Empirical test of a constrained choice discrete model: Mode choice in São Paulo, Brazil

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Vol. 218. No. 2. pp. 103-115.

0191.2615/87 0 1987 Pergamm

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EMPIRICAL TEST OF A CONSTRAINED CHOICE DISCRETE MODEL: MODE CHOICE IN SAO PAULO, BRAZIL JOFFRE SWAIT Programa de Engenharia de Transportes/COPPE, Universidade Federal do Rio de Janeiro, Brazil

MOSHE BEN-AKIVA Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. (Received 3 July 1984; in revised form 6 November 1985)

Abstract-This

paper examines the properties and empirically tests a model of discrete choice which incorporates probabilistic choice set generation. Denominated the Parametrized Logit Captivity (PLC) model, it is a generalization of the well-known “dogit” specification. The PLC model is shown to be theoretically and empirically more flexible than the latter. Work mode choice data collected in a 1977 O/D survey in SHoPaulo, Brazil, is used to obtain parameter estimates, as well as to evaluate consumer reaction to a series of perturbations in travel time, travel cost and income, for both the PLC and Multinomial Logit models. Comparisons between the two specifications are made in terms of statistical fit, rcasonableness of predictions and differences in predictions across models.

1. INTRODUCTION

15 years have witnessed the transition of discrete choice modelling from the realm of state-of-the-art to that of state-of-the-practice, especially in the field of transportation demand estimation. The economic basis for probabilistic choice has been formalized by McFadden (198 l), who is responsible for many significant theoretical and empirical advances in the field. The reader is referred to Ben-Akiva and Lerman (1985) for an extensive review of the many developments in the area of probabilistic models. A basic premise for the theoretical development and practical utilization of discrete choice models is that the analyst is correctly able to specify the set from which an individual decisionmaker chooses a given alternative. It is almost always the case that only the observed choice is known with certainty, so that the analyst is burdened with the task of specifying the remaining (unknown) alternatives, usually based upon very limited information. Some early estimation work (e.g. Rassam et al., 1971; Domencich and McFadden, 1975) basically ignored the problem of possible mis-specification by assuming that all decision-makers chose from the same choice set. Realizing the inadequacy of this simplification [see the analysis by Williams and Ortuzar (1982)], practitioners soon began to combine sample information and logical tests in an attempt to better approximate an individual decision-maker’s actual choice set (e.g. Lerman, 1975; BenAkiva and Lerman, 1974; Train, 1980), hoping thus to eliminate a mis-specification in the choice model. A natural extension of this effort was the formulation of models that explicitly incorporate a parametrized probabilistic component for generating the alternatives to be included in the choice sets. Gaudry and Dagenais (1979), for instance, presented a specification in which the individual is either captive to an alternative or is free tochoose from the full choice set according to a Multinomial Logit (MNL) model; they called this the “dogit” model. Ben-Akiva (1977), considered the situation in which individuals have the full choice set or full choice set less one alternative, as well as a combination of captivity and the latter assumption. A small number of other authors [the reader is referred to Swait and Ben-Akiva (1985~) for a review of these works] have concerned themselves with the development of choice set generation models, but generally these studies have been of a theoretical nature. Little empirical work with such models is reported in the literature [exceptions are Wermuth (1978); Gaudry and Wills (1979); Pitschke (1980); Swait and Ben-Akiva (1985a,b)]. The purpose of this pap& is to report empirical experience with a specific choice set generation formulation, which we denominate the Parametrized Logit Captivity (PLC) model. The past

103

JOF!=RE SWAITand MOSHEBEN-AKIVA

104

This generalization of Gaudry and Dagenais’ (1979) “dogit” model is calibrated on work mode choice data for the city of Sao Paulo, Brazil, and compared with a standard MNL specification in terms of statistical fit and differences in predictions under alternate simulation scenarios. In the following sections we present the PLC model and discuss its properties, as well as briefly mentioning our estimation method. Next we describe the data utilized for parameter calibration, which is followed by a discussion of estimation results for both models. The two models are then utilized to predict consumer reaction to changes in travel time and cost, as well as perturbations to decision-makers’ incomes. An analysis of the differences in predictions is then carried out. 2. THE

PARAMETRIZED

LOGIT

CAPTIVITY

MODEL

As we mentioned above, the “dogit” model of Gaudry and Dagenais (1979) hypothesizes that an individual is either captive to alternative j E C, where C is the full choice set deemed available to him/her (generally differentiated among individuals), or is free to choose from among all the alternatives of C according to a MNL model. (We have omitted the individualspecific subscript for C, and other quantities, for the sake of simplicity in notation.) The fotm of this model is pi = 2%

1+u

+

( i 1 1+u

-

PilC, Vi

E CT

(1)

where ui is the odds that the individual is captive to i E C; u =

2

uj.

jEC

Pile is the MNL choice model. The first term on the right-hand side of (1) is the probability of captivity to i E C, while (1 + u)- ’ is the probability of the decision-maker being free to choose from the full choice set C. Although the MNL model is included in (1), Gaudry and Dagenais show that the “dogit” model does not display the Independence of Irrelevant Alternatives (HA) property of the MNL choice model. Indeed, avoidance of IIA seems to have been their primary motivation for development of the former specification, rather than the choice set generation interpretation adopted by Ben- Akiva ( 1977). The “dogit” model has been applied few times empirically. Gaudry and Wills (1979) calibrated the specification for two data sets, both of an aggregate nature. The first was a model of manner of payment (having to do with the number of tickets purchased at one time) for an urban transit mode, using time-series data; the second was a model of intercity mode choice between private automobile and all other modes, based on cross-section data. Swait and BenAkiva (1985a) used the “dogit” model with disaggregate work mode choice daia for the Brazilian city of Maceio. They compared model (1) to the standard MNL model and to a second choice set formation specification, the Independent Availability Logit model [see also Swait and BenAkiva (1985~) for a development of this model]. In his own development of (1), Ben-Akiva (1977) suggested that the captivity odds parameters ui be parametrized as functions of independent variables. Specifically, since the Ui are restricted to be non-negative, Ben-Akiva suggested that U(Xi) = exp(DX,), Vi E C,

(2)

where D = (d,, . . . , d,, . . , dK) is a vector of parameters, and Xi is a vector of socioeconomic characteristics of the decision-maker and attributes of alternative i. Substitution of (2) into (1) results in what we term the Parametrized Logit Captivity (PLC), model: Pi =

eDX>

1 + 2 (EC

(3)

eDX, IEC

IEC

Empirical test of a constrained choice discrete model

105

= (b,, . . . , b,, . . . , bL) is a vector of parameters of the MNL choice model, and

whereB Yi is a vector of socio-economic characteristics of the individual and attributes of alternative i. The two vectors Xi and Yido not generally contain the same independent variables, as we shall discuss below. Swait and Ben-Akiva (198%) postulate that models of discrete choice should have two stages, that of choice set generation and that of actual choice. They introduce the intuitive notion that constraints acting upon the decision-maker define the choice set. From the point of view of the analyst, these constraints are random due to incomplete knowledge, errors in data and so forth. This concept then leads to the formulation of specific models of choice set formation, when united to assumptions about the structure of possible choice sets. This approach to choice modelling emphasizes the importance of parametrizing the odds captivity parameters Ui, as in (2). This parametrization is attempting to describe, in a more flexible manner than can be done in a “dogit” model, the aggregate impact of constraints on the availability of alternatives, given the specific hypothesis about choice set structure adopted to arrive at (3). In this light, Xi should include those variables thought to explain captivity to alternative i, whereas Yi should include variables (perhaps partially or totally overlapping with Xi) which influence choice of i from among the alternatives of C. An aggregate description of constraints, as given in (2), is an intermediate step between a simpler, more aggregate representation as given in (l), and a full-fledged specification of individual constraints (e.g. maximum travel time/cost, acceptable access distance, vehicle availability, etc.), an approach advocated in Swait and Ben-Akiva (198%). The added flexibility of (2) with respect to (1) may well be necessary in practice to, at least partially, overcome the restrictive assumption of captivity or choice from the full choice set C, thus ignoring all intermediate choice sets. The empirical work of Swait and Ben-Akiva (1985a) lends credence to this contention. They calibrated both MNL and “dogit” models by income market segments, and found that the latter were statistically superior for the data they utilized. However, predictions from the two models were very similar, indicating that perhaps the choice set structure assumption of the “dogit” model was too restrictive for the choice and population in question, and that the simple parametrization of the captivity odds (as constants) was insufficient to compensate for the deficiency. Our empirical work with the more flexible form (3) shows that it is not only statistically superior to the MNL model for the data analyzed, but also yields quite different (and reasonable) predictions, in certain simulation scenarios, than the MNL formulation. Identification problems will result when calibrating (1) if the true model is a market-share or pure MNL specification. In this case, the Uiparameters are indistinguishable from the alternative-specific constants of the MNL model, resulting in the nonidentifiability of both sets of parameters. The same type of problem can occur with the PLC model. As is true with the “dogit” model, (3) permits identification of constants in the captivity odds functions in all alternatives, though only (J - 1) constants in the MNL portion of the specification are identifiable, if a total of J alternatives are encompassed in the model. Both Gaudry and Dagenais (1979) and Ben-Akiva (1977) derived the own and cross point elasticities of the choice probabilities of the “dogit” model with respect to independent variables in the MNL model. If an independent variable in (3) is present only in the MNL portion, the elasticities of the PLC specification are of the same form as those of the “dogit” model. Otherwise, suppose that Zi = Xi, = Yi,(i.e. an independent variable, such as income, is present as the /cth variable in the captivity odds function and as the Ith variable in the utility function of alternative i). Then the direct and cross elasticities of (3) are given by

GW

E?,= - [4XiMX)l(4Zi) +

q(X)

where q(X) = (1 + z

eDi)-‘.

JOFFRE SWAIT andMOSHE BEN-AKIVA Ep, E$y are the direct and cross point elasticities of the MNL model with respect to Zi; other quantities are as previously defined. Expressions (4) reduce, in general form, to their “dogit” counterparts when Zi is present only in the utility specification of alternative i, since the first term on the right-hand side of both expressions disappear (due to the fact that a[u(Xi)q(X)]laZi will be zero). These elasticities reduce exactly to those of the MNL model, when in addition, the parameter dk = +m (depending upon the sign Of Xik)Tcausing U(Xi)+ 0. The term q(X)[PilcIPi] is always positive and less than one, which results in the second terms in (4a, b) always being of the same sign, but smaller in magnitude than Ev, the elasticity of the MNL model. This property of the “dogit” model was noted by Gaudry and Dagenais (1979) and Ben-Akiva (1977). The PLC model, however, does not necessarily have this characteristic. The elasticities of the PLC specification will have its sign and magnitude determined by the relative magnitude and signs of both terms in the expressions, if the variable in question is present in both the captivity odds and utility functions. Therefore, contrary to both the MNL and “dogit” models, the PLC model permits alternatives to behave as complements as well as substitutes (the MNL and “dogit” models restrict alternatives to the latter behavior, as shown by Gaudry and Dagenais). It should be noted here that there has been one previous empirical experience with the PLC model, utilizing the same data set used here, and it is reported in Swait and Ben-Akiva (1985b). Due to a lack of understanding of the properties of the PLC model at the time, the authors imposed unnecessary restrictions on the captivity odds and utility function specifications, which resulted in an ambiguous performance of the PLC with respect to the MNL model. This deficiency is corrected in the present work, which shows unambiguously the statistical superiority of this stochastic choice set generation model for the modeling context investigated. Before describing the data utilized to test the PLC model, we briefly mention that the estimation results presented in Section 4 were obtained by maximum likelihood, utilizing the method described in Bemdt er al. (1974). The software, described in Swait (1984), permits imposition of upper and lower bounds on parameters, a necessary precaution given the highly nonlinear nature of the likelihood function, its lack of desirable properties (e.g. concavity), and the possibility of identification problems. 106

3. DESCRIPTION OF THE DATA The choice we shall examine in our empirical application is mode choice for work trips. More specifically, we shall use home-based, morning peak period (arrival time before lo:30 AM) trips. Of the 17,088 such one way trips contained in the 1977 Sao Paulo Origin/Destination Survey (EMPLASA, 1978a, b), 1746 were randomly selected for inclusion in the estimation data set. The O/D survey provides a rich source of information on modal choice, including access, egress and principal modes. Unfortunately, the currently available impedance measures (which represent the shortest path over all access, egress and main mode combinations) have forced us to perform an aggregation of the elemental choices into the following alternatives: (1) (2) (3) (4) (5) (6)

bus, auto drive, auto passenger, train, metro, and walk.

No two zones are served by the two rail modes since the physical networks are nonoverlapping. For our study, these two modes have been combined into a single rail alternative. The availability of walk trips for Sao Paulo adds significantly to the applicability of our modelling results. In fact, the unavailability of walk trips in Maceio, Brasil, is one of the deficiencies noted by Swait et al. (1984) concerning the data usually collected in transport studies in developing countries.

Empirical

test of a constrained

choice discrete model

The deterministic allocation of alternatives to individual (1) (2)

trip

107

makers followed certain rules:

the network connection for the observed and generated modes had to exist; the maximum allowed one-way travel time was 3 h for bus, auto and walk, and 4 h for the rail mode (clearly the latter restriction applies, in practice, to the suburban train mode, not the metro alternative).

We are unable to restrict the auto drive alternative to individuals from auto-owning households, since a significant number of observed drivers declared an automobile ownership level of zero. The application of the above rules eliminated 20 observed trips, so that the final estimation data set consists of 1726 trips. For informational purposes only, the distribution of deterministic choice sets in the estimation data set is given below. Observed Trips

Choice Set Bus, Bus, Bus, Bus,

Auto Auto Auto Auto

Drive, Drive, Drive, Drive,

Auto Auto Auto Auto

Pass. Pass., Rail Pass., Walk Pass., Rail, Walk

67 231 942 486 1726

The actual estimation data set, however, has one less observation. Following initial calibration results, an outlier analysis was performed, and an observation with a miscoded income value was detected and removed from the data. Therefore, the calibration results are for a random sample of 1725 workers.

4.

MODEL

ESTIMATION

RESULTS

Table 1 presents the parameter estimates for the pure MNL and PLC models. The utility specifications include alternative-specific constants and travel impedance measures, as well as a number of socio-economic variables to capture systematic taste variations. The specifications of the various captivity odds functions includes mostly the same so&o-economic variables as found in the utility functions, though certain differences exist (e.g. the inclusion of gender and age in the walk captivity function). The MNL model includes a total of 29 parameters, while the PLC model has an additional 30 parameters in the captivity odds functions. Note that the specification of the utility functions is the same in both models, making the MNL model derivable from the PLC formulation by setting the parameters of the captivity odds functions to --a;. The log likelihoods at convergence for the MNL and PLC models are - 1437.0 and - 1394.2, respectively. We utilize the calculated x2 statistic of - 2( - 1437.0 + 1394.2) = 85.6, with a conservative 30 degrees of freedom, to test the hypothesis that the data provide no evidence for the existence of captivity to the various modes. The corresponding value from the x2 distribution is 50.9 at a 99% confidence level, from which we conclude that the hypothesis of no captivity can be rejected. Thus, contrary to prior experience with the PLC model (Swait and Ben-Akiva, 1985b), the present PLC specification is unambiguously statistically superior to the pure MNL model, despite the high “price” of 30 additional parameters. We see an increase in the scale factor of the MNL model by noting the generalized trend of utility function parameters being greater (in absolute value) in the PLC model than in the MNL. This scale parameter, unidentifiable in linear-in-parameters specifications, is inversely related to the variance of the underlying Gumbel distribution. Hence increases in the scale factor correspond to decreases in the variance of the stochastic component of the utility functions [see Ben-Akiva and Lerman (1985) for further information on the scale factor]. This scale increase can be interpreted intuitively by noting that the PLC model “removes” nonchoosers from the determination of the utility function parameters, whereas they are adversely biasing the corresponding values in the pure MNL formulation. [See Swait and Ben-Akiva (1985b) for a theoretical analysis of this effect.] For example, the alternative-specific constants of the utility functions in the PLC model TR21:28-8

JOFFRE SWAIT and MOSHE BEN-AKIVA

108

Table 1. Sb Paul0 AM peak home-based

work mode choice models Estimated Parameters (Asymptotic

Variables Utility Functions 1. Constants -Walk -Bus -Auto Drive -Auto Passenger -Rail 2. Travel Time, door-to-door, one-way, time if time > 0 and c 30 O.W. 30 -Walk timeif time > 30 30 0

3.

4.

5.

6.

7.

8.

Logit

r-ratios in parenthesis) Parametrized Logit Captivity

-o-

-O-0.98 (-3.4) -4.80(-11.0) -4.42 (- 10.7) -2.29 (-5.1)

-

-0.0844

-0.4864

(- 2.6)

-0.0583

(-4.6)

-0.0524

(- 5.2)

-0.0902 -0.0112

(-2.9) (-2.2)

12.44 (-2.2) 5006.73* 15.58 (-2.7) 15.31 (-2.7)

in minutes

-0.252

(-7.7)

(-7.4)

O.W.

-Bus (-2.5) -0.0117 -Auto Drive and Auto (-2.4) Passenger -0.0310 -Rail (-0.1) -0.0003 One-Way Travel Cost (Cr$ 1977)1 Income (defined in item 4, below) -Walk -O--Bus -647.6 (-4.2) -Auto Drive and Auto Passenger -76.4 (- 1.6) -706.1 (-3.8) -Rail Income = Personal monthly income (0% 1977) if visitor or boarder = Household monthly income (Cr$ 1977), otherwise -o-Walk - 2.58-6 (-0.2) -Bus 22.58-6 (2.2) -Auto Drive 9.OE-6 (0.6) -Auto Passenger - 10.4E-6 (-0.5) -Rail Members = 0 if visitor or boarder = # household members O.W. -o-Walk -0.082 (- 2.3) -Bus -0.128 (-2.7) -Auto Drive -0.092 (- 1.6) -Auto Passenger -0.141 (-2.6) -Rail Auto Availability = 0 if visitor or boarder = #cars/#workers O.W. -O-Walk -0.911 (-3.5) -Bus 1.220 (6.2) -Auto Drive -0.295 (-0.8) -Auto Passenger -0.350 (-0.9) -Rail Auto Ownership = 1 if h.h. owns 1 + autos = 0 O.W. 2.66 (6.8) -Auto Drive 1.86 (5.4) -Auto Passenger Head of Auto-owning Household for Auto Drive alternative = lif h.h. owns I+ autos and 0.65 (3.2) worker is head of h.h. and mode is Auto Drive = 0 O.W.

-O-282.4 (-0.8) - 106.8 (-0.7) - 1086.5 (-2.5)

-o99.9E-6 168.6E-6 94.98-6 75.9E-6

(1.6) (2.4) (1.1) (1.1)

-o-0.248 -0.636 -0.502 -0.267

(-2.6) (-2.9) (-3.1) (-2.4)

-o-1.153 3.571 -0.951 -0.748

(-1.2) (2.4) (-0.4) (-0.6)

4990.57 (809.8) 1.22 (1.0)

0.05 (0.1)

Empirical

test of a constrained

Worker with secondary or university education for Auto Drive and Passenger Modes = 1 if worker has secondary or university education and mode is Auto Drive or Passenger = 0 O.W. 10. Gender for Auto Drive = 1 if female worker and Auto Drive = 0 O.W. Captiviry Functions 1. Walk Alternative -Constant --Income -Members -Auto availability -Gender (1 if female; 0 if male) -Head of household (1 if worker is head; 0 otherwise) -Age, in years --Gender*Age 2. Bus Alternative -Constant -Income -Members -Auto availability 3. Auto Drive Alternative -Constant --Income -Members -Auto availability -Head of household -Auto owning household --Secondry or university education -Gender 4. Auto Passenger Alternative -Constant -Income -Members -Auto availability -Auto owning household -Gender 5. Rail Alternative -Constant -Income -Members -Auto availability Summary Statistics -Log likelihood for random choice -Log likelihood at convergence -rho squared -rho-bar squared (Akaike) -# parameters

109

choice discrete model

9

Sample Description-Observed Walk 409 Bus 626 Auto Drive 466 Auto Passenger 95 Rail 129 Total

1725

Choices

0.38 (2.3)

-1.21

0.56 (1.3)

(-4.9)

- 1.33 (- 1.6)

--m

--35 --31 -cc --m

-3.41 (-4.0) 17. IE-6 (0.8) -0.095 (-0.8) -0.761 (-0.8) 0.24 (0.2)

-ma

-1.63

--31

0.063 (2.9) -0.051 (-1.1)

--oo -cc -cc -02

--co

-co -co -co -02

--m --m

-cc

--m --co -cc -cc -02

--m -co

-cc -co -co --3o

- 2480.5 - 1437.0 0.4207 0.4090 29

(-2.1)

-0.95 -5.78-6 -0.069 -0.910

(-2.9) (-0.2) (- 1.1) (- 1.9)

-4.22 8.38-6 -0.010 0.568 0.70 3.02

(-8.5) (0.5) (-0.2) (2.2) (2.5) (7.2)

0.31 - 1.31

(1.3) (-3.0)

-5.25 9.9E-6 0.077 -0.397 2.93 0.18

(-5.8) (0.6) (0.9) (-0.8) (3.9) (0.5)

-0.72 9.5E-6 -0.495 -0.183

(-1.1) (0.3) (-2.0) (-0.2)

- 2480.5 - 1394.2 0.4380 0.4146 59

110

JOFFRESWAIT and MOSHE BEN-AKIVA

are significantly larger than those of the pure MNL model. Note that the constant for the auto drive mode is constrained to the reported value, due to identification problems with the auto ownership variable (item 7 of the utility function specifications) for the auto drive mode. This very large negative constant for the auto drive mode, combined with the large positive parameter for the auto drive auto ownership dummy, result in the elimination of this mode for all workers whose households own no vehicle, since the choice probability will be essentially zero. This result stands in marked contrast to our deterministic choice set allocation rule (see the previous section). We were unwilling to eliminate the auto drive mode from the choice set of individuals from households with no vehicles, since some individuals with zero auto ownership are observed to be driving, but the PLC model is indicating that this rule seems inappropriate. The travel time coefficients have magnitudes 2-6 times greater in the PLC model, as compared to the MNL specification. In the case of the rail mode, the increase is on the order of 28 times, with the PLC coefficient being significantly different from zero, which is not true for the MNL model. Of the motorized modes, the suburban rail mode (here aggregated with metro) is the one with the generally highest travel times and lowest cost (due to large fare subsidies). The MNL specification indicates that its patrons, generally of the lowest economic strata of society, are insensitive to increases in travel time; upon accounting for captivity to the various modes via the PLC formulation, however, we find that these low-income users are sensitive to travel time changes, even if at a lower level than in other modes. In the case of the travel cost coefficients, there occurs a decrease in magnitude for the bus mode and’increases for the auto and rail modes, though the changes are smaller than those of the travel time variables. In addition, the cost coefficients for the bus and auto modes are not significantly different from zero in the PLC model. Using the travel time and cost coefficients, it is possible to estimate the value of time for patrons of the various modes in the following manner:

Vi =

(b,ilb,i)I,

where i indexes the four motorized modes, ui is the value of time as a function of mode i and the individual’s income I, and b,, and bci are the estimated travel time and cost coefficients for mode i, respectively. Below we present the ratio vi/I, which expresses the rate of increase in value of time per unit of income, for both models:

Bus Auto Drive and Passenger Rail

MNL

PLC

18.1 x 1O-6 405.8 x lo+ 0.4 x 10-6

1856 x 10-6 844.6 x 10-6 10.3 x 10-6

Clearly, despite the large confidence intervals associated with most of these indices, due to the large standard errors of certain of the cost and time parameters, the PLC model will produce quite different reactions to changes in these impedance measures than will the MNL model. The alternative-specific income and number of household members variables were included in the utility function specifications to describe the effect of disposible income upon mode choice, thus accounting for fixed maintenance costs for household members. A priori, one would postulate a positive sign for the coefficients of income and a negative sign for those of number of household members. In the MNL model, with the exception of the auto drive mode, the income coefficients are not significantly different from zero, and two of the modes display a negative sign. The PLC model has income coefficients that are all positive, as expected, though only the auto drive mode’s coefficient is different from zero at an acceptable significance level. Note again the increases in magnitude of these coefficients across the two models, as well as the increase in the asymptotic r-ratios of the PLC income coefficients. The coefficients of number of household members, while displaying correct signs in both models, shows the same pattern of magnitude and accuracy increase. The mode-specific auto availability variables have the purpose of describing the effect upon choice of the competition between workers within a household for its own transportation resource,

Empiricaltestof a constrainedchoice discrete model

111

automobile. The coefficients of both models display the expected signs, but the significance level of the parameters in the PLC model are smaller than those in the MNL specification. This may be due to the inclusion of the competition variables in the captivity functions, which may be a more proper place to capture this effect. The remaining variables in the utility function specifications, which are mode-specific socioeconomic dummies, generally display the same pattern of increased magnitude and f-ratios of the PLC parameters in relation to the MNL parameters. Turning our attention now to the captivity odds functions, we note that, individually, many of the parameters are not significantly different from zero. This may be due to multicollinearity between some of the variables. For example, income coefficients are not significant in any of the captivity odds functions, perhaps due to the presence of the auto availability variables in both the utility and captivity odds functions. The auto availability coefficients are generally significantly different from zero, and of the correct sign, in all functions. Thus, as auto availability increases (i.e. more cars per worker), the odds of captivity decrease for all modes except auto drive. The strong effect of auto ownership on the odds of captivity to the auto drive and passenger modes is discernible from the large and significant coefficients of the auto ownership dummies in these two functions, especially for the auto drive mode. This effect is shown graphically in Fig. 1, which depicts the estimated

the

2 auto \_

0.25 0 auto \ 04 0

25

50

75

100

Xonthly &x.se~~EiIncome (000'sCR$ 1977)

FL3usehdACkxteristics:

4 nm&zrs,

1 xxker

Korker Chxacteristics: he&l of l-rxsehdd,male, 30 years Ad, high school ar universityalucation,all alternatives deterministically available. Fig. I. Estimated probabilities of captivity to auto drive by income and

autoownershiplevelWC model).

112

JOFFRESWAITandMosmB~~-AKIVA

probabilities of captivity [calculated according to the first term of (3)] to the auto drive mode as a function of income and auto ownership level for a household of four persons, in which the only worker is the 30-year-old male head of household, who has a university level education. The graph shows clearly that the determining factor (under the specified conditions and assuming that all modes are deterministically available) in explaining captivity to auto drive is possession of one vehicle. Additional vehicles, though they do increase the probability of captivity, have a smaller marginal effect. Approximately 45% of So Paul0 households owned one or more automobiles in 1977 (EMPLASA, 1978c), so that the effect discussed above is not applicable to a great portion of the population which does not own a vehicle. In Fig. 2 we present the probability of captivity to the bus and rail modes (supposing all modes are deterministically available) as a function of family size for a single worker household, in which the worker is a 30-year-old male head of household with a primary education and a monthly income of 1977 Cr$ 1,OOfl(approximately one minimum salary at the time the survey data were collected). For this context, the walk, auto drive and auto passenger modes have negligible probabilities of captivity (taken together, about a 5% chance of captivity to all three modes), while the bus and rail modes jointly account for a 23% to 38% chance of captivity, depending upon family size. Thus the PLC model is predicting a greater than 50% chance that the low-income worker is choosing from his full choice set. It seems unlikely that an individual in such circumstances has this flexibility of choice, which leads us to hypothesize that a choice set model which allows more intermediate

Nmha

of I-Kuseiml.d%m.tm-s

I-hsebld Characteristics:1 mrker,mnthly Wrker Characteristics:m

i-e

of 1977 cR$ 1,000

of hxseimld, male, 30 years

old,

e3ucation. Fig. 2. Estimated probabilities of captivity to bus and rail for low income worker WC model).

prirary

Empirical test of a constrained choice discrete model

113

size sets might be more appropriate for individuals with limited means. Despite this observation, it should be noted that the PLC model will produce very low probabilities of choice for the auto drive mode for this individual, since the alternative will be virtually eliminated from the full choice set by the large negative alternative-specific constant of the mode’s utility function. 5. PREDICTED

CONSUMER

REACTIONS

In this section we present and analyze the predictions generated by the MNL and PLC models for changes in travel time, travel cost and household income. The changes are applied to all individuals in the sample, and arc elasticities are estimated by sample enumeration. Table 2 shows the own and cross arc elasticities with respect to travel time, resulting from a uniform doubling of this modal attribute. The reason this large change is being imposed is to bring out the differences between the two specifications when they are subjected to perturbations in which the MNL model permits continuous changes, but the PLC model imposes a type of threshold below which no change occurs, due to the captivity assumption. Thus we see that the PLC generally predicts that consumer response to changes in the automobile and walk mode travel times is less than that predicted by the MNL model, especially in the case of the walk mode. In the case of the bus and rail travel times, however, the reverse is seen to be the case. This can be explained by the fact that the increase in magnitude of the travel time coefficients from the MNL to the PLC models is more than sufficient to compensate for the occurrence of captivity to these modes, resulting in a greater sensitivity of travellers to travel time in the PLC model. In the case of the bus mode, we note a doubling of its own elasticity, with even greater increases in the cross elasticities for auto passenger and rail, the two modes which are predicted to substitute the use of bus. The case of the rail mode is even more marked: its own elasticity is 15 times greater in the PLC model, with the principal shifts of patrons occurring to bus and auto passenger. In the case of changes in travel cost (shown in Table 3), the PLC predicts the consumers to be significantly less sensitive to travel cost changes than does the MNL model. Even the rail mode, whose travel cost coefficient (see Table 1, item 3) increases from the MNL to PLC model, is predicted to be less sensitive by the latter. This contrasts with the results for travel time. This greater sensitivity of rail mode travellers (especially when interpreted with respect to suburban train, rather than metro, users) to travel time rather than cost is supported by observation. During the 1970’s, there occurred several instances of mob violence against the suburban rail service in S&o Paul0 and Rio de Janeiro, with destruction of trains and stations. These incidents always occurred when, sometimes for reasons outside the control of the railway operator, trains suffered long delays (often of many hours duration). The patrons, almost exclusively of very low-income households, are generally employed at jobs that permit them little or no flexibility with respect to arrival time at work; hence the extreme anger at the delays. Thus policies aimed at improving suburban rail service should, according to the PLC model, be more effective if they concentrate on improving the adherence of trains to schedules and Table 2. Estimated arc elasticities for uniform 100% increase in travel time (a) Multinomial Logit Model

Mode Changed

Bus Auto Rail Walk

Bus -0.23 0.09 0 0.15

Arc Elasticity Auto Drive Auto Passenger 0.08 0.23 -0.14 -0.25 0 0 0.07 0.13

Rail 0.41 0.14 - 0.02 0.04

Walk 0.09 0.04 0 -0.35

Auto Drive 0.07 -0.10 0.01 0.02

Rail 0.88 0.13 -0.30 0.02

Walk 0.14 0.01 0 -0.14

(b) Parametrized Logit Captivity Model

Mode Changed

Bus Auto Rail Walk

Bus -0.45 0.08 0.05 0.07

Auto Passenger 0.85 -0.26 0.03 0.03

JOFFRE SWAIT and Mown

114

Table 3. Estimated arc elasticities (a) Multinomial

100% increase in travel cost

Logit Model

Mode Changed

Bus Auto Rail

Bus -0.14 0.02 0.04

(b) Parametrized

Logit Captivity

Model

Mode Changed

for uniform

BEN-AKIVA

Bus Auto Rail

Bus -0.03 0.01 0.04

Arc Elasticity Auto Drive Auto Passenger 0.03 0.12 -0.02 -0.07 0.01 0.02

Rail 0.24 0.03 -0.25

Walk 0.08 0 0

Auto Drive 0 -0.01 0

Rail 0.07 0.02 -0.22

Walk 0.01 0 0

Auto Passenger 0.04 -0.06 0.02

decreasing travel times, rather than increasing the level of fare subsidies. Purchase of modem rolling stock, traffic control equipment and track maintenance seem more promising investments than the direct operating subsidies given today. The final simulation performed with the two models is that of a doubling of income. The estimated arc elasticities are presented in Table 4, segmented by three income groups. Overall, the MNL model predicts that this drastic income distribution shift results in a loss of ridership for the bus and walk modes, with consequent gains in the others, especially the automobile drive and passenger modes. In contrast, the PLC model predicts that users of the auto passenger, rail and walk modes will transfer to bus (so that it remains in equilibrium) and auto drive. In both models, however, the aggregate elasticities are small. 6.

CONCLUSION

The Parametrized Logit Captivity probabilistic choice model, a generalization of the “dogit” captivity model of Gaudry and Dagenais (1979), has been presented and its properties discussed. The model assumes that the individual is either captive to a single alternative, or is free to choose from a choice set (specified by the analyst) according to a MNL model. The probability of captivity is expressed as a function of independent variables, which permits the PLC model to predict a more flexible range of responses to changes in independent variables than possible with either the “dogit” or MNL models. The PLC specification is compared statistically and with respect to predictions with the MNL model for work mode choice data for Sao Paulo, Brazil. For this data, the PLC is statistically superior to the MNL model, despite the large number of additional parameters in the former specification. The simulation of changes in travel time, travel cost and income result in some markedly different predicted responses by consumers. In certain cases, the PLC model predicts

Table 4. Estimated arc elasticities (a) Multinomial

Income Segment* (b) Parametrized

Income Segment*

for uniform

100% increase in household

income (CrS 1977)

Logit Model

Low Medium High Total Logit Captivity Low Medium High Total

Bus -0.68 -0.44 1.12 0

Arc Elasticity Auto Passenger Auto Drive -0.78 -0.76 -0.74 -0.64 0.43 0.42 0.10 -0.04

Rail -0.69 -0.35 1.13 -0.07

Walk -0.14 -0.57 0.64 -0.08

Auto Drive -0.78 -0.73 0.42 0.09

Rail -0.66 -0.35 1.31 0

Walk -0.73 -0.57 0.69 -0.05

Model Bus -0.70 -0.45 1.05 -0.04

Auto Passenger -0.69 -0.60 0.51 0.02

*Low: [0,4) minimum salaries (1 min. sal. = Cr$ 1,000 1977). Medium: [4,8) minimum salaries. High: [8, ) minimum salaries.

Empirical

test of a constrained

choice discrete model

115

lesser sensitivity to changes than does the MNL model (as one would expect a priori, due to the captivity assumption), but in other situations the opposite is seen to occur. Compared to other empirical applications of choice set generation models, in which the choice set probabilities are represented by simple constants (e.g. Swait and Ben-Akiva, 1985a), this application of the PLC model clearly demonstrates the great potential that parametrization of the choice set probabilities holds for better explaining observed choice behavior. REFERENCES Ben-Akiva, M. (1977) Choice Models With Simple Choice Set Generating Processes. Working Paper, Dept. of Civil Engineering, MIT, Cambridge, MA. Ben-Akiva M. and Lerman S. (1974) Some estimation results of a simultaneous model of auto ownership and mode choice to work. Transportation, 4, 357-376. Ben-Akiva M. and Lerman S. (1985) Discrete Choice Analysis: Theory and Application to Predict Travel Demand, MIT Press, Cambridge, MA. Bemdt E., Hall B., Hall R. and Hausman J. (1974) Estimation and inference in nonlinear structural models. Ann. Econ. Social Meas. 3.653-665. EMPLASA (1978a) ‘77 Origin-Destination Survey-Basic Results-Bilingual Document. Empresa Metropolitana de Planejamento da Grande S&o Paul0 S/A-EMPLASA, SHo Paula, Brazil. EMPLASA (1978b). O/D 77-Volume I-Referencias Basicas. Empresa Metropolitana de Planejamento da Grande Sao Paul0 S/A-EMPLASA, S&o Paulo, Brazil. EMPLASA (1978~) Pesquisa Chigem-Destine/77-Volume 4-Sumario de Dados. Empress Metropolitana de Planejamento da Grande SBo Paulo S/A-EMPLASA, SLo Paulo, Brazil. Gaudry M. and Dagenais M. (1979) The dogit model. Transp. Res. B 13B, 105-I Il. Gaudry M. and Wills M. (1979) Testing the Dogit Model With Aggregate Time-Series and Cross-Sectional Travel Data. Transp. Res. B 13B, 155-166. Lerman S. (1975) A disaggregate behavioural model of urban mobility decisions. Unpublished Ph.D. thesis, Dept. of Civil Engineering, MIT, Cambridge, MA. McFadden D. ( 198 1) Econometric Models of Probabilistic Choice. In Structural Mode/s of Discrete Data with Econometric Applications (Edited by C. Manski and D. McFadden). MIT Press, Cambridge, MA. Pitschke S. (1980) Choice set formulation for discrete choice models. Unpublished MSc thesis, Dept. of Civil Engineering, MIT, Cambridge, MA. Swait J. (1984) Probabilistic choice set formation in transportation demand models. Unpublished Ph.D. thesis, Dept. of Civil Engineering, MIT, Cambridge, MA. Swait J. and Ben-Akiva M. (1985a) Constraints to individual travel behavior in a Brazilian city. Transp. Res. Record, to be published. Swait J. and Ben-Akiva M. (1985b) Analysis of the effects of captivity on travel time and cost elasticities. Proc. of the 1985 International Conference on Travel Behavior, Introductory Papers, 16-19 April, 1985, Noordwijk, The Netherlands, pp, 113-128. Swait J. and Ben-Akiva M. (1985~) Incorporating random constraints in discrete models of choice set generation. Transp. Res. B, 21B. 91-102. Swait J., Kozel V., Barros R. and Ben-Akiva M. (1985). A model system of individual travel behavior for a Brazilian city. Tansp. Policy Decision-Making 2, 451-480. Train K. (1980) A structured logit model of auto ownership and mode choice. Rev. Econ. Studies XLVII, 357-370. Wermuth M. (1978) Structure and calibration of a behavioural and attitudinal binary mode choice model between public transportation and private car. Presented at PTRC Summer Annual Meeting, lo-13 July, Univ. of Watwick, England. Williams H. and Ortuzar J. (1982) Behavioural theories of dispersion and the mis-specification of travel demand models. Transp. Res. B 16B, 167-219.