3767 3.6754
0.2118 0.5049 0.5081 0.1009 0.0695 0.6851 0.2566 0.2491
0.0000*** 0.0000*** 0.0001*** 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000***
0.7086
0.0294
# of estimated parameters (N) Sample size Initial log-likelihood Convergent log-likelihood AIC (AIC/N) BIC (BIC/N) Rho-squared Adjusted Rho-squared
0.0000*** 8 1784 1959.924 1685.341 3386.683 (423.335) 3430.576 (428.822) 0.1401 0.1360
8 1784 1959.924 1741.963 3499.926 (437.491) 3543.819 (442.977) 0.1112 0.1071
***p-value <0.01; **p-value <0.05; *p-value <0.1.
Please cite this article as: X. Pan and Z. Zuo, Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice, International Journal of Transportation Science and Technology, https://doi.org/10.1016/j.ijtst.2019.12.001
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numbers of draws, from 100 to 1000 with a step size 100. The final convergent log-likelihoods are shown in Fig. 1. Fig. 1 shows that the largest difference of final log-likelihood is less than 0.9 for the LN-threshold model and less than 0.4 for the TRI-threshold model, and the final convergent log-likelihoods for both models tend to be stable with the increasing of the number of Halton draws. Therefore, the estimation results with 1000 Halton draws for both models are reported in this sub-section. The model was estimated based on simulated maximum likelihood method using R package ‘‘maxLik” (Henningsen and Toomet, 2011). All attributes of the bus services entered the models as continuous values while the socio-demographic characteristics entering the models were effects coded. Table 4 shows the estimation results with 1000 Halton draws. The adjusted Rho-squared is 0.1378 and 0.1371 for LN-threshold and TRI-threshold models, respectively, which both indicate acceptable model performance (McFadden, 1979; Louviere et al., 2000). Since these two models are not nested, the adjusted likelihood ratio test (Ben-Akiva and Swait, 1986) was applied to compare their performance statistically. The probability that the TRI-threshold model being preferred is:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P q2ln q2tri > z / 2 z LLð0Þ þ ðK tri K ln Þ
ð15Þ
where q2ln and q2tri are adjusted Rho-squared for LN-threshold and TRI-threshold models, respectively; / is the standard normal cumulative density function; LLð0Þ is the initial log-likelihood; K tri and K ln are the numbers of parameters for the two models, respectively. Finally, it turns that the probability is around 0.0271. Together with AIC and BIC indexes, we conclude that the LN-threshold model is just slightly better than the TRI-threshold model. Therefore, in the following we present and discuss the results from both models. Another issue should be noted is that some literature (e.g., Hensher and Greene, 2003; Hensher et al., 2015) argued that although log-normal distribution can constrain the sign of a random parameter or component, unreasonable or biased results may be obtained due to its very long tail – in these cases, triangle distribution is usually suggested. In this study, the emphasis is not on the comparison between different distributions, but we think the readers should realize the difference as we will show in the following. In terms of the taste parameters, all alternative-specific attributes are significant and have expected signs. Results from the LN-threshold model has lower values than those from the TRI-threshold model. If we calculate the willingness-to-pay, we will find the TRI-threshold model would generate higher values of willingness-to-pay. Additionally, we included an alternative-specific constant in the utility function of the new bus service. This is principally not necessary for unlabeled choice context. However, considering the choice context in our experiment, which is not exactly an unlabeled choice originally, we believe that there might be bias if an alternative-specific constant is not taken into account. Besides, Hensher et al. (2015) also suggests to include this constant at the beginning even in unlabeled choice context and later removed it if it is found to be statistically insignificant. The estimation results from both models confirm a significant alternative-specific constant. In terms of the utility-difference threshold, the estimation results from both models confirm its observed and unobserved heterogeneity, which support our assumption that the utility-difference threshold varies across population. To present parsimonious models, the insignificant variables are removed. The 95% (average) interval of threshold ranges from 0.0082 to 14.0460 for LN-threshold model and from 0.1658 to 1.3174 for TRI-threshold model, which means a change of unit in terms of the attributes only has influence on a part of the population. Based on the results of the LN-threshold model, it tells that averagely one-minute change in in-vehicle time, wait time, walk time or transfer time would make the overall utility increased/decreased by 0.0526, 0.0612, 0.0470, or 0.0661, respectively, which implies only 16.33%, 18.37%, 14.91%, or 19.47% of the population could feel the change; a ¥1 change in ticket price would make the overall utility increased/decreased by 0.4709, implying 56.86% of the population could feel the change; a unit change in standing duration (e.g., from standing for the entire trip to sit for the entire trip or the opposite) would make the overall utility increased/decreased by 0.9157–69.94% of the population could feel the change. Based on the results of TRI-threshold model, it tells that averagely for
Fig. 1. Convergent log-likelihoods based on various numbers of Halton draws.
Please cite this article as: X. Pan and Z. Zuo, Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice, International Journal of Transportation Science and Technology, https://doi.org/10.1016/j.ijtst.2019.12.001
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X. Pan, Z. Zuo / International Journal of Transportation Science and Technology xxx (xxxx) xxx Table 4 Estimation results of the utility-difference threshold model with 1000 Halton draws. Model with log-normal dist.
Alternative-specific attributes In-vehicle time (h) Wait time (h) Walk time (h) Standing duration Ticket price (¥) Transfer time (h) Constant (new bus service) Utility-difference threshold Constant Std. deviation Education High school or below Master/PhD Bachelor
Model with triangular dist.
coef.
std. err.
p-Value
coef.
std. err.
p-Value
3.1533 3.6744 2.8207 0.9157 0.4709 3.9650 0.6801
0.3071 0.5574 0.5552 0.1228 0.0896 0.7971 0.0781
0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000***
2.6108 3.3143 2.4964 0.7872 0.3488 3.0909 0.5501
0.1914 0.4561 0.4505 0.0939 0.0615 0.6072 0.0488
0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000***
1.0812 1.8998
0.4180 0.7585
0.0097*** 0.0123**
0.7416
0.0440
0.0000***
0.3324 0.4170 0.0846
0.1920 0.1161
0.0299** 0.4663
0.1373 0.1793 0.0420
0.0659 0.0468
0.0065*** 0.3698
# of estimated parameters (N) Sample size Initial log-likelihood Convergent log-likelihood AIC (AIC/N) BIC (BIC/N) Rho-squared Adjusted Rho-squared
11 1784 1959.924 1678.819 3379.638 (307.2398) 3439.991 (312.7265) 0.1434 0.1378
10 1784 1959.924 1681.159 3382.317 (338.2317) 3437.184 (343.7184) 0.1422 0.1371
***p-value <0.01; **p-value <0.05; *p-value < 0.1.
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one-minute change in in-vehicle time, wait time, walk time, or transfer time, only 0.17%, 0.28%, 0.16%, or 0.24% of the population could feel the change; 11.06% of the population could feel a ¥1 change in ticket price; 55.96% of the population could feel a unit change in standing duration (e.g., from standing for the entire trip to sit for the entire trip or the opposite). Some conclusions can be made here. First of all, when calculate the willingness-to-pay, the utility-difference threshold must be taken into consideration – otherwise the results may be largely overestimated. Second, some attributes have limited improvement space, such as standing duration. Even the bus service in these attributes has got the best performance, there are still a part of population do not care about the changes. In this sense, policy makers should not pay much attention to such attributes. The estimation results also show that this heterogeneity property of utility-difference threshold is less related to consumers’ socio-demographic characteristics. According to our results, only consumers’ education level has significant (at the 95% probability level for the LN-threshold model and at the 99% probability level for the TRI-threshold model) influence on the effect of utility-difference threshold. In detail, consumers with a higher education level has a higher utility-difference threshold. A plausible reason is that consumers with a higher education level is more likely to simplify the choice tasks to save process efforts. Note that the sample used in this study cannot reflect the real distribution of population in Dalian. Although the estimated coefficients of the threshold may be biased from those for the whole population, this study already confirmed the observed and unobserved heterogeneity of utility-difference threshold based on the specific data set. Note the main objective of this study is to explore the role of utility-difference threshold rather than measure the effect of the threshold in population. Policy makers may be also interested in the elasticities associated to specific attributes. Therefore we also compute the average direct and cross elasticities, using the estimation results of LN-threshold and TRI-threshold models, when the performance of attributes of the new bus service are deteriorated by 1%. The results are presented in Table 5, which indicate that the loss of market share of the new bus service would not entirely goes to the original one – some of the sample cannot feel the changes therefore would in fact stick to the one that they preferred originally. In terms of elasticity with respect to a specific attributes, Table 5 indicates that deterioration in in-vehicle time is the severest, followed by ticket price. The elasticities based on different models are slightly different, which again confirms the importance of the assumption of the distribution of utility-difference threshold.
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5. Conclusions
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Due to various reasons, consumers may not recognize or does not care about a small difference among alternatives. The previous studies introduced the concept of threshold to indicate insensitivity of consumers if the difference of alternatives is smaller than the threshold. The threshold could be attribute-specific or alternative-specific. In addition, there may be heterogeneous threshold among population. This assumption is so importation that its ignorance may cause bias when calculate
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Please cite this article as: X. Pan and Z. Zuo, Exploring the role of utility-difference threshold in choice behavior: An empirical case study of bus service choice, International Journal of Transportation Science and Technology, https://doi.org/10.1016/j.ijtst.2019.12.001
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Table 5 Elasticities when attributes of the new bus services deteriorated by 1% (unit: %). New bus service
Original bus service
The almost same option
LN-threshold model In-vehicle time Wait time Walk time Standing duration Ticket price Transfer time
0.426 0.077 0.063 0.075 0.099 0.047
0.382 0.070 0.055 0.067 0.087 0.042
0.043 0.007 0.008 0.008 0.013 0.006
TRI-threshold model In-vehicle time Wait time Walk time Standing duration Ticket price Transfer time
0.430 0.085 0.068 0.078 0.090 0.046
0.368 0.074 0.056 0.067 0.073 0.038
0.062 0.011 0.011 0.011 0.017 0.008
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willingness-to-pay and elasticity associated with specific attributes. This study provides an empirical case study of utilitydifference threshold model proposed in Cantillo et al. (2010) in the context of bus service choice to explore the role of utilitydifference threshold. The stated preference experiment was applied to collect data and models were estimated using maximum likelihood method. The model with utility-difference threshold was compared with the one with quasi-opt-out option. Further, the utilitydifference threshold was assumed to be log-normally or symmetrically & triangularly distributed and also a function of consumers’ socio-demographic characteristics. Results provided acceptable goodness of fit and confirmed the existence and heterogeneity property of the utility-difference threshold. Moreover, results showed that the LN-threshold model was only slightly better than the TRI-threshold model while the estimated coefficients of the model were quite different both for taste parameters and utility-difference threshold parameters. Therefore, one should carefully determine the distribution of the threshold. In addition, results also showed that the utility-difference threshold was related to consumers’ education level that a higher education level would lead to a higher utility-difference threshold. We also analyzed consumers’ sensitivity to marginal change in terms of each alternative-specific attribute. Since the existence of utility-difference threshold, a marginal change in a certain attribute cannot trigger full awareness of the population. This sensitivity varies across attributes – to some attribute, due to its limited improvement space, even an alternative has already improved to its best performance, there are still a part of population choose to ignore this change. In addition, we also computed the direct and cross elasticities associated with all alternative-specific attributes, which also confirmed the importance of utility-difference threshold. This study also provides policy implications to the policy makers. It reminds the policy makers to make a single policy with a big change in some cases. For instance, if the transportation department is going to make a policy to release the road congestion, the results should be significant so that the most of the travelers could feel the improvements. However, in other cases it reminds the policy makers to not make a single policy with a big change. For instance, if the transit company is going to increase the ticker fare from ¥1 to ¥2, it is better to do it in several steps, such as increasing ¥0.5 in the first year and then increasing ¥0.5 in the second year, to avoid severe responses from travelers. There are still many things to do towards this topic. Here we just list a few. First of all, empirical applications in different choice context are always needed, both from stated and revealed preference data sets. Second, extension of the model to multiple choice context could make the model more attractive and universal. This extension may make the model choiceset dependent. In this sense, the random regret model (Chorus et al., 2008) and the relative utility model (Zhang et al., 2004) may could be served as references. A challenge on this topic is that a multiple choice case makes consumers’ choice behavior much more complicated as they have to compare each pair of alternatives in the choice set. Third, deep investigation of elements that could influence the utility-difference threshold is also needed. Our results show consumers’ sociodemographic characteristics are less related, which leads us to think about other elements, such as consumers’ knowledge level, perception ability, even personality and behavior habits. This may suffer from the difficulty of data collection but it seems promising.
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Conflict of Interest
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Acknowledgements and declarations
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The authors thank the anonymous reviewers for their comments, which help to improve this paper. On behalf of all authors, the corresponding author states that there is no conflict of interest.
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