Content uploaded by Shlomo Bekhor
Author content
All content in this area was uploaded by Shlomo Bekhor
Content may be subject to copyright.
Transportmetrica, Vol. 4, No. 2 (2008), 155-177
155
DISCRETE CHOICE MODELING OF COMBINED MODE AND
DEPARTURE TIME
SHAMAS BAJWA1, SHLOMO BEKHOR2, MASAO KUWAHARA3 AND EDWARD CHUNG4
Received 15 January 2007; received in revised form 27 September 2007; accepted 8 January 2008
Typical daily decision-making process of individuals regarding use of transport system involves mainly three
types of decisions: mode choice, departure time choice and route choice. This paper focuses on the mode and
departure time choice processes and studies different model specifications for a combined mode and departure
time choice model. The paper compares different sets of explanatory variables as well as different model
structures to capture the correlation among alternatives and taste variations among the commuters. The main
hypothesis tested in this paper is that departure time alternatives are also correlated by the amount of delay.
Correlation among different alternatives is confirmed by analyzing different nesting structures as well as error
component formulations. Random coefficient logit models confirm the presence of the random taste
heterogeneity across commuters. Mixed nested logit models are estimated to jointly account for the random
taste heterogeneity and the correlation among different alternatives. Results indicate that accounting for the
random taste heterogeneity as well as inter-alternative correlation improves the model performance.
KEYWORDS: Departure time choice, mode choice, nested logit, cross-nested logit, error component logit,
mixed logit, value of time, value of schedule delay
1. INTRODUCTION
Advent of the advanced transport control and communication technologies has made it
possible to implement time-varying demand management policies for example, time-
varying road pricing. It is necessary to develop the departure time and mode choice
behavioural models to assess the impact of these policies.
This paper discusses the development of a combined mode and departure time choice
model for morning commuters. Typical daily decision-making process of individuals
regarding use of transport system involves mainly three types of decisions: departure
time choice, mode choice and route choice. In this paper, we focus on the first two
components of the choice behavior.
Most commuters have preferred arrival times at their destinations due to constraints of
work (school) starting times. These arrival times are generally concentrated in a short
time period indicating peak demand during rush hours. Time-dependent demand
management policies attempt to spread the peak demand on a longer time period, by
providing commuters a trade-off between arriving early/late at destination than the
preferred arrival time and thus spending less time in congestion (cost) or arriving on-
time but spending more time in congestion (cost). Hence, the parameter of interest is not
only the value of travel time savings which is usually estimated in traditional mode
choice models but also the value of early/late schedule delay to establish the trade-off
among different alternative departure/arrival times available to commuters.
In this paper, our main focus is the departure time choice behavior of road traffic
network users. However, it is also known that mode choice always stands as a viable
option if alternative public transport modes provide extensive spatial coverage with high
1 School of Civil, Environmental and Chemical Engineering, RMIT University, Melbourne, Australia.
Corresponding author (E-mail: shamas.bajwa@rmit.edu.au).
2 Transportation Research Institute, Technion – Israel Institute of Technology, Haifa, Israel.
3 Institute of Industrial Science, The University of Tokyo, Tokyo, Japan.
4 Institute of Industrial Science, The University of Tokyo, Tokyo, Japan.
156
frequency which is usual for metropolitan areas. Need to include mode choice in
commuters’ choice behavior models is further highlighted if we want to test some travel
demand management strategies e.g. road pricing which may be prohibitively expensive
for some section of the commuters. Hence, mode choice can be considered to provide a
viable alternative to travel. Mode choice can also be considered as a stand-in for elastic
demand in the traffic networks by assuming a constant travel time between origin and
destination as well as a fixed cost and schedules will keep its utility constant, giving it
the role of a null option in the elastic demand models.
Important considerations for behaviour modelling include appropriate specification of
the utility functions associated with all alternatives. Users' choices depend not only on
their socio-economic characteristics but also on the level of service attributes of the
network used for commuting. An appropriate utility function specification should
include a mix of socioeconomic characteristics and level of service attributes explaining
the maximum variance among the user’s behaviour. However, it is still possible that the
taste variations among the commuters are not captured due to some unknown
characteristics or measurement errors. These random (unmeasured and/or non-
quantifiable) taste variations among commuters can significantly affect the performance
of the model, and should be taken into account. The progress in the field of the mixed
logit models allows the estimation of such taste variations by assuming a distribution of
parameters over the commuters’ population instead of a single value. In addition,
flexibility of the model structure allows capturing correlation among different
alternatives. In the case of discrete choice models of departure time, the consecutive
departure time intervals can be highly correlated due to the continuity of the underlying
variable (i.e. time) and inability of the commuters to distinguish between the close by
alternative departure times.
In this paper, our aim is to identify appropriate utility specifications and capture the
random taste heterogeneity across the commuters by using the mixed logit class of
models. In addition, correlation among alternatives is investigated by specifying
different nesting structures as well as error components. Different models are compared
and discussed to highlight their relative strengths and weaknesses.
Two main contributions can be outlined in this paper as follows. First, it provides a
comprehensive comparative study of different model structures based on the same
dataset, by showing the relative improvement over successive flexible models. Second,
the paper shows that departure time alternatives are also correlated by the amount of
delay, by examining different correlation structures among alternatives.
The remaining of this paper is organised as follows: Next section describes in brief
previous attempts to model departure time choice behaviour as well as a brief
introduction to the mixed logit and nested logit models. Section 3 details the
methodology used in this study, including the survey design, data collection and model
specifications. Section 4 shows the results of the estimated models and discusses the
improvements, relative merits and demerits as well as problems encountered in different
specifications. Finally, section 5 concludes the paper with a summary of the findings and
outlines the future research directions.
2. LITERATURE SURVEY
Nearly all the existing models about departure time choice are based on the trade-off
between early, late or on-time arrival first proposed by Vickrey (1969). Assuming a
single bottleneck on the route, the departure time choice is analyzed by evaluating the
157
trade-off between waiting in queue and arriving on-time or starting early/late and
arriving earlier/later than a preferred arrival time. Hendrickson and Kocur (1981)
reformulated the same problem and elaborated the analysis using the queuing theory
notation. Henderson (1974), Hendrickson et al. (1981), Hurdle (1981), Fargier (1981)
also independently solved the departure time choice problem for a single bottleneck case
(i.e. route choice is not considered). Kuwahara (1985) and Kuwahara and Newell (1987)
extended the analysis of departure time choice in a network to a many-to-one origin
destination network, where each commuter passes only one bottleneck. de Palma et al.
(1983) used a similar construction of one origin and destination with a single bottleneck
as used by Vickrey (1969) and others for a stochastic equilibrium model of departure
time choice. All models above consider the interaction among supply and demand and
provide equilibrium solution. Same trade-off principle is used in all these models and is
also employed for discrete choice model estimation.
Most discrete departure or arrival time models developed are based on the multinomial
logit (MNL) model: for example, Small (1982) used the data collected from the car
commuters in the San Francisco bay area to model the arrival time choice, Hendrickson
and Plank (1984) estimated a combined mode and departure time choice model.
Abkowitz (1981) also used the same data set as used by Small (1982) including
additional socio-demographic variables as determinants of commuter’s departure time
choice behavior. Chin (1990) modeled the departure time choice of morning commuters
using the data collected in Singapore. Shimizu and Yai (1999) carried out a survey to
gauge the reaction of commuters to a variable peak period toll in Tokyo Metropolitan
area. The departure time choice of the users was modeled as a discrete logit choice in
half an hour intervals including the shift to public transport or alternative un-tolled route
as a choice at the same level as departure time choice.
Use of MNL models ignores any correlation among the consecutive discrete departure
time intervals. If the departure time interval becomes small, it is for the decision-makers
to distinguish between the adjacent time intervals resulting in a higher correlation
between alternatives. Small (1987) proposed an OGEV model for the departure time
choice which has a more flexible correlation structure than MNL model by allowing for
the correlation parameter to exist for pairs of alternatives which depends on the distance
among the alternatives based on time-of-day natural ordering. The number of correlated
alternatives needs to be specified before-hand. Bhat (1998a) used MNL for modeling
mode choice and an ordered generalized extreme value (OGEV) for departure time
choice. The proposed MNL-OGEV model was applied to data obtained from the 1990
San Francisco Bay area travel survey data and was found to perform better than the
MNL and nested logit models. Results indicate that the MNL and nested logit models
lead to biased level-of-service estimates and to inappropriate policy evaluations of
transportation control measures. Polak and Jones (1994) used a nested logit structure to
model the departure time choice in a tour based context.
Cross-nested logit models extend and generalize the correlation structure among the
alternatives. Instead of each alternative belonging to a single nest in nested logit models,
cross-nested models allow alternatives to belong to more than one nest thus resulting in a
flexible correlation structure. These models define the share of each alternative
belonging to different nests. Recently similar flexible correlation structures have been
developed and used as cross-nested, generalized nested and paired combinatorial logit by
many researchers (Vovsha, 1997; Vovsha and Bekhor, 1998; Koppelman and Wen, 2000;
Wen and Koppelman, 2001; Papola, 2004).
158
Mixed logit is a highly flexible model that can approximate any random utility model
(McFadden and Train, 2000). It obviates the three limitations of standard logit model by
allowing for random taste variations, unrestricted substitution patterns, and correlation
among unobserved factors over time. Unlike probit, mixed logit is not restricted to
normal distributions, have straightforward derivation and choice probabilities can be
simulated easily (Train, 2003). The first application of mixed logit was apparently the
automobile demand models created by Boyd and Mellman (1980) and Cardell and
Dunbar (1980).
Bhat (1998b) used mixed multinomial logit model for analysis of travel mode and
departure time choice for home-based social–recreational trips using data drawn from
the 1990 San Francisco Bay Area household survey. The empirical results highlighted
the need to capture unobserved attributes for mode as well as departure time which not
only improved model fitness but also resulted in realistic evaluations of transportation
control measures. de Jong et al. (2003) also developed an error component logit model
for the joint choice of time-of-day and mode using stated preference data for car and
train travelers in The Netherlands. The results indicate the time-of-day choice is
sensitive to the peak travel time and cost. A different approach to model departure time
has been to use continuous time models instead of discrete time intervals (Wang, 1996;
Bhat and Steed, 2002).
In most researches, the multinomial logit is used as the kernel for the mixed logit
model. Recently, Hess et al. (2004) have applied different model structures such as
mixed nested logit model and mixed cross-nested logit model for the mode choice. They
proposed these modeling structures to capture the effect of the random taste
heterogeneity as well as the inter-alternative correlation. This study showed that use of
mixed GEV models improves the performance compared to basic models.
Review of the past literature reveals that different model structures have been applied
for disaggregate departure time choice models but a comprehensive study comparing
different model structures estimated for the same data set has not been carried out.
Furthermore, most of the existing departure time choice models are developed solely for
departure time choice of the commuters while it may be reasonable to assume that
commuters decide their mode and departure time concurrently. In this research, a
comprehensive combined departure time and mode choice modeling study is carried out
in order to compare the relative pros and cons of different modeling structures.
3. METHODOLOGY
Discrete choice models are used to replicate the choices made by decision-makers (i.e.
commuters) from a discrete number of alternatives which constitute the choice set
depending on the availability. The methodology used to specify and estimate combined
departure time and mode choice models are described in the subsequent sections,
organised as follows. The first section briefly presents the model structures used in this
paper. Data describing users’ behaviour is presented next. After that choice set definition
is described. The last subsection presents the specification of the utility functions and
correlation structures.
3.1 General model structure
The usual form of the utility function is
159
ij ij ij
UV
=
+ε , (1)
where ij
U is the utility of the individual j for alternative j
iC∈, ij
V is the deterministic
part of the utility of the alternative i for individual j, and ij
ε
is the random component of
the utility of the alternative i for individual j.
()
,
ij ij
Vfx=β , (2)
where ij
x
is a vector representing the attributes of an alternative i as well as the socio-
economic characteristics of the decision-maker j, and β is vector of coefficients which
needs to be estimated from the data. Depending on the assumed form of the random
component of the utility, different models can be developed.
In the random coefficient logit model, some of the elements of the vector β
described
in equation (2) are declared as random variables to capture the taste heterogeneity in the
population. In the error component logit model, instead of some elements of vector
β
corresponding to the vector ij
x
being random, separate random term is introduced in
the utility function,
()
,
ij ij
Vfx=β +ξ, (3)
where, ξ is a random disturbance, generally assumed to follow a multivariate normal
distribution with a mean zero and covariance matrix Ω, where Ω is generally assumed to
be diagonal (Walker, 2001).
3.2 Survey design and data collection
A stated preference survey was conducted in Tokyo Metropolitan Area to elicit the
responses of the users corresponding to different hypothetical scenarios specifying
different departure times as well as travel times and costs to reach destination at a
preferred arrival time. Households were randomly selected and data about the primary
morning commuter was collected using a mail-back survey. A total of 1324 valid
responses were used for model estimation.
The problem posed to the users was as follows: given a specific arrival time,
commuters have different departure options from home; they can select the mode as well
as the departure time. Commuters are assumed to choose their mode between car and rail
(the only public transport mode presented in the survey). Departure time was modeled in
15 minute intervals and this option was only available if commuters chose car as mode.
Car commuters can trade-off between arriving early or late with less congestion (i.e.
shorter travel time) or arriving on-time with higher congestion (i.e. longer travel time) on
the road. Different levels of toll were also introduced. This allowed us to measure the
trade-off among the monetary costs, travel times and schedule delay penalties. It was
assumed that rail users can reach their destination without any schedule delay and with a
fixed cost. This assumption is quite reasonable due to the high frequency of trains in the
region. The only aspect of the public transport not accounted for in this study is the
congestion levels inside the train as the commuters can choose a different departure time
from their home while using public transport to commute in order to avoid severe
congestion inside the trains but anecdotal evidence suggests to the contrary.
The questionnaire presented to the users consisted of two sections: in first section,
socio-economic characteristics of the household as well as the commuter were collected.
160
The collected information consists of the household type (single, couple, couple with
children etc.), dwelling type and number of cars in household. Personal information
collected includes the personal characteristics such as age, gender, income, work
location (post-code) and information about morning commute on a typical day. In
second part, stated preference scenarios were presented and users were asked to choose
departure time as well as mode. Cost and travel time of the rail were fixed at 500 yen
(~$4) and 60 minutes respectively while for the car, cost was varied between three levels
of 500, 700 and 1000 yen and the travel time was varied at five levels from 40 to 60
minutes at 5 minute intervals. The early and late arrival delay was automatically
deducted from the interaction of the departure time, travel time and preferred arrival time
at the destination.
To ensure the statistically efficient information retrieval from the collected data while
not cognitively burdening the users excessively, a standard statistical experiment design
procedure named factorial design was used in this study. A fractional factorial design,
which can cater for the main effects as well as some first-order interactions among the
attribute levels of different alternatives, was used. Fractional factorial design means loss
of some statistical efficiency by ignoring second and higher order effects i.e. interactions
among two or more than two attributes. But the loss of information is not critical as it
has been shown that more than 80% of the information is explained by main effects,
15% by the first order effects and remaining 5% by the second and higher order effects
(Louviere et al., 2000). Hence, proper fractional factorial design can be used to design
experiments with over 95% statistical efficiency.
3.3 Choice set definition
Based on the information collected from the survey respondents in the stated
preference survey, the selected alternatives as well as the alternative set presented to the
subjects in each scenario can be aggregated as proposed by Cascetta and Papola (2003).
The number of choices presented in each scenario were limited to three to avoid the
cognitive load to the survey respondents where one alternative was always rail indicating
its availability to all the users independent of their location. Remaining two alternatives
were different departure time options using car. As each respondent was presented with a
maximum of either two early arrivals or two late arrivals due to rail being constrained to
on-time arrival, hence the alternatives can be aggregated into following six alternatives
available to each user without any loss of information in data:
• Earliest Early Arrival Car (EEA)
• Latest Early Arrival Car (LEA)
• On-time Arrival Car (OT)
• Earliest Late Arrival Car (ELA)
• Latest Late Arrival Car (LLA)
• Rail (RL)
This aggregation can be justified because many of the alternatives may never be
chosen in the sample because of its size and consequently not included in the final
choice set. No departure time option is available for the rail because of its frequency.
The frequency of rail in Tokyo area in the morning is high and all the rail users can
choose rail which allows them to reach their destination without any schedule delay.
Table 1 shows a summary of the choices and availabilities of each alternative in the
sample.
161
TABLE 1: Choices and availabilities of alternatives in the sample
Alternatives Choices Availabilities
Earliest Early Arrival Car (EEA)
Latest Early Arrival Car (LEA)
On-time Arrival Car (OT)
Earliest Late Arrival Car (ELA)
Latest Late Arrival Car (LLA)
Rail (RL)
Total
28
332
24
50
4
886
1324
345
1241
130
880
52
1324
3.4 Model structure specifications and selected attributes
As the correlation among the alternatives is not known in advance, we need to
hypothesize and test different correlation structures to identify the best fitting and
explanatory model. The basic structure tested is an MNL model assuming that no
correlation exists between any of the alternatives. The nesting structure as well as
covariance matrix form for this formulation is as shown in Figure 1.
FIGURE 1: Multinomial logit correlation structure and relevant covariance matrix
Figure 2 shows a nested logit model structure in which the alternatives are grouped
together by mode i.e. rail is a separate nest while the departure time options
corresponding to car are grouped together in a single nest. The covariance matrix
corresponding to this structure is also shown in Figure 2 indicating that alternatives EEA,
LEA, OT, ELA and LLA are correlated with each other while the rail is not.
FIGURE 2: Nested logit (mode based) correlation structure and relevant covariance
matrix
Figure 3 shows another nesting structure in which the alternatives are assigned to three
nests. Rail is in a separate nest i.e. is not correlated with any other alternative while the
departure times using car resulting in early or on-time arrival at the destination are
grouped in one nest while the alternatives depicting the departure by car for late arrivals
are grouped together in one nest. The corresponding covariance matrix structure is as
shown in the Figure 3.
162
FIGURE 3: Nested logit (3nests-a) correlation structure and relevant covariance matrix
Figure 4 shows another nesting structure indicating that alternatives are grouped
together based on the arrival time at the destination irrespective of the mode. Three nests
are formed each corresponding to early, on-time and late arrivals. Each nest has two
alternatives. The corresponding covariance matrix is also shown in the Figure 4.
FIGURE 4: Nested logit (3nests-b) correlation structure and relevant covariance matrix
Figure 5 shows the nesting structure with four nests, one corresponding to the early
arrivals using cars, another corresponding to late arrivals using car while the remaining
two indicate the on-time arrivals at the destination but using different modes. The
corresponding covariance matrix is also shown in the Figure 5.
FIGURE 5: Nested logit (4nests) correlation structure and relevant covariance matrix
The attributes used in the modeling stage can be divided into two distinct groups: level
of service attributes and personal characteristics of each individual user. Different
attributes tried for the model estimations are defined as follows:
βtravel time : Coefficient for travel time, where travel time is given in minutes
βcost : Coefficient for cost of travel, where cost is in yen
βearly arrival : Coefficient of early arrival penalty, where early arrival penalty is
calculated based on the departure time as well as preferred arrival time
at the destination, in case of random coefficient model, this represents
the mean of the coefficient distribution
σearly arrival : Variance of the early arrival penalty in random coefficient model
βlate arrival : Coefficient of late arrival penalty, where late arrival penalty is
calculated based on the departure time as well as preferred arrival time
at the destination, in case of random coefficient model, this represents
the mean of the coefficient distribution
163
σlate arrival : Variance of the late arrival penalty in random coefficient model
βcar availability : Coefficient of car availability, where Car Availability is a dummy
variable equal to 1 if commuter owns a car; 0 otherwise
βold age : Coefficient representing the effect of old age, where old age is a
dummy variable equal to 1 if commuter is more than 70 years old and
0 otherwise
βhigh income : Coefficient representing the effect of high income, where high income
is a dummy variable equal to 1 if commuter’s annual income is more
than 15 million yen and 0 otherwise
βyoung : Coefficient representing the behaviour of young people, where young
is a dummy variable if commuter is younger than 30 years of age or is
a student and 0 otherwise
βwork in suburbs : Coefficient representing the effect if the commuter’s work place is not
in central Tokyo, where work in suburbs is a dummy variable equal to
1 if the commuter’s work place is in suburbs and 0 otherwise; in case
of random coefficient model, this represents the mean of the
coefficient distribution
σwork in suburbs : Variance of the work in suburbs coefficient in random coefficient
model
ξcar : Random error component constrained to be same for all the
alternatives using car as a mode
ξrail : Random error component constrained to be same for the alternative
using rail as a mode
ξon-time : Random error component constrained to be same for the alternative
resulting in on-time arrival
ξearly arrival : Random error component constrained to be same for the alternative
resulting in early arrival
ξlate arrival : Random error component constrained to be same for the alternative
resulting in late arrival
ξearly arrival/on-time : Random error component constrained to be same for the alternative
resulting in early or on-time arrival
4. MODEL ESTIMATION RESULTS
Maximum likelihood and simulated maximum likelihood methods were used for
estimating the parameters of the closed-form GEV and mixed GEV models respectively.
These methods try to maximize the log-likelihood function. As stated earlier, the data
collected in a survey of the morning commuters in the Tokyo Metropolitan area is used
in this study. Estimation software BIOGEME is used for model estimations (Bierlaire,
2005)
4.1 MNL models
MNL model as depicted in Figure 1 is the first estimated model structure for combined
mode and departure time. Different utility specifications were tested to find the best
possible utility function explaining the maximum variance in the data. Two alternative
utility specifications are mentioned below. One only includes the level of service (LOS)
attributes while other also includes the personal characteristics of the commuters. The
164
model using the personal characteristics in addition to the LOS attributes shows a
marked improvement over the LOS only model. The log-likelihood value increases by
about 70 points by addition of the 5 personal characteristics. The personal characteristics
chosen for the inclusion in the model are those which have been found to be significant
during different trial estimations. The LOS only model is good for the network-wide
applications where the detailed data about the personal characteristics of the commuters
is not available. All the utility functions are linear in attributes as well as parameters.
The estimation results of the MNL models are reported in the Table 2 with t-statistics
shown in brackets. All the parameters are found to be significant at a level greater than
95%. The value of travel time savings is about 38 yen/min or about 2300 yen/hour
(about US$20). The value of early and late arrival penalty is about 22 yen/min and 110
yen/min respectively which is about 60% and 300% of the value of travel time savings.
These values are quite similar to what have been reported elsewhere in literature for
different geographic locations.
TABLE 2: Estimation results of MNL model of Figure 1
Coefficients Level of service attributes Level of service + socio-demographic attributes
ASCRail 0.7040 (5.13) 1.8450 (9.38)
βtravel time -0.0300 (-3.84) -0.0314 (-3.91)
βcost -0.0008 (-3.15) -0.0008 (-3.21)
βearly arrival -0.0167 (-7.03) -0.0180 (-7.28)
βlate arrival -0.0844 (-7.78) -0.0883 (-7.92)
βcar availability 0.9535 (6.64)
βold age 1.3211 (2.48)
βhigh income 1.1410 (4.20)
βyoung -0.5656 (-2.91)
βwork in suburbs -1.0320 (-7.90)
No. of observations 1324 1324
No. of parameters 5 10
Null-log likelihood -1454.56 -1454.56
Final-log likelihood -1052.76 -982.91
Rho-Squared 0.276 0.324
Rho-Squared bar 0.273 0.317
VTTS(yen/min) 38.0 38.5
VEAP(yen/min) 21.2 22.0
VLAP(yen/min) 107.1 108.2
VTTS = Value of Travel Time Savings
VEAP = Value of Early Arrival Penalty
VLAP = Value of Late Arrival Penalty
The utility functions for the LOS only model is
car,EEA traveltime EEA cost EEA earlyarrival EEA latearrival EEA
car,LEA traveltime LEA cost LEA earlyarrival LEA latearrival LEA
car,OT traveltime OT cost OT earlyarrival
VTTCOSTEALA
VTTCOSTEALA
VTTCOSTE
= β +β +β +β
= β +β +β +β
= β +β +β OT latearrival OT
car,ELA traveltime ELA cost ELA earlyarrival ELA latearrival ELA
car,LLA travel time LLA cost LLA earlyarrival LLA latearrival LLA
Rail Rail traveltime Rai
ALA
VTTCOSTEALA
VTTCOSTEALA
VASC TT
+β
= β +β +β +β
= β +β +β +β
=+β lcost Rail
COST+β
165
While for the model including level of service as well as personal attributes, the best
possible utility function specification is
car,EEA traveltime EEA cost EEA earlyarrival EEA latearrival EEA
caravailability highincome
car,LEA traveltime LEA cost LEA earlyarrival LEA latearrival
VTTCOSTEALA
CarAvailability HighIncome
VTTCOSTEAL
= β +β +β +β
+β +β
= β +β +β +β LEA
caravailability highincome
car,OT traveltime OT cost OT earlyarrival OT latearrival OT
caravailability highincome
car,ELA travel ti
A
CarAvailability HighIncome
VTTCOSTEALA
CarAvailability HighIncome
V
+β +β
= β +β +β +β
+β +β
=β me ELA cost ELA earlyarrival ELA latearrival ELA
caravailability highincome
car,LLA travel time LLA cost LLA earlyarrival LLA latearrival LLA
caravailabil
TT COST EA LA
CarAvailability HighIncome
VTTCOSTEALA
+β +β +β
+β +β
= β +β +β +β
+β ity highincome
Rail Rail traveltime Rail cost Rail oldage
young workinsuburbs
CarAvailability HighIncome
V ASC TT COST OldAge
Young WorkInSuburbs
+β
=+β +β +β
+β + β
A positive alternative specific constant for rail indicates an inherent preference to
choose rail over other mode which is quite understandable owing to the chronic
congestion on the roads even with the reduced traffic demand and a good spatial and
temporal coverage provided by the railway network. Positive values for βcar availability and
βhigh income indicate that people owning a car or having higher incomes prefer to use car as
their mode of choice as expected. A positive βold age in rail utility function indicates that
old people prefer to use railway over car which is as expected. A negative value of
βwork in suburbs in the rail utility function indicates that people working in suburbs prefer to
use car over the rail. This is also plausible due to the fact that they mostly commute to
industrial areas out of the city sparsely populated and with lesser railway coverage than
the central Tokyo. A negative value of the βyoung in railway utility function indicates that
young people prefer to use car over the railway. The definition of young in this case is
people less than 30 years of age, which are mostly either students or company workers
just starting their careers. This trend can be explained as a counter to the old people’s
preference for rail.
4.2 Nested logit models
The MNL models estimated in the previous section, assume no correlation among the
alternatives but some of the choices especially in case of departure time choice may be
intrinsically correlated. To capture the effect of these correlations, different nested logit
structures as depicted in the Figure 2 to Figure 5 are estimated using the same data and
the utility specifications including the level of service and personal attributes.
The results of these four nested logit models are reported in Table 3. The results were
estimated using the MNL parameters as initial values. All the parameters are significant
in all the four models at more than 95% significance level except the βold age in nested
model with 4 nests where it is significant at 94% level. Results indicate that all the four
nesting structures are significant. Statistical tests indicate that nesting parameter in all
166
the four models is significantly different from null and unit hypothesis values.
Likelihood ratio tests comparing all the nesting models to corresponding MNL model are
satisfied at more than 99th percentile of a χ2 random variable with one degree of freedom.
The value of travel time shows some differences from the MNL model, for example in
case of 2 nests it is higher while in all the other nests it is lower than the MNL model.
Values for early arrival and late arrival penalties are also found to be resilient to the
changes in correlation structure.
TABLE 3: Estimation results of NL models of Figures 2, 3, 4, and 5
Coefficients 2 Nest NL model
(Figure 2) 3 Nest NL model
(Figure 3) 3 Nest NL model
(Figure 4) 4 Nest NL model
(Figure 5)
ASCRail 2.7988 (5.04) 5.1608 (3.66) 2.9832 (5.24) 5.0572 (3.25)
βtravel time -0.0414 (-3.78) -0.0785 (-3.54) -0.0495 (-3.48) -0.0767 (-3.15)
βcost -0.0010 (-3.30) -0.0025 (-3.53) -0.0014 (-3.22) -0.0020 (-3.07)
βearly arrival -0.0231 (-6.51) -0.0463 (-4.68) -0.0309 (-4.63) -0.0458 (-3.99)
βlate arrival -0.1026 (-7.15) -0.2616 (-4.04) -0.1587 (-4.31) -0.2455 (-3.60)
βcar availability 1.4848 (4.54) 2.8604 (3.43) 1.5469 (4.68) 2.7300 (3.08)
βold age 2.0729 (2.36) 3.9534 (2.00) 2.2378 (2.20) 3.7227 (1.92)*
βhigh income 1.7708 (3.41) 3.4333 (2.94) 2.0178 (3.44) 3.2418 (2.75)
βyoung -0.8750 (-2.65) -1.6697 (-2.29) -1.0335 (-2.47) -1.6117 (-2.18)
βwork in suburbs -1.6012 (-4.74) -3.1012 (-3.58) -1.6217 (-5.54) -2.9553 (-3.28)
μ(0) 0.6275 (5.67) 0.3254 (3.98) 0.5486 (4.67) 0.3461 (3.49)
(1) (-3.37) (-8.26) (-3.84) (-6.60)
No. of observations 1324 1324 1324 1324
No. of parameters 11 11 11 11
Null-log likelihood -1454.6 -1454.6 -1454.6 -1454.6
Final-log likelihood -978.9 -969.8 -977.4 -974.6
Rho-Squared 0.327 0.333 0.328 0.330
Rho-Squared bar 0.319 0.326 0.321 0.322
VTTS(yen/min) 42.2 31.1 35.8 37.7
VEAP(yen/min) 23.5 18.4 22.4 22.5
VLAP(yen/min) 104.5 103.8 114.9 120.6
Nested logit model shown by the correlation structure in Figure 3 performs better than
the other nesting structures. The fact that other nesting structures are also significant
indicates the presence of cross-nesting which is investigated in next section.
4.3 Cross-nested logit models
As reported in Section 4.2, four different correlation structures are found to be
significant indicating the possibility of the cross-nesting among different alternative sets.
To test this hypothesis, we developed different cross-nesting structures by combining the
nesting structures shown in Figures 2-5 and two of these have been found to be
significantly better than the nested models (see Figure 6).
2134
EEA LEA OT ELA LLA RAIL
RAI L
CAR- EA CAR-LA
Max. Sch. Delay
(A) (B)
FIGURE 6: Two cross-nesting structures
167
Cross-nesting structure (A) shows that departure time alternatives using car are not
only grouped together based on their order but also by schedule delay associated with
them On-time arrival alternative also belongs to two nests i.e. early and late arrival nests
indicating its proximity to both. Cross-nesting structure (B) also indicates that departure
time alternatives using car are not only grouped together based on their order but also by
schedule delay associated with them. For example, nest 1 shows the alternatives of early
arrival and on-time arrival using car as grouped together and nest 4 shows that
alternatives having late arrival time are grouped together but at the same time Earliest
Early Arrival (EEA) and Latest Late Arrival (LLA) are also found to be nesting together
in nest 3 indicating that departure time alternatives are not only correlated due to their
proximity to each other but also due to the schedule delay associated with them. This
nesting by schedule delay hypothesis is further strengthened by the observation that on-
time arrival by car which shares a nest with early arrival using car options also share nest
with the on-time arrival using rail.
Table 4 shows the estimation results for the two cross-nesting structures shown in
Figure 6. All the attribute parameters, nesting parameters (μ's) as well as cross-nest share
parameters (α's) of alternatives are found to be significant. The log-likelihood values
show significant improvement over the nested logit models shown in Section 4.2 with
one and three extra parameters for cross-nesting structure (A) and (B).
TABLE 4: Estimation results of cross-nested logit models
Coefficients Cross-nesting structure (A) Cross-nesting structure (B)
ASCRail 10.2 (3.44) 1.037 (3.15)
βtravel time -0.172 (-2.76) -0.1692 (-2.8)
βcost -0.00541 (-2.73) -0.0046 (-3.28)
βearly arrival -0.0853 (-3.62) -0.0830 (-3.82)
βlate arrival -0.516 (-3.52) -0.516 (-3.57)
βcar availability 5.46 (3.29) 5.5140 (3.10)
βold age 7.61 (2.12) 7.7750 (2.13)
βhigh income 6.38 (2.66) 6.7083 (2.68)
βyoung -3.2 (-2.26) -3.0507 (-2.12)
βwork in suburbs -5.92 (-3.35) -5.9807 (-3.22)
μ(0) 0.168 (3.57) 0.166 (3.58)
(1) (-17.6) (-17.95)
αEEA,1(1) 0.734 (-2.86) 0.762 (-2.6)
αEEA,2(1) 0.266 (-7.91) -- --
αEEA,3(1) -- -- 0.238 (-8.3)
αOT,1(1) 0.505 (-6.28) 0.745 (-3.8)
αOT,2(1) -- -- 0.255 (-11.1)
αOT,3(1) 0.495 (-6.41) -- --
αLLA,2(1) -- -- -- --
αLLA,3(1) 0.556 (-2.94) 0.464 (-3.5)
αLLA,4(1) 0.444 (-3.67) 0.536 (-3.04)
No. of observations 1324 1324
No. of parameters 17 14
Null-log likelihood -1454.56 -1454.56
Final-log likelihood -958.7 -958.56
Rho-Squared 0.341 0.341
Rho-Squared bar 0.329 0.329
VTTS(yen/min) 31.8 36.8
VEAP(yen/min) 15.8 18.0
VLAP(yen/min) 95.4 112.2
168
Values of travel time savings for cross-nesting structure (A) is 28 yen/min while the
value of early arrival penalty and late arrival penalty are 19 and 98 yen/min respectively.
Nesting structure (B) reflect a value of travel time savings of around 37 yen/min and
values of late and early arrival penalties of around 18 and 112 yen/min. Cross-nesting
parameters (α's) are all significantly different from zero and one, hence indicating that
they are not dominantly contained in a single nest. It also confirms the fact that different
correlation structures co-exist among different alternatives as indicated separately by
different nesting structures being significant at the same time in previous section.
In order to evaluate the relative improvement in model performance due to an
additional nest for maximum schedule delay, cross-nested model shown in Figure 6 (A)
is compared with another model without the cross-nesting structure. These model
structures are shown in Figure 7. Estimation results for these models are provided in
Table 5.
TABLE 5: Estimation results for comparison of cross-nested logit models
Coefficients Cross-nesting structure
Figure 7 (A) Cross-nesting structure – relaxed
Figure 7 (B)
ASCRail 10.2 (3.44) 3.14 (4.91)
βtravel time -0.172 (-2.76) -0.0588 (-3.7)
βcost -0.00541 (-2.73) -0.00198 (-3.48)
βearly arrival -0.0853 (-3.62) -0.0193 (-4.55)
βlate arrival -0.516 (-3.52) -0.139 (-5.16)
βcar availability 5.46 (3.29) 1.61 (4.31)
βold age 7.61 (2.12) 2.20 (2.37)
βhigh income 6.38 (2.66) 1.91 (3.19)
βyoung -3.2 (-2.26) -0.944 (-2.55)
βwork in suburbs -5.92 (-3.35) -1.75 (-4.59)
μ(0) 0.168 (3.57) 0.58 (5.54)
(1) (-17.6) (-4.01)
αEEA,1(1) 0.734 (-2.86) -- --
αEEA,2(1) 0.266 (-7.91) -- --
αEEA,3(1) -- -- -- --
αOT,1(1) 0.505 (-6.28) 1.0 --
αOT,2(1) -- -- 0.0 --
αOT,3(1) 0.495 (-6.41) -- --
αLLA,2(1) -- -- -- --
αLLA,3(1) 0.556 (-2.94) -- --
αLLA,4(1) 0.444 (-3.67) -- --
No. of observations 1324 1324
No. of parameters 17 13
Null-log likelihood -1454.56 -1454.56
Final-log likelihood -958.7 -968.47
Rho-Squared 0.341 0.334
Rho-Squared bar 0.329 0.325
VTTS(yen/min) 31.8 29.7
VEAP(yen/min) 15.8 9.8
VLAP(yen/min) 95.4 70.2
Results in Table 5 compare the two models with and without the maximum schedule
delay nest and the comparative results indicate that the model with the maximum
schedule delay nest is better than the model without maximum schedule delay nest. This
result is also confirmed using the likelihood ratio test. Similar analysis was also
conducted for smaller schedule delays. It has been found that addition of another nest for
smaller schedule delays does not improve significance of the model and the small early
169
and late schedule delays are not correlated at all. This can be explained owing to
different perceptions attached to small early or late schedule delays. For example, being
early by 15 minutes may be less onerous compared to being late for 15 minutes and
people prefer to arrive a little earlier than being late indicating different perceptions and
hence no correlation for smaller early and late schedule delays. On the other hand, the
longer schedule delays are equally undesirable no matter whether early or late and are
hence correlated.
2134
EEA LEA OT ELA LLA RAIL
RAI L
CAR- EA CAR- LA
Max. Sch. Delay
134
EEA LEA OT ELA LLA RAIL
RAI L
CAR- EA CAR-LA
(A) (B)
FIGURE 7: Cross-nesting structures with and without maximum schedule delay nest
4.4 Random coefficient multinomial logit models
As discussed before, it is important to account for the random taste variations across
the individuals. We tried to explore the attributes that are perceived and treated
differently among the population of the commuters. The variables indicating late arrival
and work in suburbs have significant random coefficients. It is assumed that these
random coefficients are normally distributed.
Results of random coefficient estimation for different random coefficient specifications
are shown in Table 6. First mixed MNL model is built using work in suburbs as random
variable. All the parameters are significant at more than the 95% significant level but the
overall improvement in the model fitness is not very high (2(L(MMNL)-L(MNL)) = 3.0
> 2.706, 90th percentile of a χ2 random variable with one degree of freedom). The value
of travel time savings as well as value of early arrival and late arrival penalties remains
same as the MNL model.
Second model is estimated using work in suburbs as well as the late arrival as random
variables. All the parameters are significant at the 95% significance level. The
improvement in log-likelihood is about 25 units over the MNL model indicating that this
mixed MNL is better than MNL model at 99% significance level indicating important
gains in model performance obtained by using the random coefficient model. The
normally distributed work in suburbs parameter has a distribution of N(-0.994,2.14)
indicating that about 32% of the commuters who work in suburbs have positive utility
for rail. This effect was not captured by using the fixed parameter MNL model. On the
other hand, it can be noted that use of normal distribution for the late arrival penalty
results in a parameter distribution of N(-0.3023,0.17), which indicates that about 4% of
the commuters get positive utility from being late which is counter-intuitive. Although
this number is not very high but it would be better if a log-normal distribution is tried
which is constrained to remain in a single sign domain. Another interesting result
depicted by this model is a very high value for the late arrival penalty which is more than
double the value in MNL model with a very broad distribution. The Value of late arrival
penalty has a distribution of N(252, 148). Higher variance in late arrival distribution can
be explained by differences of personal preferences as the value of late arrival penalty
may be very high for an office worker or executive while can be low for a student or
170
people with flexible work schedules. This effect can only be modelled using the random
coefficient models.
Third model described in Table 6 shows the results for mixed logit estimation; early
and late arrival and work in suburb parameters are normally distributed. Results indicate
a decrease in the significance level for some of the parameters below 95% level. Also, it
is clear that the early arrival penalty has a very narrow though significant distribution
N(-0.0217,0.0011) indicating that a point estimate of the parameter is enough to
represent it. Hence, model including the work in suburbs as well as late arrival penalty as
random coefficients is retained for the further investigations.
TABLE 6: Estimation results of mixed MNL model of Figure 1
Coefficients Work in suburbs as
random variable Work in suburbs + late
arrival as random variable Late arrival + early arrival
as random variable
ASCRail 1.8588 (8.59) 1.6035 (6.24) 1.6090 (7.83)
βtravel time -0.0337 (-3.85) -0.0380 (-3.87) -0.0372 (-4.35)
βcost -0.0009 (-3.32) -0.0012 (-3.73) -0.0009 (-3.25)
βearly arrival -0.0198 (-6.94) -0.0273 (-8.30) -0.0217 (-8.63)
σearly arrival -- -- -- -- 0.0011 (2.04)
βlate arrival -0.0936 (-7.65) -0.3023 (-5.07) -0.2545 (-5.10)
σlate arrival -- -- -0.17 (2.99) -0.1504 (-4.65)
βcar availability 1.0421 (6.34) 1.1042 (5.86) 0.8612 (5.96)
βold age 1.5960 (2.52) 2.0469 (2.99) 0.5516 (1.53)*
βhigh income 1.1968 (4.16) 1.1761 (3.80) 0.5058 (1.68)*
βyoung -0.5784 (-2.60) -0.5792 (-2.25) -0.2860 (-1.39)*
βwork in suburbs -0.9832 (-6.13) -0.9942 (-5.16) -1.013 (-7.50)
σwork in s uburbs 1.4377 (2.26) 2.1410 (2.60) -- --
No. of observations 1324 1324 1324
No. of parameters 11 12 12
Null-log likelihood -1454.6 -1454.6 -1454.6
Final-log likelihood -981.3 -958.84 -968.32
Rho-Squared 0.325 0.341 0.334
Rho-Squared bar 0.320 0.333 0.326
VTTS(yen/min) 37.4 31.7 41.3
VEAP Mean (yen/min) 22 22.8 24.1
VEAP Variance (yen/min) -- -- 1.2
VLAP Mean (yen/min) 104 251.9 282.8
VLAP Variance (yen/min) -- 147.7 167.1
4.5 Random coefficient nested logit models
Nested logit models described in Section 4.2 provide an improvement over the simple
MNL model by capturing the correlation among the alternatives while the mixed logit
models described in Section 4.3 improve upon the MNL model by accounting for the
random taste heterogeneity. In this section, we combine these two types of models to
jointly account for the correlation among the alternatives as well as the random taste
variations among the population of the commuters. Same nesting structures as described
in Section 3.4 and used in Section 4.2 are employed here and corresponding mixed
nested logit model are estimated.
Table 7 details the results of the estimations for the four nesting structures. All the
parameters are statistically significant at 95% confidence level except the variance for
the work in suburbs random variable which loses significance at any suitable level for
two of the nesting structures.
171
TABLE 7: Estimation results of mixed NL models of Figures 2, 3, 4, and 5
Coefficients 2 nest NL model
(Figure 2) 3 nest NL model
(Figure 3) 3 nest NL model
(Figure 4) 4 nest NL model
(Figure 5)
ASCRail 5.1259 (3.45) 3.2958 (4.65) 4.1796 (3.41) 3.8949 (3.39)
βtravel time -0.0810 (-3.73) -0.0701 (-3.50) -0.0804 (-3.61) -0.0783 (-3.25)
βcost -0.0025 (-3.54) -0.0023 (-3.84) -0.0027 (-3.65) -0.0023 (-3.42)
βearly arrival -0.0531 (-5.53) -0.0525 (-5.65) -0.0525 (-5.27) -0.0528 (-4.74)
βlate arrival -0.5833 (-4.02) -0.6575 (-3.82) -0.7150 (-3.23) -0.6589 (-3.42)
σlate arrival -0.3609 (-3.66) -0.3884 (-3.62) -0.4235 (-3.05) -0.3894 (-3.25)
βcar availability 3.1643 (3.45) 2.1580 (4.50) 2.7010 (3.66) 2.5783 (3.41)
βold age 4.6176 (2.28) 4.3912 (2.64) 4.1469 (2.47) 4.3479 (2.47)
βhigh income 3.6260 (2.86) 2.4600 (3.64) 3.0031 (2.95) 2.7488 (2.89)
βyoung -1.8084 (-2.25) -1.3416 (-2.24) -1.5085 (-2.16) -1.3895 (-2.04)
βwork in suburbs -3.3823 (-3.57) -2.0117 (-4.17) -2.7152 (-3.15) -2.3780 (-3.35)
σwork in suburbs 0.3484 (0.21)* 4.6790 (2.87) 2.5022 (0.7)* 3.9974 (1.96)
μ(0) 0.2976 (3.88) 0.4394 (5.37) 0.3711 (3.77) 0.4157 (3.99)
(1) (-9.16) (-6.85) (-6.39) (-5.61)
No. of observations 1324 1324 1324 1324
No. of parameters 13 13 13 13
Null-log likelihood -1454.6 -1454.6 -1454.6 -1454.6
Final-log likelihood -945.4 -948.4 -948.3 -951.8
Rho-Squared 0.350 0.348 0.348 0.346
Rho-Squared bar 0.341 0.339 0.339 0.337
VTTS(yen/min) 32.4 30.5 29.8 34
VEAP(yen/min) 21.2 22.8 19.5 23
VLAP(yen/min) 233.3 286 265 287
VLAP Variance
(yen/min) 144.4 169 157 169
Comparison of the estimated model results with the corresponding mixed MNL model
indicates an improvement in the overall model fitness at a significance level of over
99.5% with a single degree of freedom indicating the gains in performance of the model.
Similar observations can be made by comparing the mixed nested logit models with
corresponding nested logit models in Section 4.2. The improvement in log-likelihood
over the nested logit models of the Section 4.2 with 2 degrees of freedom is statistically
significant at 99.5% level.
The value of travel time savings is reduced in comparison to MNL and NL models and
is around 30 to 35 yen per min in this case while the value of early arrival penalty is
consistent at about 20 to 23 yen/min. The value of late arrival penalty show a distribution
with a mean of around 250 to 300 yen and corresponding variance of around 140-170
yen which is consistent with the results obtained for the mixed MNL model in the
previous section.
Mixed NL model corresponding to nesting structure shown in Figure 2 has better
goodness-of-fit compared to other models but one of the parameters is insignificant.
Mixed NL models corresponding to nesting structures shown in Figures 3 and 4 show
similar goodness-of-fit measure. In case of NL models, nesting structure corresponding
to Figure 3 had better goodness-of-fit than other models and in case of mixed NL models,
this nesting structure again has the best goodness-of-fit with all parameters being
significant. One of the trends observed in all the above proposed models is a gradual
decrease in the significance levels of the parameters though they are still significant at
95% confidence level. This is quite expected as each subsequent modelling structure
introduced above, decomposes the error term further than the previous models.
172
4.6 Error component logit models
Mixed logit models can also be explained from another interpretational viewpoint and
that is to model the correlation structures among the alternatives. Error component logit
models can virtually approximate any other modelling structure for discrete choices
provided appropriate specifications are used. This section describes four error
component logit models corresponding to four nested logit structures depicted in Section
3.4 and used in Sections 4.2 and 4.4. Random error terms are introduced across
alternatives which remain constant for a given set of alternatives hence, indicating the
correlation among them.
Table 8 shows the results of the estimations for the error component logit models
corresponding to different nested logit correlation structures. Model results indicate that
for three out of four correlation structures (namely three and four nest models) error
component logit outperforms the corresponding nested logit models while for one (2 nest
model) it is statistically same at a higher significance level. This shows that error
component logit models can better capture the correlation structures as compared to the
nested logit models.
TABLE 8: Estimation results of error component logit models
Coefficients Corresponding to 2
nest NL model
(Figure 2)
Corresponding to 3
nest NL model
(Figure 3)
Corresponding to 3
nest NL model
(Figure 4)
Corresponding to 4
nest NL model
(Figure 5)
ASCRail 2.7484 (4.84) 7.5034 (2.90) 3.1556 (7.98) 7.6417 (2.74)
βtravel time -0.0409 (-3.72) -0.0794 (-3.24) -0.0430 (-3.51) -0.0824 (-2.98)
βcost -0.0010 (-3.28) -0.0027 (-3.47) -0.0015 (-3.54) -0.0024 (-3.24)
βearly arrival -0.0230 (-6.38) -0.0510 (-4.58) -0.0388 (-5.44) -0.0570 (-4.51)
βlate arrival -0.1023 (-7.10) -0.1253 (-3.28) -0.0593 (-3.81) -0.1413 (-2.92)
βcar availability 1.4354 (4.43) 3.4943 (2.81) 1.4040 (5.87) 3.5820 (2.56)
βold age 1.8382 (2.37) 3.6231 (1.5)* 1.9122 (1.8)* 3.8950 (1.54)
βhigh income 1.7812 (3.24) 4.5224 (2.69) 1.9833 (4.51) 4.7666 (2.48)
βyoung -0.8291 (-2.55) -2.1808 (-2.12) -0.9903 (-3.03) -2.4483 (-2.05)
βwork in suburbs -1.5874 (-4.51) -2.92 (-2.92) -1.4818 (-7.06) -3.9019 (-2.62)
ξcar -0.1145 (-0.2)*
ξrail 1.9681 (3.08) 4.4354 (2.66) 4.9559 (2.32)
ξon-time 0.0134 (0.12)* -0.0116 (-0.12)*
ξearly arrival 3.7514 (4.42) 6.4209 (2.85)
ξlate arrival 0.5276 (0.9)* -0.0123 (-0.3)* -0.1413 (1.6)*
ξearly arrival/on-time 6.5455 (2.87)
No. of observations 1324 1324 1324 1324
No. of parameters 12 13 13 14
Null-log likelihood -1454.6 -1454.6 -1454.6 -1454.6
Final-log likelihood -979.6 -965.3 -967.7 -961.6
Rho-Squared 0.326 0.336 0.335 0.339
Rho-Squared bar 0.318 0.327 0.326 0.329
VTTS(yen/min) 41 29.4 29 34.3
VEAP(yen/min) 23 19 26 24
VLAP(yen/min) 102 46 40 59
All the random error component terms are assumed as normally distributed with mean
zero i.e. N(0,σ) where σ
is estimated. Results indicate that not all the random error
terms are statistically significant. The other parameters are usually significant at 95%
confidence level except βold age whose significance reduces to 85% and 90% confidence
levels for two of the models. Value of travel time savings as well as value of early arrival
173
penalty remains almost same as in previous models while the value of late arrival penalty
reduces significantly in this model. This may have resulted due to interaction among the
late arrival penalty parameter and the error components.
Comparison of these error component logit models with the mixed logit model shows
them at par but inferior to the mixed nested logit models. This may be explained by the
fact that proposed error component structures just capture the correlation across the
alternatives while the mixed nested logit models also account for the random taste
heterogeneity in addition to inter-alternative correlation structures.
5. OVERVIEW OF THE ESTIMATION RESULTS
All model estimation results are summarized in Table 9. The table shows the final log-
likelihood values and the number of estimated parameters.
TABLE 9: Summary of estimation results
Model Final log-likelihood Number of estimated
parameters
Null -1454.6 0
MNL with LOS variables only -1052.8 5
MNL with both SE and LOS variables -982.9 10
NL_1 (structure defined in Figure 2) -978.9 11
NL_2 (structure defined in Figure 3) -977.4 11
NL_3 (structure defined in Figure 4) -969.8 11
NL_4 (structure defined in Figure 5) -974.6 11
Cross nesting structure (A) -958.7 17
Cross nesting structure (B) -958.5 14
Mixed MNL (work in suburbs) -981.3 11
Mixed MNL (work in suburbs + early arrival) -958.8 12
Mixed MNL (early arrival + late arrival) -968.3 12
Mixed_NL_1 -945.4 13
Mixed_NL_2 -948.4 13
Mixed_NL_3 -948.3 13
Mixed_NL_4 -951.8 13
Error components_NL_1 -979.6 12
Error components_NL_2 -965.3 13
Error components_NL_3 -967.7 13
Error components_NL_4 -961.6 14
It can be inferred from the results that inclusion of both socio-demographic i.e.
personal characteristics and level of service attributes greatly improves the model
performance. This result was verified for the simple MNL model. Four different nesting
structures were tested to account for correlation among alternatives and all of them were
found to be significantly better than the simple MNL structure. Significance of all the
four nesting structures indicated the presence of cross-nesting. Different cross-nesting
structures were tried and two structures were found significant. All the attribute
coefficients, nesting parameters and cross-nest share parameters of the alternatives were
found to be significant.
Random coefficient models using few normally distributed parameters were estimated
and were found to perform better than corresponding MNL models indicating that they
can capture the taste heterogeneity across the users. Only two attributes namely late
arrival penalty and work in suburbs were found to have significant random coefficients.
Taste heterogeneity was found to be significant in case of work in suburbs attribute
where the distribution of the parameter indicated that about 32% of the commuters will
174
have a positive utility for using trains to work while in the case of the MNL model the
parameter estimated for work in suburbs had a negative value indicating the reduction in
utility for all the commuters in the population.
Taste heterogeneity also indicated distributed value of late arrival penalty across the
individuals, indicating differences between the commuters. This can be explained by the
existence of different commuter groups such as office workers or executives for whom it
is important to arrive on-time in contrast to a student or a worker with a flexible arrival
time indicating a lower value of late arrival penalty.
Mixed nested logit models perform better than the nested logit and mixed MNL
models, indicating that they can jointly capture the correlation structures as well as
random taste heterogeneity.
Error component logit models were developed corresponding to the correlation
structures of nested logit models and mostly perform better than corresponding nested
logit models. However, these models did not outperform the mixed nested logit models.
6. CONCLUSIONS
In this paper, results from a comprehensive study of mode and departure time choice
problem are presented. The database was formed from a SP study performed among
Tokyo commuters. The paper tested both closed-form GEV structures (MNL, NL and
CNL) and more advanced mixed logit forms. In this aspect, the main contribution of the
paper is to provide a step-by-step comparison of the models, in an attempt to illustrate
the relative improvement of the more general model forms.
As expected, nested logit models perform better than the MNL. With respect to the
four proposed nesting structures, nesting structure shown in Figure 3, having three nests
of rail, car with early/on-time arrival and car with late arrival performs better than other
nesting structures but the other nesting structures are also found to be found significant
indicating the possibility of different alternatives belonging to more than one nest.
Further analysis using CNL models with different nesting structures confirmed it.
Reported results show that cross-nesting of the alternatives is significant and cross-
nested logit models show significant improvement in model fitness over corresponding
nested logit models. It should be stressed that as with the NL models, several possible
CNL model structures can be specified, and not all specifications result in statistically
significant nesting coefficients.
This paper tested different logit kernel specifications for the mixed logit, namely the
MNL and NL models. The mixed MNL performs better than the MNL by accounting for
the random taste variations, and similarly the mixed nested logit models perform better
than NL models. In addition, the mixed NL model outperforms the mixed MNL model
by accounting for both correlation structure as well as random taste variations jointly.
Finally, Error component logit corresponding to the correlation structures of the NL
models were developed and although they perform better than corresponding NL models,
they were not found superior than mixed NL models.
Level of service variables such as travel time, cost and schedule delay as well as
socioeconomic characteristics such as age, income, car availability and work locations
are found to be significantly affecting the mode and departure time choices of
commuters. It has been found that the inclusion of socioeconomic characteristics not
only significantly improves the statistical significance of the estimated models but also
greatly enhances the explanatory power of these behavioural models. The paper shows
that departure time alternatives are correlated by the amount of delay. Results indicate
175
that the late arrival penalties for the commuters are higher in the Tokyo Metropolitan
Area. Choice of departure time is found to be sensitive to the schedule constraints as
well as congestion levels and costs while the choice of mode is found sensitive to the age,
income level, car availability as well as work locations of the commuters.
The models estimated in this paper will be incorporated in a dynamic multi-modal
transport simulation model. Given the results obtained in this paper, it has been found to
be important to include the behavioural characteristics of the commuters in the
multimodal network assignment models. Such behavioural multimodal network
assignment models are important to bridge the gap between disaggregate behavioural
models and the network assignment models which are mostly based on level of service
variables. Although the mixed multinomial logit models have been found to perform
slightly better than the advanced closed-form models such as cross-nested logit models
but for practical application closed-form models may be advantageous due to relative
simplicity and ease to incorporate them in network assignment models. This dynamic
multi-modal transport simulation model will be applied to the multi-modal transportation
network in the Tokyo Metropolitan Area.
ACKNOWLEDGEMENTS
Authors are thankful to the two anonymous referees who provided insightful
comments to help improve this paper.
REFERENCES
Abkowitz, M. (1981) An analysis of the commuter departure time decision.
Transportation, 10, 283-297.
Bhat, C. (1998a) Analysis of travel mode and departure time choice for urban shopping
trips. Transportation Research Part B, 32, 361-371.
Bhat, C. (1998b) Accommodating flexible substitution patterns in multi-dimensional
choice modeling: formulation and application to travel mode and departure time choice.
Transportation Research Part B, 32, 455-466.
Bhat, C. and Steed, J. (2002) A continuous-time model of departure time choice for
urban shopping trips. Transportation Research Part B, 36, 207-224.
Bierlaire, M. (2005) An Introduction to BIOGEME (Version 1.4).
http://biogeme.epfl.ch/doc/tutorial.pdf.
Boyd, J. and Mellman, J. (1980) The effect of fuel economy standards on the U.S.
automotive market: a hedonic demand analysis. Transportation Research Part A, 14,
367-378.
Cardell, N. and Dunbar, F. (1980) Measuring the societal impact of automobile
downsizing. Transportation Research Part A, 14, 432-434.
Cascetta, E. and Papola, A. (2003) A joint mode-transit service choice model
incorporating the effect of regional transport service timetables. Transportation
Research Part B, 37, 595-614.
Chin, A. (1990) Influences on commuter trip departure time decisions in Singapore.
Transportation Research Part A, 24, 321-333.
de Jong, G., Daly, A., Pieters, M., Vellay, C., Bradley, M. and Hofman, F. (2003) A
model for time of day and mode choice using error components logit. Transportation
Research Part E, 39, 245-268.
176
de Palma, A., Ben-Akiva, M., Lefevre, C. and Litnas, N. (1983) Stochastic equilibrium
model of peak period traffic congestion. Transportation Science, 17, 430-453.
Fargier, P. (1981) Effects of the choice of departure time on road traffic congestion.
Proceedings of the Eighth International Symposium on Transportation and traffic
Theory, Toronto, Canada.
Henderson, J. (1974) Road congestion: a reconsideration of pricing theory. Journal of
Urban Economics, 1, 346-365.
Hendrikson, C. and Kocur, G. (1981) Schedule delay and departure time decisions in a
deterministic model. Transportation Science, 15, 62-77.
Hendrickson, C. and Plank, E. (1984) The flexibility of departure times for work trips.
Transportation Research Part A, 18, 25-36.
Hendrickson, C., Nagin, D. and Plank, E. (1981) Characteristics of travel time and
dynamic user equilibrium for travel-to-work. Proceedings of the Eighth International
Symposium on Transportation and Traffic Theory, Toronto, Canada.
Hess, S., Bierlaire, M. and Polak, J. (2004) Development and application of a mixed
cross-nested logit model. European Transport Conference, Strasbourg.
Hurdle, V. (1981) Equilibrium flows on urban freeways. Transportation Science, 5, 255-
293.
Koppelman, F.S. and Wen, C.H. (2000) The paired combinatorial logit model: properties,
estimation and application. Transportation Research Part B, 34, 75-89.
Kuwahara, M. (1985) A time-dependent network analysis for highway commute traffic
in a single core city. Ph.D Thesis, University of California, Berkeley.
Kuwahara, M. and Newell, G. (1987) Queue evolution on freeways leading to a single
core city during the morning peak. Proceedings of the 10th International Symposium
on Transportation and Traffic Theory, Boston, pp. 21-40.
Louviere, J., Hensher, D. and Swait, J. (2000) Stated Choice Methods: Analysis and
Applications in Marketing, Transportation and Environmental Valuation. Cambridge
University Press, Cambridge.
McFadden, D. (1978) Modeling the choice of residential location. In A. Karlqvist et al.
(eds.) Spatial Interaction Theory and Residential Location, North-Holland, Amsterdam.
McFadden, D. and Train, K. (2000) Mixed MNL models for discrete response. Journal
of Applied Econometrics, 15, 447-470.
Papola, A. (2004) Some developments on the cross-nested logit model. Transportation
Research Part B, 38, 833-851.
Polak, J.W. and Jones, P.M. (1994) A tour-based model of journey scheduling under
road pricing. 73rd Annual Meeting of the Transportation Research Board, Washington,
DC.
Small, K. (1982) The scheduling of consumer activities: work trips. The American
Economic Review, 72,, 467-479.
Small, K. (1987) A discrete choice model for ordered alternatives. Econometrica, 55,
409-424.
Shimizu, T. and Yai, T. (1999) An analysis of the effect on peak period toll in Tokyo
Metropolitan Expressway. Journal of the Eastern Asia Society for Transportation
Studies, 3, 305-318.
Train, K. (2003) Discrete Choice Methods with Simulation. Cambridge University Press,
New York.
Vickrey, W. (1969) Congestion theory and transport investment. The American
Economic Review, 59, 251-261.
177
Vovsha, P. (1997) Application of a cross-nested logit model to mode choice in Tel Aviv,
Israel, Metropolitan Area. Transportation Research Record, 1607, 6-15.
Vovsha, P. and Bekhor, S. (1998) The link-nested logit model of route choice:
overcoming the route overlapping problem. Transportation Research Record, 1645,
133-142.
Walker, J. (2001) Extended discrete choice models: integrated framework, flexible error
structures and latent variable. Ph.D. Thesis, Massachusetts Institute of Technology.
Wang, J. (1996) Timing utility of daily activities and its impact on travel. Transportation
Research Part A, 30, 189-206.
Wen, C.H. and Koppelman, F.S. (2001) The generalized nested logit model.
Transportation Research Part B, 35, 627-641.
