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How to Incorporate the Spatial
Dimension in Destination Choice
Models: The Case of Antwerp
a
Hakim Hammadou , Isabelle Thomas
d
Verhetsel & Frank Witlox
a b c
, Ann
e
a
Department of Geography , Université Catholique de
Louvain , Louvain-la-Neuve, Belgium
b
Centre of Operational Research and Econometrics
(CORE) , Louvain-la-Neuve, Belgium
c
National Fund for Scientific Research , Brussels,
Belgium
d
Department of Transport and Regional Economics ,
University of Antwerp , Belgium
e
Department of Geography , Ghent University ,
Belgium
Published online: 13 Mar 2008.
To cite this article: Hakim Hammadou , Isabelle Thomas , Ann Verhetsel & Frank
Witlox (2008) How to Incorporate the Spatial Dimension in Destination Choice Models:
The Case of Antwerp, Transportation Planning and Technology, 31:2, 153-181, DOI:
10.1080/03081060801948126
To link to this article: http://dx.doi.org/10.1080/03081060801948126
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Transportation Planning and Technology, April 2008
Vol. 31, No. 2, pp. 153181
ARTICLE
How to Incorporate the Spatial
Dimension in Destination Choice
Models: The Case of Antwerp
HAKIM HAMMADOU*, ISABELLE THOMAS*,**,$,
ANN VERHETSEL$$ & FRANK WITLOX§
*Department of Geography, Universite´ Catholique de Louvain, Louvain-la-Neuve,
Belgium; **Centre of Operational Research and Econometrics (CORE), Louvain-laNeuve, Belgium; $National Fund for Scientific Research, Brussels, Belgium;
$$Department of Transport and Regional Economics, University of Antwerp, Belgium
& §Department of Geography, Ghent University, Belgium
(Received 3 May 2006; Revised 10 January 2008; In final form 17 January 2008)
ABSTRACT This paper presents and estimates destination choice models based on
a large sample of intra-urban trips. Particular attention is paid to incorporating
the effects of the spatial dimension. The data used relate to non-work trips in the
agglomeration of Antwerp (Belgium). A geographical analysis is performed in
order to represent the city and its suburbs by a limited set of zones of destinations
and to characterize these zones in terms of land use. Different types of discrete
choice model are compared in terms of utility function, global formulation and
performance. The mixed nested logit formulation with random coefficients
appears to be the most attractive. The results confirm the difficulty of grasping
spatial realities by simple quantitative measurements but also illustrate the
importance of ‘space’ when choosing a destination. The empirical results also
show that land use and urban development policies clearly have their effect on
urban mobility.
KEY WORDS: Destination choice model; mixed nested logit; trip-based approach;
land use; urban mobility; Antwerp
Correspondence Address: Frank Witlox, Department of Geography, Ghent University, Belgium.
Email: [email protected]
ISSN 0308-1060 print: ISSN 1029-0354 online # 2008 Taylor & Francis
DOI: 10.1080/03081060801948126
154 H. Hammadou et al.
Introduction
Travel demand analysis is intrinsically spatial: the spatial separation of
activities is indeed the essence of travel demand. Although this assertion
is self-evident, in travel analysis and even more in travel behaviour
modelling, the incorporation of the effects of the spatial distribution of
activity-based travel has only recently been taken explicitly into account.
Important contributions in this respect are by Dijst and Vidakovic
(1997), Badoe and Miller (2000), Ewing and Cervero (2001), Stead
(2001), Bhat and Zhao (2002), Kitamura (2004) and Handy et al.
(2005). In each case an attempt was made to sort out the extent to which
the characteristics of the built environment and land use have an impact
on travel behaviour. The present paper sets out to achieve an identical
goal. It aims at representing, modelling and understanding travel
behaviour and destination choices, but differs from previous analyses
in a number of aspects: (i) the number of so-called spatial variables is
extended; (ii) different types of destination choice models are being
compared; (iii) use is made of activity-based data (instead of crosssectional data); and (iv) the emphasis is on explaining destination choice
(instead of trip length, trip frequency, or modal choice).
First, our paper attempts to incorporate a ‘real’ value of space in
destination choice models. Given that there is no unique measurement
that summarizes the complexity of space this is a truly challenging effort.
Unlike Dijst and Vidakovic (1997), for instance, who focus on people’s
action space within a city using only the spatial variable ‘distance
between locations of activity bases’, we assume, like Handy
et al. (2005) that the choice of destination depends upon the characteristics of the zones (i.e. the built environment, land use, neighbourhood
characteristics) as well as of the travellers (socio-economic characteristics, attitudes). Therefore, it is necessary to have information on
attributes of the zones and data on individual and household characteristics. With respect to the former, geographical information systems
(GIS) techniques can be used in order to provide information on land
use, density characteristics and accessibility of each zone (Slavin, 2004).
Second, an important problem in destination choice modelling relates
to the modelling approach itself. Statistical theory and methods often
assume independent observations, but due to spatial dependence this
condition is rarely met when analysing spatial data (Miller, 1999). As a
result, the nature of spatial data conditions the model structure. Our
search for the most appropriate modelling procedure consists of two
steps: first, the performance of two types of discrete choice model
structures is tested: the multinomial logit (MNL) and the nested logit.
Second, the most convincing form of the utility function must be
defined. Related to the selection of the choice model, is the issue of how
Destination Choice Models
155
to deal with a very large number of spatial choice alternatives. BenAkiva and Lerman (1985) suggested to use a restricted set of
alternatives rather than a full set. In this paper, however, the study
area is divided into a limited number of zones (33 in all) representing
the alternative areas of destination. There is no sampling: the total set
of spatial destination alternatives for each individual is considered, but
aggregated into zones. The number of zones is large enough for
representing reasonably well the urban reality, and small enough to
comprehend spatial patterns. The model based on spatially aggregated
data is also compared to a disaggregated formulation.
Third, given the need to have information on attributes of the zones
as well as on individual and household characteristics, we opt to use to
work with trip chaining data. The data used stem from individual travel
surveys describing daily activities (about 30,000 trips collected in 1999)
in the city region of Antwerp (Belgium).
Fourth, and finally, the objective of the current paper is on
explaining destination choice and not, for example, modal choice.
Hence, an intrinsic spatial variable is used as target or dependent
variable. The results of the spatial model to be developed are to give an
insight into the factors that explain the respondent’s probability to
choose a certain destination within a set of destination zones,
contributing to a better understanding of travel behaviour.
This paper is structured as follows. Section ‘Methodology and Data
Requirements’ is dedicated to the discussion of the methodological
framework and the specific data requirements, including model
specification and estimation procedure, defining the study area and
the destination zones, introducing the travel data, and discussing the
construction of the spatial variables (land use, density and accessibility). In Section ‘Results’ the estimation results of the destination
choice models are presented. Finally, in Section ‘Conclusions and
Future Research’, conclusions and research perspectives are reported.
Methodology and Data Requirements
If we want to explain the way in which people choose among different
destination zones, we need to choose a method and a model as well as
an adequate set of data. With respect to the model specification it is
common sense to advance a discrete choice model. However, given
the large number of model possibilities, specific attention is paid to the
econometric model structure (i.e. the type of choice model) and the
functional form (i.e. the type of utility function). The data requirements
refer to the issue of the definition of the destination zones and the
selection and construction of the geographical variables.
Model Specification and Estimation Procedure
Modelling destination choices implies using and selecting a discrete
choice model. Discrete choice models assume that the overall utility of a
choice alternative is composed of a fixed (i.e. systematic or deterministic) utility value and a random or error utility component. Depending on
the assumptions made regarding the distribution of the error terms,
several discrete choice model structures have been developed in the
literature (Domencich & McFadden, 1975; Hensher & Johnson, 1981a;
Cascetta, 2001; Train, 2003; Quinet & Vickerman, 2004). Logit is by
far the most widely used discrete choice model. It is derived under the
assumption that the random utility elements are independently and
identically distributed (IID) (McFadden, 1973, 1976). In other words, it
is assumed that the unobserved factors are uncorrelated over the choice
alternatives, and have the same variance for all alternatives. If IID can be
defended, the Type I extreme value distribution seems the most suitable
distribution (Johnson & Kotz, 1970), resulting in the well-known MNL
model (Cramer, 1991; Ortuzar & Willumsen, 2001).
It is true that logit models have a strong theoretical base, a simple
mathematical structure, and are quite easy to estimate. Their popularity
is due to this convenience. However, given the restrictive assumption
that a MNL model can only be applied to situations in which
alternatives are totally independent, and that this is certainly not the
case for spatial alternatives (our concern), the use of a simple MNL
model seems inappropriate. A common solution to relaxing the
assumptions of independence among alternatives is to introduce a
nested model structure (Koppelman & Sethi, 2000; Papola, 2004).
In such a nested logit model, choice alternatives are segmented and
structured in branches or nests that are more similar (Suarez et al.,
2004). Indeed, if the destination choice process is hierarchical and
similar alternatives are grouped into the same branches of the choice
hierarchy, then alternatives within each branch are more likely to follow
the IID assumption. In the present paper, destinations will be grouped
according to urban level (subregions mentioned in Section ‘Studied Area
and Defining Destination Zones’ and defined by Van der Haegen et al.,
1996). This means that each individual is assumed to first choose an
urban level (e.g. urban versus suburban) and then, within that broad
spatial zone, to choose a precise destination. The result is a nested logit
formulation with two levels of decision for destination (Figure 1).
The mathematical formulation of the nested logit with two levels of
decision can be briefly described as follows. We assume that the utility
function of the destination choice j can be split into one part that
characterizes the urban level (Vl) and another part that varies with the
choice within that level (Vj(l))
157
Origin
Urban level
l = 1,…,4
City centre
l=1
19th century area
l=2
Destination
j = 1,…, 33
Dest j= 1 ,.., 5
Dest j = 1
Suburbs
l=3
Dest j = 1,…, 21
Urban fringe
l=4
Dest j = 1, …, 6
Figure 1. Decision-making structure according to the nested logit model developed in
this paper
Uij Vl Vj(l) o j
(1)
where oj is the stochastic part of the utility (error term). We assume that
the individual specific error terms o1, o2,. . .,oj are random and IID
distributed. Hence, the probability that an individual chooses a
destination j is given by
ll 1
PJl
exp(Vj =ll )
exp(V
=l
)
j(l)
l
j1
P(j) (2)
PL
PJl
exp(V
=l
)
j(l)
l
j1
l1
which corresponds to a nested logit formulation with two levels of
decision for destination. The parameter ll is a measure of the degree of
independence in unobserved utility among the alternatives within urban
subregion l. Its value must be between zero and one for the model to be
consistent with utility maximization behaviour for all possible values of
the explanatory variables (Train, 2003; Bhat & Guo, 2004).
The nested logit is appealing in terms of its ability to accommodate
differential degrees of interdependence (i.e. similarity) between subsets
of alternatives in a choice set (Hensher & Greene, 2002), but many
published applications display a lack of attention to the very precise
form that these models must take to ensure that the resulting model is
consistent with utility maximization. Thus, next to the assumptions
regarding the unobserved utility components that lead to different
econometric model structures, we also have to focus on the functional
form of the utility function.
In its usual form the utility function of a discrete choice model is given
by Uij Vijoij where Vij is the deterministic part of the utility and oij the
random unobserved component of the utility. The deterministic term is
usually specified as a linear function of the attributes of the alternative
and the characteristics of the decision-maker, and is denoted by
ak bjk Xijk o ij where bjk is the coefficient for attribute k of alternative
j, and Xijk is a vector of observable attribute values. Although a linear
function of the parameters is most commonly applied, it is not always
the most accurate representation. Therefore, two alternatives for the
linear utility function are considered here: BoxCox transformation and
random coefficients. If a BoxCox transformation (c.f. Gaudry &
Dagenais, 1979; Picard & Gaudry, 1998; Hensher & Johnson, 1981b) is
done, it can be shown that the regression residuals are usually more
homoscedastic and closer to a normal distribution. Just as for models
with linear utility functions, models with BoxCox transformation also
assume that the coefficients are constant (fixed for all individuals): the
explanatory variables have the same effect for each individual. However, the population is in general rather heterogeneous and the effect of
each explanatory variable can vary from one individual to another. In
order to account for this characteristic the b-coefficients should be
allowed to vary randomly instead of being fixed over the decisionmakers. The result is a so-called model with random coefficients, which
accounts for the heterogeneity in the population. The functional form of
the utility function is then somewhat different (Train & McFadden,
2000). The utility is Unj b?nk Xnjk o nj ; where b?n is the vector of
coefficients for decision-maker n representing that person’s characteristics. Suppose b?n is normally distributed in the population with mean b
and covariance W:b?n N(b; W): The goal of the research is to estimate
the parameters b and W (Train, 2003, p. 111). Assessment of what type
of utility function specification should be used depends on the choice
problem being analysed. In Section ‘Determining the Functional Form’
this is discussed in more detail.
Studied Area and Defining Destination Zones
The area studied in this paper is that of metropolitan Antwerp. Antwerp
is the second largest city of Belgium, located in the northern part of the
country, close to the border with The Netherlands. It has approximately
500,000 inhabitants and is characterized by a large harbour bound to
the North Sea by the river Scheldt. As in most cases, the city sprawls far
away from its historical centre. We here consider the city region from
centre to periphery: the city centre, the 19th century centre (core), the
suburbs and the urban fringe. These subregions were defined independently by means of morphological and socio-economic variables (Van
der Haegen et al., 1996). In a former paper, we showed that the friction
of distance (distance decay) is different in each of these subregions
159
(Hammadou et al., 2003). The city region consists of 608 statistical
sectors (also called neighbourhoods or wards equivalent to 32 zip
codes and 18 communes) (Figure 2).
Clearly, as also pointed out by Ben-Akiva and Lerman (1985),
modelling destination choices at a ward level is quite difficult: the
number of spatial alternatives is too high (in our case N 608) and
wards vary in size and shape. In order to solve this problem we
aggregated the 608 statistical sectors into a smaller, more workable
number of zones (Figures 3 and 4). As in most quantitative geographical analyses (Openshaw & Taylor, 1979), spatial aggregation also
substantially affects travel demand modelling results. Miller (2004)
mentioned some major spatial analytical issues related to zoning:
spatial dependency, spatial heterogeneity, boundary problems and scale
effects. Unfortunately, there is no predefined method for aggregation in
order to avoid these analytical pitfalls. In our approach we attempt to
tackle the spatial aggregation problem by constructing a so-called
zoning algorithm that aims at reducing the number of destinations by
maximizing internal homogeneity and external heterogeneity of the
aggregated zones. By aggregating places, we want to better control the
main spatial trends and avoid ‘noise’ in the spatial model.
Methodological details about the zoning algorithm are reported in
Van Hofstraeten and Verhetsel (2004). The algorithm uses a traditional
methodology: 26 variables are measured on 608 sectors. The variables
describe land use, attractiveness and accessibility. Some of these
19th century area
City centre
Suburbs
Urban fringe
Figure 2. The urban division of Antwerp
grouping input variables into
factors
grouping statistical sectors into
clusters
=
cluster
factor
spatial zoning algorithm
1
densely-built, large employment
1
infrastructure
2
residential built-up area
2
densely-built, large employment
3
industry, port
3
industry, port
4
green area
4
agriculture
5
parks
5
residential built-up area
6
infrastructure
6
green area
7
parks
8
green residential area
9
open area with scattered housing
grouping adjointsectors
based on
cluster value of sector
line infrastructure
administrative borders
Figure 3. Spatial zoning algorithm: a synthesis.
variables are highly correlated; hence, a principal component analysis is
applied. It summarizes the 26 variables into six components that are by
definition uncorrelated. The number of components is determined by
the eigen value ( 1.0); this guarantees that the number of factors is
reduced, but still explains a large part of variance (70%). A clustering
SZA
N
10 km
Figure 4. From 608 sectors to 33 destination choice zones in Antwerp by applying a
spatial zoning algorithm (SZA)
Source: Van Hofstraeten and Verhetsel (2004).
161
method is then applied on the 608 sectors in order to group wards that
are similar in terms of the scores on the six components. The 608 wards
are grouped into nine clusters of similar land use (Figure 3).
The final step consists of defining a reasonable number of destination
zones, keeping in mind the observed reality of the study area. The basic
underlying assumptions of the applied spatial zoning algorithm (SZA)
are:
1. neighbouring statistical sectors with similar land use are grouped
together;
2. major linear infrastructures (i.e. river Scheldt, highways and ring
roads) serve as major spatial barriers. These physical barriers are the
primary borderlines, except at slip roads where developments (i.e.
industry) occur at both sides of the highway;
3. other main roads, railways and administrative borders are considered as secondary borderlines;
4. open spaces without major developments are not considered as
separate destination zones. They are grouped with other sectors
within the corresponding municipality or neighbourhood.
In fact the zoning algorithm is a step-by-step process grouping
contiguous statistical sectors; it is based on the clustering results,
administrative borders and infrastructure elements. By following these
steps, we obtain destination zones, which are rather homogenous. We
end up with 33 zones (Figure 4). This level of aggregation is not
excessive for Antwerp: it better corresponds to the geography of the
city (fieldwork), and hence to the understanding of the attraction
factors influencing destination choice. The aggregation process is well
defined and can be adapted over time or extrapolated to other cities,
but as in each aggregation process, strong criticisms can be addressed:
heterogeneity exists within each zone, zones can evolve over time, etc.
The obtained destination zones will form the input for our choice
model that aims at explaining the destination choice and the activity
patterns (Section ‘Results’): the dependent variable is the probability of
selecting one of these 33 zones as destination (J 33).
Travel Data, Activity and Tour Structure
The travel data set used is the so-called OVG travel data set collected in
1999 as a result of the Flemish Travel Behaviour Research project. In
this survey, each person above the age of five and being a member of
the selected sample of households was asked to fill in a travel diary for
two consecutive days. This resulted in a large data set, including data
on each trip (e.g. activity, mode, distance, and duration) as well as
socio-demographic information on each person and household (e.g.
age, income, household type, gender). The original OVG data set of
Antwerp contains information on about 30,000 trips made by 5613
different persons (Witlox and Tindemans, 2004; Tindemans et al.,
2005; Witlox, 2007).
Constructing a destination choice model always starts with the
definition of the activity patterns (Kitamura, 2004). There are
numerous possible activity and tour structures that a person can build,
and there is no unique alternative for the modeller for simplifying and
aggregating the various structures to a reasonable (limited) number of
choice alternatives. In our modelling approach we only considered the
destination of the main stop in each tour, each tour starting and ending
‘at home’. We are aware of the fact that by doing this our approach
becomes in fact trip-based. The main stop is here associated to the
‘main activity’; the latter corresponds to the activity that lasts the
longest in each tour. This definition is by essence arbitrary and can be
criticized. Work is assumed to be a mandatory activity fixed in space.
Due to the spatial deterministic character of work trips, only non-work
stops are considered in this paper. We are aware that these choices may
introduce biases in the estimations. The four possible tour structures
are reported in Table 1.
The model limits itself to the choice of the main destination of the
tour. The alternatives in the destination choice models determine
the tour structure and stops for various purposes. Our model estimates
the probability that a person, making a stop in a tour, chooses a specific
place (zone) as destination. The model includes stops made before and
after the main activity in the tour (these stops are called Intermediate).
Note that in the end only a relatively small number of trips are used as
data input. This data reduction results from the fact that work trips
were not analysed, trips with more than three stops were excluded
(only a very small percentage), and a certain number of trips had to be
excluded because they could not be geocoded due to a lack of spatial
information.
Table 1. Distribution of tour types in Antwerp
Non-work tour type
HomeMainHome
HomeIntermediateMainHome
HomeMainIntermediateHome
HomeIntermediateMainIntermediateHome
Total
Source: OVG Antwerp city region (1999).
Frequency
%
2711
396
268
122
3497
77.5
11.3
7.7
3.5
100.0
163
Construction of Geographical Variables
Van Wee (2002) stressed the importance of the introduction of spatial
variables in the analyses of travel behaviour. Badoe and Miller (2000),
Stead (2001) and Handy et al. (2005) also indicated that land use, built
environment, density characteristics and accessibility are important
types of spatial variables that explain the travel choice behaviour.
Hence, several spatial variables were collected for each of the 608
statistical wards, bearing in mind that an individual can not only be
influenced by the attractiveness of one particular ward, but also by its
surrounding wards.
Spatial variables are incorporated into our choice model in two ways:
the studied city region is redefined by a limited set of homogeneous
destination zones (Section ‘Studied Area and Defining Destination
Zones’) and the spatial characteristics of destination zones are
introduced in the model as explanatory variables. Three types of
geographical variables were constructed: land use (Section ‘Land use
variables’), attractiveness (density) (Section ‘Attractiveness variables’),
and accessibility (Section ‘Accessibility variables’).
Land use variables. Land use variables are created drawing from two
databases. In 1996, OC-GIS Flanders developed a first digital land use
map for the Flemish. The data set is based on satellite images, soil
information and road network information. By using an automatic
classification procedure, satellite information is converted into 19
different categories of land use (OC-GIS Vlaanderen, 2002).
A second land use data set is MultiNet (2001) collected by TeleAtlas.
This data set contains information on administrative borders, road
network and specific land use characteristics such as ‘built-up area’.
Unfortunately, most details about land use pertain to non-built-up
surfaces: hence little is known within urban areas.
The surface occupied by each type of land use is expressed in square
metres as well as in percentage of the total surface, and is computed for
each of the 608 statistical wards. The most interesting variables for
Antwerp are: (i) housing development (density, built-up, green residential);
(ii) industrial, commercial and port development; (iii) green areas and open
spaces; and (iv) infrastructure (highways, district roads, airports, ports).
Attractiveness variables. A second group of spatial variables relates to
‘size variables’, giving an indication of the importance of a place of
destination in terms of population, employment, shopping opportunities or presence of schools. The expected effect on travel behaviour is
obvious: the higher the density, the more destinations within the
activity range, and hence, the more trips and multipurpose trips (Van
Wee, 2002; Van Acker et al., 2007).
Cervero (1996), Cervero and Knockelman (1997) and Badoe and
Miller (2000) showed that, based on the relationship between population density and travel behaviour, a larger concentration of population
leads to shorter trip distances and the discouragement of the use and
possession of cars. That is why it is interesting to attach the number of
inhabitants to each origin and each destination sector. The data used
are those provided by the National Institute of Statistics (NIS, 2001).
The relationship between employment, travel behaviour and activity
patterns is well known. In this respect, Badoe and Miller (2000), p.
251) demonstrated that a higher concentration of employment has a
significant impact on travel behaviour. The data used here are the
number of jobs provided by the Regional Development Agency of
2001. This is only a rough (but workable) estimation of the exact
employment figures of the National Census of 2001, but as we estimate
non-work trips, these data are well enough.
A third attractiveness variable is the presence of schools. The data
used here were provided by the Department of Education of the
Flemish Government, and consisted of a list of addresses of all Flemish
schools (primary and secondary schools, colleges, and universities).
After geocoding the addresses, it was possible to compute the number
of schools in each of the 608 statistical sectors. A high number of
schools will no doubt have an impact on the modal choice (i.e. bringing
or getting children to or from school) and on the distance of school
trips. Unfortunately, the size of each school is unknown.
A last variable is shopping. This is a crucial variable since shopping
trips are very frequent in our (non-commuting) travel data set. We
assume that a large number of shopping alternatives leads to shorter
trips and to a lower use of cars. An Internet data source, called SCOOT
(www.scoot.be), provided us with the addresses of most shopping
alternatives. In total approximately 6000 stores (large and small) in the
city region of Antwerp were included. Geocoding these addresses made
it possible to assign this information to each of the 608 sectors. For
practical reasons, we limited ourselves to the presence/absence of
stores; no reference is made to their relative importance (surface,
turnover, and employees). Hence, we mainly computed a number of
proxies for different variables.
Accessibility variables. A third important geographical variable is
accessibility. This variable was included by calculating the shortest
distance path distance and shortest time path distance for each centroid
of the 608 sectors by means of the StreetNet 2001 network and
ArcView Network Analyst. The StreetNet road network includes
information on traffic regulations, such as closed streets, one-way
streets, underpass and overpass, and travel surplus. We did however
165
not account for congestion, waiting time at traffic lights or extra time
to take turns, but in order to partly compensate some network segments
were assigned slightly lower speed levels than the actual maximum
authorized speed. For the car mode a network distance and a deduced
travel time were calculated. For other transport modes (foot and bike)
network distances are used to estimate travel time and a speed factor of
4 km/h for walking and 15 km/h for biking.
Briefly recapitulating, Table 2 summarizes the explanatory variables
used in the models. Data are rather complex: different types of
variables are combined for different data levels. However, by doing
so, spatial planners are able to analyse sensitivities at different levels.
The model can also explore socio-economic behaviour and urban
development simultaneously.
Results
In this section our main findings are discussed. First, Section
‘Determining the Functional Form’ reports on the methodological
issue of what type of functional form should be used for the utility
Table 2. Variables selected for modelling destination choices (definition and types)
Data
source
Levels
OVG data Individual
Attributes
Age
Gender
Household Income
Location
Household type
Trip
GIS
Zone
Trip
Types
Continuous
Binary (two categories)
Discrete (five categories)
Discrete (four categories)
Discrete (eight
categories)
Discrete (four categories)
Discrete (five categories)
Continuous percentage
of geographical area by
land use type
Purpose
Transport mode
Land use
Built-up, housing
Industrial/commerce/port area
Agriculture and meadowland
Housing and other developments
Attractiveness variables
Continuous frequencies
aggregated by zone
Inhabitants
Shops
Schools
Jobs
Travel time
Continuous accessibility
measure
function (linear, BoxCox transformation, or random coefficients).
Section ‘Modelling Destination Choices’ discusses the modelling
results for all non-work trip purposes, whereas Section ‘Modelling
by Purpose: Shopping and Leisure’ is devoted to show the model
results for leisure and shopping trips separately. Finally, in Section
‘Aggregated versus Disaggregated Model’ our aggregate findings are
compared to the disaggregated approach proposed by Ben-Akiva and
Lerman (1985), which consists in composing individual choices from
the choice alternative and a sample of the non-selected alternative.
All models were estimated using the Biogeme software (Bierlaire,
2007).
Determining the Functional Form
Prior to analysing the overall modelling results we first concentrate on
a particular important methodological issue in discrete choice modelling: i.e. the selection of the type of utility function. Table 3 shows the
results. For the sake of clarity, the analysis is limited to one key
explanatory variable: travel time. By doing so, we avoid estimation
biases and/or changes in the conclusions due to the transformation of
the time variable. Time is indeed one of the most important elements
in the explanation process. Other tests were also performed (not
reported here) with other variables taken individually or together:
they all confirmed the results proposed here. Table 3 compares the
Table 3. Estimation of the destination choice model: results for different utility
functions for the variable travel time
Linear utility
Accessibility variables
Time
Mean
Variance
Lambda (l)
Number of estimated
parameters
Sample size
Null log-likelihood
Final log-likelihood
Likelihood ratio test
Rho-square
BoxCox transformation
Value
t-Test
Value
t-Test
0.55
74.03
1.84
20.42
0.28
Random coefficient
Value
t-Test
0.93 44.38
0.70
14.70
9.10
1
2
2
3497
12,227
7373
9708
0.397
3497
12,227
7130
10,195
0.417
3497
12,227
7069
10,316
0.422
167
results obtained with a linear MNL formulation to that of the Box
Cox estimation and of mixed logit with random coefficients. In the
BoxCox logit, the exponent of the transformation (l) is equal to 0.28
and is significantly different from zero. This means that the
transformation of travel time is not linear. For the model with
random coefficients, the estimated variance of the distribution of the
coefficient of travel time is significantly different from zero; hence, we
accept the hypothesis of a random coefficient of time: the perception
of travel time varies randomly from one individual to another. These
two alternative solutions to the standard MNL lead to a significant
reduction of the absolute value of the log-likelihood (high chi-square
values): it drops from 7373 in the standard MNL to 7130 in the
BoxCox logit and to 7069 in the random coefficient mixed logit
model. As a result the random coefficient formulation is preferred but
the utility function is kept linear. Hence, we end up with a nested
logit model with random coefficients, often referred to as a mixed
nested logit (MXNMNL).
Modelling Destination Choices
In Table 4 the estimation results for destination choice for all non-work
trip purposes are presented for two types of models.
As expected, travel time appears to be one of the most important
variables in the explanation process (0.64). Hence, destinations
located further away from the place of residence or intermediate stop
location, have a smaller probability of being selected as destination for
an activity. Let us add here that travel time may hide many socioeconomic and demographic disparities in a city like Antwerp.
The choice of a destination is additionally and significantly
influenced by other spatial and socio-economic variables; the signs of
the coefficients meet the expectations. Spatial variables include attractiveness as well as land use information. The influence of the number of
shopping and employment alternatives is positive: they increase the
attractiveness of a destination, and hence its probability of serving as a
destination. Moreover, the higher the percentage of surface affected to
housing, parks or green area, the higher the probability of choosing a
zone as destination. In contrast, industrial land use has a negative effect
on the attractiveness of the destination. Note that ‘agriculture’ also has
a positive impact on the probability to choose a zone: open ‘agricultural
spaces’ are attractive in terms of recreation and leisure, especially in an
urban agglomeration.
Most socio-economic variables as well as the characteristics of the
tours themselves are less or not important in the explanation of the
destination, with the exception of the location (suburbs) and the type
(two children and more) of household. This means that personal
characteristics are not very important in the modelling process. In other
words, an individual chooses the destination that optimizes his/her
utility under the constraint of the characteristics of the destination
without reference to its personal characteristics with the exception of
the transportation mode and some types of households (Table 4). This
is a quite important result in terms of town planning: planning new
attractive shopping malls or other shopping/service alternatives will
have a drastic effect on transportation fluxes within the city. Let us also
mention here the importance of the households’ location in the
explanation (suburbs).
Note further the weak impact of the socio-economic variables, which
can be explained by the gross zone sizes and aggregation over purposes,
but also by the fact that socio-economic characteristics are less
important in non-work trips than for commuting. In most choice
modelling approaches, socio-economic characteristics are introduced
with specific effects: they hence obtain for each variable as many
estimated coefficients as alternatives. In our case, the high number of
spatial alternatives necessitates to constrain the coefficients not to vary
from one alternative to the other. Each socio-economic variable has as
many coefficients as there are alternatives.
It seems to be quite difficult to use this model as a deterministic,
predictive tool: one variable can counterbalance effects of other
variables. This is especially true for land use variables. In order to
solve this problem, elasticities are computed. Table 5 and Appendices A
and B present direct and indirect (cross) elasticities for the spatial
variables and for the four levels of urbanization. By definition, elasticity
measures the relative importance of the response of the choice
probability to marginal changes in the explanatory variables. It is
computed by the weighted average of the individual elasticities for each
spatial variable. If, for instance, the number of shops in the ‘19th
century area’ increases by 1%, then the probability of choosing a
destination in that zone increases by 0.65% and that of the other
destinations decreases by 0.035%.
It can be noted that for variables such as ‘percentage of land surface
devoted to agriculture’ or ‘percentage of land surface devoted to
industry’, the observed hierarchy in the elasticities corresponds to the
hierarchy in the level of urbanization of the destination zone: the
elasticities are higher in the periphery of Antwerp (suburbs and urban
fringe) than in the centre (city centre and 19th century area). As
expected, the destination zones close to the city centre have higher
direct elasticities for the variable ‘number of shops’. The opposite is
observed for indirect elasticities. These results support the relationship
between the geographical factors and the destination choice.
169
Table 4. Parameter estimates for the multinomial logit and the mixed logit
Mixed multinomial logit
Value
t-Test
Time: Mean
0.75
34.41
Variance
0.51
12.30
Land use variables
Agriculture
0.014
3.63
Industry
0.021
5.81
Housing
0.003
2.30
Parks
0.010*
1.79
Green area
0.103
6.12
Size variables
Employment
0.00002
3.12
Number of shops
0.00305
22.02
Socio-demographic variables
Age
0.005*
0.83
Income
Income B500 euro
0.75*
0.96
Household type
Single parent with two or
1.25
2.09
more children
0.88*
1.48
Couple with two or more
children
Household location
Suburb
1.46
4.21
Characteristic of chains
Purpose: Service
1.11
2.03
Mode: bike
0.40*
0.80
l1
l2
l3
l4
Number of estimated
18
21
parameters
Sample size
3497
3497
Null log-likelihood
12,227
12,227
6554
6498
11,347
11,458
Rho-square
0.46
0.47
*Not significant at 5% level.
Mixed nested logit
Value
t-Test
0.64
0.38
29.14
11.82
0.011
0.013
0.009
0.013
0.092
3.18
4.59
10.02
3.01
6.81
0.00002
0.00173
3.23
12.37
0.006*
1.30
0.71*
0.99
1.10
2.02
0.90*
1.71
0
Fixed
1.06
0.31*
1.00
0.62
0.82
0.63
2.07
0.70
Fixed
7.50
5.43
3.48
Table 5. Direct and cross elasticity on land use and size variables
Urban level
City centre
Spatial
variables
Direct
Land use variables
Agriculture
0.013
(%)
Parks (%)
0.000
Green area
0.130
(%)
Housing (%) 0.003
Industry (%) 0.000
Size variables
Shopping
1.542
alternatives
Jobs
0.279
Cross
19th Century
area
Direct
Cross
Suburb
Direct
Cross
Urban fringe
Direct
Cross
0.001
0.078 0.005
0.247 0.007
0.275 0.005
0.000
0.010
0.016 0.001
0.385 0.020
0.007
0.000
1.145 0.031
0.000
0.000
1.055 0.015
0.000
0.388 0.024
0.325 0.010
0.315 0.006
0.000 0.015
0.001 0.117
0.002 0.127
0.001
0.115
0.646 0.035
0.194 0.010
0.128 0.003
0.021
0.154 0.007
0.100 0.003
0.048 0.001
Modelling by Purpose: Shopping and Leisure
In the previous section we considered all non-work trip purposes
together. Common practice is however to consider subgroups of trips,
for instance by type or purpose; this reduces heterogeneity in the data
and hence avoids misspecification of the global estimators and increases
the quality of the model. In this section, the behaviour of two
subgroups is considered; we test if greater homogeneity enhances the
modelling results.
Four types of trip purposes were originally considered separately:
leisure, shopping, visiting someone and services. Leisure and shopping
were however the only one for which there were enough observations
for obtaining significant statistical results. The modelling results are
reported in Table 6.
The results obtained for subgroups are close to those reported in
Table 4 (all data): socio-economic characteristics of the individuals
have only a very small impact on the destination choice; transportation
time and spatial characteristics remain important explanatory factors in
destination choice. However, the analyses by subgroups give a better
fit, as far as shopping is concerned; the Rho-square increases from 0.47
(i.e. the result taken from Table 4) to 0.53. This is not the case for
leisure trips where Rho-square decreases significantly from 0.47 to
0.37. In a way this can be explained by the definition of the activity:
‘leisure’ is vague and refers to sport, cinema, etc. as well as simply
171
Table 6. Mixed nested logit for shopping and leisure trips
Shopping
Value
t-Test
Time: Mean
0.67
20.72
Variance
0.32
7.27
Land use variables
Agriculture
0.01
2.93
Industry
0.02
3.60
Housing
0.01
6.50
Parks
0.02
2.67
Green area
0.11
6.01
Size variables
Employment
0.00001*
1.14
Number of shopping
0.00183
8.91
Age
0.02
2.86
Income: IncomeB500 euro
1.41*
1.44
Household type
Single parental with two or
0.40*
0.52
more children
0.12*
0.17
Couple with two or more
children
Mode: bike
0.11*
0.18
l1
1.00
0.00
l2
0.56
14.28
l3
0.81
21.07
l4
0.92
8.56
Number of estimated parameters
17
Sample size
1881
Null log-likelihood
6577
3068
7018
Rho-square
0.53
Leisure
Value
t-Test
0.52
0.33
15.43
6.65
0.01*
0.01*
0.01
0.02
0.05
1.65
1.30
5.95
2.36
2.27
0.00002
0.00177
2.63
7.49
0.01*
1.65*
1.47
0.00
1.88*
1.82
1.61*
1.78
0.04*
1.00
0.63
0.80
0.41
0.05
0.00
10.18
17.48
4.81
17
1090
3811
2397
2828
0.37
*Not significant at 5% level.
‘walk’. Moreover, the location of this kind of activities is also often
vague. Even if our data set is quite large, it is not large enough to
consider separate modelling of smaller subgroups of leisure activities.
Leisure activities tend to occur at random. ‘Shopping’ corresponds to a
more homogeneous definition; that is why results are much better in
terms of Rho-square. Compared to Table 4, we observe that the
number of jobs does not affect the destination choice for shopping
purposes (at the level of analysis, jobs are indeed concentrated in other
places than shops), and age enters positively and significantly in the
equation. Shopping seems in our case to affect more adults and even
elderly people than youngsters. One could also argue here that the
choice set for shopping should be different to that of leisure; this is not
the case here because of the spatial aggregation adopted. At this level of
aggregation most purposes are encountered in each zone.
Aggregated versus Disaggregated Model
Throughout this paper we adopted a nested logit structure for handling
the IID restrictions. We are aware that this approach can be seen as
extremely rigid and afflicted by the arbitrariness of the definition of the
spatial aggregates at successive scales as well as by the implicit
assumption that all the individuals perceptually delineate the alternatives at various scales in the same fashion. This justifies our rigourous
zoning of the city (Section ‘Studied Area and Defining Destination
Zones’) and the comparison of several modelling results. In this final
section we compare the results obtained on spatially aggregated data to
spatially disaggregated choice alternatives with a model formulation
suggested by Ben-Akiva and Lerman (1985) for all trip purposes (see
also Appendix C). On the average and as expected, the Rho-square is
higher in the aggregated process (the well-known phenomenon in
spatial statistics). The sign of the coefficients tend to be similar with the
exception of that variable ‘suburbs’. It has a positive effect in the
disaggregated model and the inverse is observed in the aggregated
model. This can easily be explained by the fact that the intra-zone
homogeneity is larger in suburban areas after the aggregation process.
Last but not least, the values of the coefficients obtained for
disaggregated data are difficult to compare to those obtained with
spatially aggregated data: scale does not only affect the statistical
computation of the coefficients but also generates differences due to the
spatial reality. Let us give one example: single parent households with
at least two children have a much stronger coefficient in disaggregated
models. This is due to the fact that these types of family are
concentrated in some small wards characterized by social housing.
At this stage, we note that aggregation has advantages but also leads
to biases in the models. Advantages are the reduction of the number of
alternatives, all destinations are taken into account in the modelling
process, the easiness of modelling, the clearness of the interpretation,
and the ‘better’ statistical level of explanation (Rho-square). The
weaknesses of the aggregated destination choice model mainly depend
upon the aggregation rule adopted: the delineation of the new zones
can differ with the aggregation rule, and hence influence the modelling
173
results. This is a well-known geographical problem linked to the socalled modifiable areal unit problem (MAUP). Moreover, the intrazone homogeneity will highly depend upon the aggregation rule, and it
will hence affect the quality of the adjustments.
Conclusions and Future Research
This paper has compared several destination choice models for and
particularly with the introduction of the characteristics of space. The
application was limited to the city region of Antwerp and to one data
set (OVG). The results seem to be quite promising, both methodologically and empirically.
When considering space in destination modelling, the main methodological problems encountered are: (i) summarizing the spatial reality
by a few variables; (ii) defining independent spatial alternatives; (iii)
disposing of GIS (adequate data, software and ‘life ware’); and (iv)
choosing an adequate formulation for the model choice. This explains
why only a few approaches of this problem are to be found in the discrete
choice literature. In our case, several variables were created in order to
‘measure’ space and spatial attractiveness. Several modelling methodological formulations were also developed and compared in order to
avoid the numerous methodological pitfalls of discrete choice modelling;
in our case, the MXNMNL seems to be the best formulation.
The application consisted in comparing different formulations of the
model and interpreting the parameters for the present situation in
Antwerp. We showed how difficult it was to represent space in the
analysis of travel patterns and how little the socio-economic characteristics of the individuals/households explain the observed behaviour
compared to spatial characteristics. The choice of model estimation
influences the results only to a small extent. We are aware that the set
of variables introduced in the models is very limited. At the light of the
results presented in this paper, we do think that the major challenges in
future understanding destination choice are not methodological but
behavioural: developing better measurements of the factors that
influence destination choice and developing better understanding of
the processes (i.e. dynamics) underlying destination choices.
Let us add here that the models used were also tested for simulating
planning effects: we analysed the impact of a specific change in the
choice alternatives. Two new urban development projects were
compared. According to the simulation results, one project is more
attractive to inhabitants of the city region than the other. This can be
explained by the fact that a project containing new shopping facilities, a
municipal park, housing and offices obviously attracts more people
than an industrial project containing no residential or shopping
facilities (see for more details Verhetsel et al., 2005). By using this
approach, public stakeholders can encourage developments in specific
areas and study the impact of their policy measures on the overall
mobility. They can also play an active role in investing in land
development, housing, infrastructure, etc. Whatever the simulation,
the characteristics of space seem to be a decisive variable in destination
choice modelling.
Further analyses have to be done. Particular attention has to be paid
to testing the sensitivity of the model to changes in spatial and
behavioural measures. Avenues for future research also consist in
adding new variables or new measurement of the urban reality in the
explanatory process. Furthermore, the techniques developed in this
paper for generating spatial variables, defining destination choice zones
and destination choice modelling will be applied to other city regions
(Gent, Mechelen and Leuven) in order to compare the empirical results,
to further test the robustness of the proposed methods, and to test if the
results depend upon the size of the city.
Acknowledgements
The research on which this paper is based was part of the 2nd Scientific Support Plan
for a Sustainable Development Policy (SPSD II) for the Belgian Federal Office for
Scientific, Technical and Cultural Affairs (OSTC). The authors would like to thank
Hans Tindemans and Dries Van Hofstraeten for helping in the analysis and Peter Goos
for providing critical remarks on a previous draft of this paper. All remaining errors are
the responsibility of the authors.
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Appendix A: Direct and cross elasticity of spatial variables on each zone
Percentage zonal Percentage zonal
of agriculture
of housing
Urban level
City centre
19th Century area
Suburbs
Zone Direct Indirect Direct Indirect
0.013
0.031
0.064
0.023
0.128
0.116
0.084
0.071
0.188
0.127
0.268
0.301
0.334
0.247
0.294
0.498
0.488
0.312
0.348
0.271
0.286
0.424
0.028
0.042
0.111
0.031
0.357
0.256
0.171
0.187
0.108
0.321
0.458
0.166
0.130
0.222
0.381
0.263
0.550
0.707
0.197
0.496
0.488
0.663
0.004
0.023
0.419
0.012
0.570
0.486
0.832
0.565
0.333
0.573
0.558
0.347
0.232
0.965
0.665
0.922
0.521
0.276
0.583
0.552
0.444
0.270
0.008
0.031
0.724
0.016
1.593
1.069
1.694
1.481
0.191
1.449
0.955
0.191
0.091
0.868
0.861
0.487
0.587
0.625
0.329
1.012
0.756
0.423
Percentage zonal
of built-up area
Direct
0.482
0.710
0.619
0.245
0.302
0.317
0.492
0.345
0.216
0.367
0.344
0.138
0.028
0.677
0.282
0.429
0.320
0.050
0.000
0.273
0.335
0.196
Percentage zonal
of industry
Number of
shopping
Number of
employment
Indirect Direct Indirect Direct Indirect Direct Indirect
1.015
0.969
1.069
0.340
0.844
0.697
1.001
0.904
0.124
0.926
0.588
0.076
0.011
0.609
0.365
0.226
0.361
0.113
0.000
0.501
0.572
0.307
0.000
0.000
0.000
0.000
0.015
0.016
0.000
0.028
0.921
0.056
0.009
0.367
0.701
0.082
0.011
0.162
0.000
0.001
0.028
0.042
0.044
0.003
0.000
0.000
0.000
0.000
0.041
0.036
0.000
0.072
0.528
0.141
0.015
0.202
0.274
0.074
0.014
0.085
0.000
0.003
0.016
0.077
0.074
0.005
1.333
1.315
0.749
0.149
0.427
0.410
0.646
0.417
0.093
0.584
0.312
0.145
0.099
0.130
0.098
0.294
0.102
0.108
0.014
0.087
0.367
0.455
2.806
1.795
1.294
0.206
1.193
0.903
1.315
1.091
0.054
1.476
0.534
0.080
0.039
0.117
0.127
0.155
0.114
0.246
0.008
0.159
0.626
0.712
0.029
0.035
0.030
0.031
0.005
0.008
0.013
0.005
0.046
0.019
0.022
0.024
0.027
0.005
0.003
0.021
0.001
0.005
0.006
0.006
0.004
0.003
0.061
0.047
0.051
0.042
0.014
0.018
0.027
0.013
0.026
0.048
0.038
0.013
0.011
0.005
0.004
0.011
0.001
0.012
0.003
0.011
0.007
0.005
177
1
2
3
4
5
7
8
11
6
9
10
12
13
14
15
16
19
20
21
22
24
25
Size variable
Land use variable
Appendix A. (Continued)
Land use variable
Percentage zonal Percentage zonal
of agriculture
of housing
Urban level
Urban fringe
Zone Direct Indirect Direct Indirect
26
27
29
30
33
17
18
23
28
31
32
0.251
0.825
1.055
0.431
0.191
0.257
0.706
0.190
0.985
0.448
0.193
0.777
0.233
0.211
0.068
0.369
0.581
0.130
0.503
0.330
0.002
0.537
0.359
0.650
0.833
0.279
0.448
0.397
0.888
0.358
0.787
0.061
0.422
1.114
0.184
0.167
0.044
0.866
0.899
0.164
0.948
0.263
0.000
1.174
Size variable
Percentage zonal
of built-up area
Direct
0.206
0.453
0.255
0.008
0.215
0.137
0.143
0.132
0.453
0.000
0.288
Percentage zonal
of industry
Number of
shopping
Number of
employment
Indirect Direct Indirect Direct Indirect Direct Indirect
0.639
0.128
0.051
0.001
0.417
0.309
0.026
0.348
0.152
0.000
0.801
0.000
0.006
0.000
0.214
0.013
0.005
0.031
0.014
0.000
1.192
0.035
0.000
0.002
0.000
0.034
0.025
0.012
0.006
0.038
0.000
0.005
0.096
0.249
0.674
0.104
0.041
0.096
0.200
0.135
0.195
0.240
0.000
0.033
0.771
0.190
0.021
0.006
0.185
0.452
0.025
0.517
0.080
0.000
0.092
0.003
0.007
0.001
0.154
0.002
0.015
0.003
0.001
0.005
0.015
0.003
0.008
0.002
0.000
0.024
0.004
0.033
0.001
0.003
0.002
0.000
0.009
Table (Continued)
Urban level
City centre
SHOPPING
Land use variables
Percentage zonal of agriculture
Percentage zonal of green area
Percentage zonal of housing
Percentage zonal of industry
Percentage zonal of parks
Size variables
Number of shopping alternatives
LEISURE
Land use variables
Percentage zonal of green area
Percentage zonal of parks
Percentage zonal of housing
Size variables
Number of shopping alternatives
Number of jobs
19th Century area
Suburbs
Urban fringe
Direct
Indirect
Direct
Indirect
Direct
Indirect
Direct
Indirect
0.013
0.159
0.003
0.000
0.000
0.001
0.012
0.000
0.000
0.000
0.078
0.474
0.359
0.018
0.021
0.005
0.024
0.023
0.001
0.001
0.247
1.409
0.301
0.140
0.009
0.006
0.039
0.009
0.002
0.000
0.275
1.299
0.292
0.152
0.000
0.005
0.018
0.006
0.001
0.000
1.603
0.121
0.671
0.038
0.202
0.010
0.134
0.003
0.073
0.000
0.003
0.006
0.000
0.000
0.218
0.031
0.367
0.011
0.001
0.023
0.648
0.015
0.308
0.018
0.000
0.009
0.598
0.000
0.298
0.008
0.000
0.006
1.577
0.278
0.119
0.021
0.660
0.154
0.037
0.007
0.199
0.100
0.010
0.003
0.131
0.049
0.003
0.001
Appendix B: Directs and cross elasticity of spatial variables on each urban level for leisure and shopping
179
Multinomial logit Ben-Akiva type
Value
Time
Land use variables
Agriculture
Industry
Housing
Parks
Green area
Size variables
Employment
Number of shopping
Age
Income
Income B500 euro
Household type
Single parent with two or more children
Couple with two or more children
Household location
Suburb
Purpose
Service
Mode
Bike
0.56
0.013
0.001
0.000
0.001
0.012
0.00015
0.01303
t-Test
66.45
6.62
0.35*
0.27*
0.28*
3.74
7.07
26.40
Multinomial logit zonal approach
Value
t-Test
0.43
45.52
0.02
0.02
0.01
0.03
0.09
6.62
4.93
12.12
5.45
5.75
0.00002
0.00256
3.60
18.02
0.03
1.30*
0.01
1.43*
3.10
0.08*
0.73
1.06*
20.13
29.59
6.62
0.00*
1.00
1.01
1.76*
1.76*
20.56
34.65
1.45
4.56
1.65
0.05*
1.02
2.00
6.04
0.04*
0.37
0.86*
Appendix C: Aggregated versus disaggregated multinomial logit
Appendix C. (Continued)
Multinomial logit Ben-Akiva type
Value
Number of estimated parameters
Sample size
Null log-likelihood
Rho-square
*Not significant at 0.001 level.
t-Test
15
3489
10,612
6220
8784
0.41
Multinomial logit zonal approach
Value
t-Test
15
3497
12,227
6743
10,968
0.45
Table (Continued)
181

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