formulating a methodology for modelling revealed preference discrete choice data—the selectively...

Pergamon

Transpn Res.-B. Vol. 31, No. 6, PP. 463472. 1997 Q 1997 Elsevier Science Ltd

All rights reserved. Printed in Great Britain Ol91-2615197 $17.00+0.00

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FORMULATING A METHODOLOGY FOR MODELLING REVEALED PREFERENCE DISCRETE CHOICE DATA-THE SELECTIVELY

REPLICATED LOGIT ESTIMATION

MARCO A. F. CALDAS* Departamento de Engenharia de ProduGo, Universidade Federal Fluminense-Brazil Rua Passo da Patria 156,

Sala 309, S. Domingos, 24.210-240, Niteroi, Rio de Janeiro, Brazil

and

IAN G. BLACK Centre for Logistics and Transportation, School of Management, Cranfield University, Cranfield, Bedford, UK, MK43 OAL

(Received 15 May 1996; in revisedform 12 April 1997)

Abstract-There are a number of studies on modelling with Revealed Preference (RP) data. It is a traditional technique and it is based on actual market data. The method has been extensively used in transportation as a tool for predicting travel demand. Although the method constitutes a relevant analysis on the process of modelling, it suffers from limitations, mainly associated with the lack of control over the experiment, that sometimes overwhelm the model results. This work proposes and tests a methodology for estimating a more efficient binary RP sample set. The objective is to develop and test a methodology that identifies and elimi- nates potentially irrational choices made. Responses are evaluated according to the set of trade-offs in values of time. Having identified these individuals they are eliminated from the original sample and a new sample is created, the selectively replicated (SR) sample. Original and SR samples are then re-estimated in a tree nested logit structure. 0 1997 Elsevier Science Ltd

Keywords: revealed preference (RP), stated preference (SP), demand forecesting, mode choice, multinomial logit model.

I. INTRODUCTION

In the transportation research field, methods of Revealed Preference (RP) have been developed for the analysis of travel demand and applied to various aspects of travel behaviour. The strength when modelling with RP data is that the data set is based on an individual’s actual market behaviour. On the other hand, the method has disadvantages that may introduce imprecision in the set of trade-offs leading parameter estimates to be biased and/or with high variances. One of these disadvantages is that pre-specified boundaries for the error term related to respondents’ percep- tions on the set of variables cannot be incorporated in the questionnaire design. In other words, the interpretation and the choice made against the actual set of trade-offs depends exclusively upon the respondents’ market perception.

Alternatively, respondents’ choices can be modelled with Stated Preference (SP) data. The main characteristic when collecting this type of data is the previous experimental study that sets up alternatives and trade-offs in variables. This characteristic may optimise the questionnaire by introducing trade-offs’ research background in the set of alternative choices, leading to a more controllable task.

The latest method has been the focus of research in the last few years with some important new issues on the minimisation of the variance in parameter estimates. Scaling and enrichment of

*Author for correspondence.

463

464 Marco A. F. Caldas and Ian G. Black

parameters, when estimating with SP and RP data simultaneously by using nested logit structure, is one of these new issues. It constitutes an excellent method of combining data drawn from sources of different reliability. It scales the variance of the error term associated with SP data by jointly estimating a nested logit model with RP and SP data (Morikawa, 1989; Bradley and Daly, 1991, 1992). Another breakthrough in SP research is the study of the individual boundary line for determining efficient SP experimental non-orthogonal designs. These lines define the geometric place where individuals are indifferent between each pair of alternatives. The set of boundary lines is called a boundary map. In a three variable or two value-of-time experiment case (i.e. value of in- vehicle time and value of access time), a two dimensional map of boundary lines can be drawn. Holden et al. (1992) have shown that SP non-orthogonal designs based on boundary maps may reduce the variance of the error term and parameters may be estimated more efficiently. The method encloses areas of the map in which most probable to occur values of time of a population lie, and the choices are more consistent with these values of time. Optimised pre-specified values of time reflect optimised trade-offs in pairs of time and money.

This work presents a methodology to deal with RP data in an experimental way. It is named Selectively Replicated Logit Estimation (SRLE). The methodology draws the boundary map for RP observations. It also incorporates the ranges of most probably to occur values of time for the population. These ranges are the limits of a geometric form (an area in the case of two values of time) that will embody the set of most rational choices that individuals have made related to the revealed trade-offs in time and money. The lines that do not cross the form are withdrawn from the sample and a new sample is created. It is important to point out that, if the values of time for the population are not known a priori, an assumption must be made on the range in which they should lie.

The objective of the work is to create a more efficient RP sample, in the sense of lower parameter estimates’ variances and lower error term when modelling individuals’ preferences. This gain in efficiency is achieved without enlarging the sample size. In other words, it raises sample perfor- mance without raising the costs of data collection. The methodology presented may become a new strategy for sampling.

2. THE FRAMEWORK FOR THE SRLE

2.1. Creating the selectively replicated (SR) sample For a sample of N binary choices comprising three variables, two variables for time and one

variable for cost the utility functions for the alternatives 1 and 2 are:

UI = PO + Bt Cost1 + &Time1 1 + BsTime2t + s1 (1)

u2 = BI Cost2 + /%Timelz + BjTime22 + a2 (2)

where:

& = Alternative Specific Constant (ASC); pi = Coefficient for Cost (Cost,-level of Cost for alternative 1 and Costz-level of Cost

for alternative 2); 82 = Coefficient for Time1 (Time1 i-level of Time1 for alternative 1 and Timelz-level of

Time1 for alternative 2); B3 = Coefficient for Time2 (Time2i--level of Time2 for alternative 1 and Time22-level of

Time2 for alternative 2); ~1.2 = the influence of unobserved effects or the error term for alternatives 1 and 2.

The subjective values of time for Time1 and Time2 are, respectively, the ratios of time and cost coefficients, /?z//$ and /&/fit.

The above notations for the utility functions of alternatives 1 and 2 can be re-written if all parameters are normalised by the cost coefficient and the forms are:

The selectively replicated logit estimation 465

UI = PO + BI(Costl + h/BITimell + MB1Time21) + EI (3)

U2 = /?l(Cost2 + /&/plTimel2 + jh//hTime22) + ~2 (4)

If an individual is indifferent to alternatives 1 and 2, Ui = U2:

/IO + BI(Costl + jb/#$Timell + &/BlTime21) = Bl(Cost2 + &/&Time12 + /b/jhTime22)

and making j%/fil =Value of Timel-VOTl and /Is/p, = Value of Time2-VOT2, both in cost units, the above equation becomes:

/30//31 + (Cost, - Costz) + VOTl(Time11 - Time12) + VOT2(Time21 - Time22) = 0 (5)

0

VOTl = (Cost2 - Costt) (Time22 - TimeZr)

(Timeli -Timel2)+(Timelr -Timelz) * VOT2 - f: 1

(Time1 1 - Time12) (6)

The above equation represents a straight line, relating two values of time, VOTl and VOT2. The generalised cost function for any pair of alternatives under the above indifference condi-

tions for an individual define a line where the VOTl and VOT2 lie. If there is a preference for one alternative the values-of-time must lie on one side of the line, while preference for the other indi- cates that the values lie on the other side. Figure 1 is an example of a boundary map for a two dimensional case. The rectangle ABCD is the area that embodies the most likely to occur values for the population.

The SR sample set comprises individuals whose respective boundary lines cross the rectangle. The above structure for the presented methodology takes into account only two values of time

and two alternatives, creating a particular and simplified case. However, the structure could be generalised for more than two values of time as well as more variables (other than time and money) and more choices. In other words, the primary structure does not preclude that the only willingness to pay of interest is the value of time. It does not preclude either that it is applicable only for a binary case. More than two values of time will lead the boundary map to have more than two axes. In the case of three values of time, a geometric volume will embody the lines, creating the SR sample. More relevant variables other than time and money will add to eqn (6)

Fig. I. A representation of the two dimentional boundary map.


constants obtained from the relation between the coefficients estimated for these variables and the coefficient estimated for the cost variable. Finally, the structure for a multinomial case follows the generalisation of the estimation of the binary logit model (Ben-Akiva and Lerman, 1985).

2.2. Capturing the relationship between the variances of the error terms If the reduced or SR sample is more reliable (lower error term) than the full or original sample,

the relationship between the respective variances of the error term can be captured in a special type of parameter estimate. A mixed structure, in the sense of using two different sets of data with presumed different error variances, can be built using an artificial tree logit structure.

The artificial tree logit structure follows Bradley and Daly (1991). It shows, in Fig. 2: The utility functions for both original (0) and SR samples are:

uo = px+eo (7)

USR = gx + -%R (8)

where:

B is a vector of coefficients (unknown parameters) to be estimated in both specifications;

X are shared variables that occurs in both original and replicate data sets;

CO, &SR denote the influence of unobserved effects related to original and replicate data sets (error terms).

The error components so and &sR are assumed to be independently and identically Gumbell distributed with zero means and variances:

var(&sR) = -$ 1

Var(co) = -$ 2

(10)

The factors 111 and /..Lz are the scale factors or scale parameters that are associated with both the magnitudes of parameter estimates and the variance of the error term (Ben-Akiva and Lerman, 1985) in both utility functions. It cannot be distinguished from the overall scale of the /3 in ordin- ary logit estimation and the usual procedure is to set it as 1. This assumption leads the estimation to have constant variance across observations related to both data sets. However, the SRLE assumes so and &sR to have different distribution and co is expected to hold more random effects than &sR.

In the tree structure the samples are estimated separately. The utility function for the replicated sample is estimated as a standard logit model while the utility function for the original sample is

SR Sample

Fig. 2. The artificial tree logit structure for capturing the relationship in data variances.


estimated by the tree logit structure. Standard logit estimation procedures set the scale factor that is included in eqn (9) as 1, so, the ratio of the variances given by the ratio of eqns (9) and (10) is:

Var(.so)p{ = Var(s& (11)

In the artificial tree structure, the mean utility of each of the dummy composite alternatives assumes a logsum form in which the sum of the logarithm is taken over all of the RP original alternatives (Daly, 1987). The deterministic utility of the composite alternatives becomes:

V camp = I-L log C exp( V0) (12)

For just one dummy alternative in the tree logit and considering that VO comprises only the deterministic part of the eqn (7) (without the error term), eqn (12) can be re-written as:

vcomp = /-4x0 (13)

Software which estimates full information multinomial nested logit models-FIML-NL such as the ALOGIT-Hague Consulting Group can solve the above tree structure and estimate the I_L parameter in the eqn (13).

3. A CASE STUDY ON THE APPLICATION OF THE SRLE

The RP survey was carried out in Rio de Janeiro, Brazil in November-December 1993 on a suburban train line that is part of the whole train network that serves the Rio de Janeiro metro- politan area. Two transportation modes are available for the target population: buses and train. Figure 3 represents the region as well as the ‘state of the art’ of the train and bus lines.

The dotted lines represent actual bus services for the target area. There are two bus lines that can be used to reach the city centre-one with no connections and another one where commuters have to connect to another bus line in the connection point C. The bold lines represents the actual electric train services. The actual electric train service runs between the city centre and the connection point C where commuters have to connect to an actual diesel train line if they want to proceed to the target area. The diesel train operation is represented by the straight line between C and the target area. The distance between the central point of the target area and the city centre is about 50 km with the connection point situated 15 km from the central point of the target area.

Actual Direct

,’ ,’

/’ /’

,’ ,’ Actual Bus Line

Actual Diesel

Electric Train

Fig. 3. A representation of the actual services in the surveyed area.


Table 1. Modal split for the samples

Modal

Rail Bus

Household (%)

24 76

Original (%)

49 51

SR (%) Joint (%)

54 51 46 49

The survey collected two different types of data in three different places. Face to face questionnaires were randomly administered to households inside the target area, represented in Fig. 2 (household survey). Face to face questionnaires were randomly administered to passengers using train and buses. The numbers of respondents to the household and choice based surveys were 184 and 907, respectively.

The household survey revealed information about commuters and trips in the area. The average household income is approx. Cr%22.000 or US$200 month. The average in-vehicle time for total trips is 84 min for buses and 66 min for trains. There is one connection for commuters travelling by train and one connection for commuters travelling by one category of buses. The average transfer time is 16 min for buses and 32 min for trains. The modal split for the household sample, as well as Original, SR and Joint choice based samples are shown in Table 1.

3.1. Model estimation with the original set of RP data The binary RP model specification includes:

Train dummy (TD) = 1: if the mode is train

Invtime, Invtimeb Connect, Connectb cost, Costb

= Total in-vehicle time related to individuals using the train model = Total in-vehicle time for individuals using bus = Total connection time for train users = Total connection time for bus users = Total cost for the train trip =Total cost for the bus trip

The binary utility functions for train t and bus b for an individual n are:

ut = Bo + /% Invtime, + B*Conne&, + fi3COSt, + El (14)

ub = Bl Invtimeb + /%onnectb + p3costb -i- Eb (15)

The software ALOGIT (Hague Consulting Group-Netherlands) is used to estimate the model parameters. The results are in Table 2 below.

3.2. IdentifVing and eliminating observations from the RP sample The generalised cost functions for identifying the boundary lines follow eqns (3)-(6). Software is

created to draw as many lines as the number of observations in the original RP sample set. The final two dimensional boundary map is found to be similar to the one represented in Fig. 1,

Table 2. Estimation results with the original RP data set

Variable Coefficient Standard Error t-statistic

Train dummy’ 1.744 0.516 3.4 Invtime -0.003315 0.00218 -1.5 Connect -0.07745 0.00916 -8.5 cost -0.1153 0.0454 -2.5

$=0.1241 N=907 y(0) = -628.68 y(/¶) = -550.66 j2=0.1237

*The base mode is ‘bus’.


comprising the set of 907 lines. Finally, an area delimited by the most likely to occur values of in-vehicle and connection time for the population is included. Previous study showed that, for the target population, the most likely value of connection time lies around 3 Cr% min-’ and the most likely value for in-vehicle time lies around 2 Cr$ min-’ (Caldas, 1995).

The methodology identifies 270 lines that do not cross the area out of 907 original lines (29.77% of the original sample is eliminated) resulting in a SR sample of 637 observations.

Intuitively, the new sample seems to be less reliable than the full sample, because of its smaller size leading to possible statistical weaknesses. However, the empirical results show the contrary. The next section brings these results to comments and it also compares parameter estimates with all possible sample arrangements.

3.3. Standard iogit estimation with the SR sample set Firstly, the SR sample with 637 observations is re-estimated in a binary logit model form with

the utility functions in eqns (14) and (15). Table 3 presents the results of the estimation. Looking at the magnitudes and t-ratio of parameter estimates, there is no significant difference

between the two samples apart from the cost variable. In the estimation with the selective sample this variable is 38.70% greater than its estimation with the original sample. Its t-ratio is also more significant for the selective sample. This is rather important for values of time.

3.4. Nested logit estimation with original and SR samples This procedure aims to investigate the two sets of data which may have different reliability. If it

is true, the ratio of variances of the disturbance terms can be detected and it can be used to cali- brate the final set of parameters.

The utility functions for trains t and buses b associated with the original and SR samples are:

Ulo = Bo + Bi Invtime,o + BzConnectm + /3$ost,o + s,o (16)

UbO = j$ hvtimebo + pzconnectbo + pjcostb(, + &bO (17)

UlsR = @I Invtime,SR + &connect[SR + p3costtsR + &lsR (18)

UbSR = #$ InvtimebSR -t p2ConnectbSR + p3CoStbSR -t EbSR (19)

where /3 is a vector of common coefficients (unknown parameters) to be estimated in both specifications; Invtimea, Connect,o and Costto are the set of relevant variables for the train mode in the original sample; Invtimem, Connectbo and Cost bo are the set of relevant variables for the bus mode in the original sample; Invtime,sa, Connect,sa and Cost ,sn are the set of relevant variables for the train mode in the replicated sample; Invtimebsa, Connectbsn and Costbsa are the set of relevant variables for the bus mode in the replicated sample; E denote the influence of unobserved effects related to original and replicate data sets (error terms) which are assumed to be independently and identically distributed across all observations within each respective data set.

The results for the estimation of the tree structure with the original sample comprising 907 observations and the SR sample comprising 637 observations are shown in Table 4.

Table 3. Binary logit estimation with the SR data set

Variable

Train dummy’ Invtime Connect cost

N=637

Coefficient Standard Error

I .408 0.594 -0.003395 0.00246 -0.07814 0.0111 -0.1599 0.0560

y(O)=-441.53 y(B) = -378.36

r-statistic

2.4 - I .4 -7.0 -2.9

p*=o.1431 7=0.1397



Table 4. Estimation results for the tree logit specification after eliminating observations

Variable

Train dummy’ Invtime Connect cost

Coefficient

I .26 -0.004305 -0.07296 -0.1529

Standard Error f-statistic

0.359 3.5 0.00158 -2.7 0.00682 -10.7 0.0339 -4.5

F 0.8465 0.0322 26.3

p2=0.0733

N= 1544 y(O)= -2140.44 y(B) = - 1983.56 p’ = 0.0620


The parameter p is tested against 1:

0.8465 - 1

0.0322 = -4.767 (20)

The test shows that the parameter is significantly different from 1. Therefore, the estimation supports the proposed methodology.

3.5. Correcting for sample size The values for the standard errors and z-statistic found in the above estimation are increased by

the additional and virtual number of observations that the SRLE takes into account. It is easy to see this effect, for example, when duplicating each observation of the original sample:

The estimated variance (Var) of the sample mean x,v in a sample set of N observations is known as:

(21)

Replicating each observation of the sample leads the variance (Var) of the sample mean of the replicated sample X~N to be:

3&f 2 Etxl- Xl* Var(BN) = ShN =

i=l

2N

For large samples, it can be assumed that:

&fJXi-T)2=&2fjXi-%2

El r=l

thus,

S& _ 2N

S&-N =2

(22)

(23)

(24)

(25)

A final comparison among parameter estimators is shown in Table 5.


Table 5. A comparison among estimations

Original model SR model Joint model

Train dummy’ 1.744 (3.4) Invtime -0.0033 (-1.5) Connect -0.077 (-8.5) cost -0.115 (-2.5)

VOI 0.029

voc 0.67 0.49 0.46

907 637 1544 -628.6845 -441.5348 -2140.4385 -550.6626 -378.3627 -1983.5616

0.1241 0.1431 0.0733

0.1237 0.1397 0.0620

1.408 (2.4) -0.0034 (- 1.4) -0.078 (-7.0) -0.160 (-2.9)

0.021

1.26 (3.5) -0.0043 (-2.7) -0.073 (- 10.7) -0.160 (-4.5)

0.027

0.8465

Table 6. Final values for the train dummy

Original model SR model Joint model

Train dummy 0.634 0.10 0.07

The vector of parameters estimated by the three model specifications can be compared and it can be seen that the joint estimation reproduces similar values to the standard logit estimation with the SR data.

McFadden (1974) has proved that the maximum likelihood estimator for a choice-based sample is an efficient estimator except for the Alternative Specific Constant (ASC). For a certain set of alternatives where W, is the percentual share of each alternative in the market related to whole population and Hg is the percentual share of the same alternative related to the choice-based sample, the correct ASC for each alternative estimated from a choice-based data set is:

Correct ASC = Estimate ASC - Ln(H,/ W,) 126)

The values of W,, as well as the values of Hg for the three choice-based samples are obtained from Table 1. The final ASC for the three samples are presented in Table 6.

The calibrated mode shares as well as the remain set of variables in Table 5 show that the joint estimation reproduces similar values to the standard logit estimation with the SR data. The results indicate that the joint sample successfully replicates the SR sample.

As pointed out before, the set of results contradicts statistical expectations. The SR sample is just part of the original sample and, a priori, reductions in sample size would exclude important observations that could be vital for sample efficiency.

However, the actual application of sample segmentation based in Values of Time limits has proved that, although variances in the error term may still exist in the remain SR sample and, it is assumed to be constant along SR observations, they are lower than the variances in the original sample. This may constitute one more contribution to sampling strategy.

4. CONCLUSION AND COMMENTS

This work develops and tests a methodology for estimating a more efficient set of parameters with a sample of binary RP data. It is named Selectively Replicated Logit Estimation (SRLE). The methodology identifies inefficient responses by investigating individual’s boundary lines. The lines are drawn from a set of two sampled subjective values of time estimated with the full RP data set. Ranges of most probable to occur subjective values of time for the related population are obtained


from previous work and they define an area where choices should be more consistent with the two considered values of time. The lines that do not cross the area are eliminated from the original sample and the SR sample is created. Original and SR samples are re-estimated in a tree nested logit structure. Although actual research supports that efficiency in RP samples drops when observations are lost, here it is shown that it is possible to speculate over these observations. Generally, an indiscriminate elimination of observations in a sample causes a vital degeneration in parameter estimates. On the other hand, maintaining inaccurate observations should also lead parameters to be biased and with some additional levels of variances. The boundaries between the two assignments is the central focus and the objective of this research.

It is demonstrated here that an experimental procedure can be incorporated into the market- based environment that characterises a model of RP data. A laboratory tool for eliminating undesired observations is used in order to gain efficiency in parameter estimates without raising the sample size and the costs of data collection. A case study developed in Brazil demonstrates the applicability of the methodology and the following issues are achieved:

(i) some detected observations are assumed to be the cause of noises in RP data. These observations are eliminated from the sample and a new sample is created;

(ii) the relationship between the variances of parameter estimated with original and new samples is captured and the assumptions on the noises are confirmed;

(iii) parameter estimates are re-scaled; (iv) the efficiency of the sample is improved without enlarging the sample size.

The scope of this work takes into account two subjective values of time and two transportation modes. However, the structure could be generalized for more than two values of time as well as more variables (other than time and money) and more choices.

Acknowledgemenfs-This research was funded by the CNPq-Conselho National de Desenvolvimento Cientifico e Tecnolbgico. a Brazilian Research Council.

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Proceedings of&h International Conference on Travel Behaviour, pp. 117-133, Quebec. Bradley, M. and Daly, A. (1992) Uses of the logit scaling approach in stated preference analysis. Proceedings of 7th World

Conference on Transportation Research, Lyon. Caldas, M. (1995) Assessing the efficiency of revealed and stated preference methods for modelling transport demand. PhD

Dissertation, Centre for Logistics and Transportation, School of Management, Cranfield University, Cranfield, UK. Daly, A. (1987) Estimating tree logit models. Transportation Research-B 21, 251-267. Holden, D., Fowkes, T. and Wardman, M. (1992) Automatic stated preference design algorithms. Proceedings of Xxth

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