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Institute of Transport Studies University of Leeds

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Published paper Ortuzar, J.D. (1979) Testing the Theoretical Accuracy of Travel Choice Models Using Monte Carlo Simulation. Institute of Transport Studies, University of Leeds, Working Paper 125

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UNIVERSITY OF LEEDS Institute for Transport Studies

ITS Working Paper 125

November 1979

TESTING THE THEORETICAL ACCURACY OF TRAVEL CHOICE MODELS USING MONTE CARLO SIMULATION

BY J. D. ORTUZAR

ITS Working Papers are intended to provide information and encourage discussion on a topic in advance of formal publication. They represent only the views of the authors, and do not necessarily reflect the views or approval of the sponsors.

- .

CONTENTS

Abstract 1.

Introduction

2.

The Generation of Random U t i l i t y Models

3.

A t t r i b u t e Correlation and Model Structures

4.

The Probit Model: S t r u c t u r e and Transformations

5.

The Theoretical Accuracy o f Alternative Logit Model Structures

6.

Conclusions

References Acknowledgements Figures Appendices

1

ABSTRACT

ORTUZAR, J. de D. (1979) Testing the theoretical accuracy of travel choice models using Monte Carlo simulation. Leeds: University of Leeds, Inst. Trans. Stud., WP 125 (unpublished)

In recent years a considerable advance has been made in the construction of micro-travel demand models from choice theoretic principles. Within random utility theory, the structure of models may be shown to relate to the perceived similarity between discrete choice alternatives, and this aspect may be interpreted mathematically in terms of the correlation between the components of random utility functions. Several possible model structures have now been proposed, varying from the multinomial logit model (uncorrelated) through the partly correlated structures (hierarchical and cross-correlated logit kctions) to the most general form of probit model which allows an arbitrary variance-covariance matrix. In this paper, these model structures are discussed using a geometric interpretation of random utility theory, and the possibility of invoking transformations on the general probit model is examined. Monte Carlo simulation methods are then used to investigate some aspects of the trade-off between the generality and accuracy of correlated structures (the cross-correlated logit model in particular) and the greater ease with which less consistent structures may be implemented. In this way, the theoretical accuracy of the multinomial logit model is assessed. It is concluded that where the $enera1 probit model is too complex to implement, the practice of comparing the multinomial logit model with alternative hierarchical logit structures is unlikely to lead to significant errors in forecasting.

TESTING CHOICE

THE

THEORETICAL ACCURACY

MODELS USING MONTE

CARLO

OF

TRAVEL

SIMULATION

INTRODUCTION

1.

I n recent years considerable i n t e r e s t has centred on t h e r e l a t i o n s h i p between t h e s t r u c t u r e of a t r a v e l demand model and t h e behavioural p r i n c i p l e s associated with its formation. -

This has a r i s e n not only

because of t h e need t o underpin models with a c o n s i s t e n t t h e o r e t i c a l r a t i o n a l e , but a l s o from t h e recognition of s t r u c t u r a l ambiguities i n e x i s t i n g models

-

a s , f o r example, with t h e r e l a t i v e p o s i t i o n s of

d i s t r i b u t i o n and modal s p l i t models i n t h e conventional planning system

-

which can give r i s e t o s i g n i f i c a n t l y d i f f e r e n t r e s u l t s i n policy

analysis

en-Akiva, 1974; Williams and Senior, 1977).

One p a r t i c u l a r

framework within which t h i s r e l a t i o n s h i p has been sought is t h a t provided by random u t i l i t y theory ( f o r a review, see Dcanencich and McFadden, 1975). I n t h i s quanta1 choice theory individuals a r e considered t o a s s o c i a t e with each member An; n = l , u t i l i t y Un; n = l , of U.

...,N,

...,N of

a d i s c r e t e s e t of options A , a net

and t o s e l e c t t h a t member with t h e highest value

To account f o r i n t e r p e r s o n a l v a r i a t i o n i n t h e value of a t t r i b u t e s

incorporated i n t h e u t i l i t y functions, a n d t h e i n f l u e n c e of unobserved f a c t o r s , t h e modeller considers t h e v a r i a b l e s (U1,

..., Un, ..., UN)

t o be

The

randomly d i s t r i b u t e d over t h e population confronted by a choice.

p r o b a b i l i t y P t h a t an i n d i v i d u a l with p a r t i c u l a r c h a r a c t e r i s t i c s s e l e c t s n an a l t e r n a t i v e An i s then simply expressed i n terms of t h e p r o b a b i l i t y t h a t Un be g r e a t e r than those values associated with a l l o t h e r options. A formal choice model may be derived when t h e density function f ( U-) = f (U1,

..., UN)

of t h e u t i l i t y components i s specified.

It has r e c e n t l y been recognised t h a t t h e a n a l y t i c s t r u c t u r e of a model i s c r u c i a l l y r e l a t e d t o t h e interdependency, o r s t a t i s t i c a l c o r r e l a t i o n , between t h e u t i l i t y functions associated with each a l t e r n a t i v e with t h e s t r u c t u r e of Zachary, 1978;

f(U)

(Williams, 1977; Langdon

McFadden, 1979).

1976;

-

that is,

Daly and

A s e t of formal models now e x i s t s which

accommodates varying degrees of "similarity" o r c o r r e l a t i o n between a l t e r n a t i v e s ranging from t h e widely used multinomial l o g i t model (MNL),

generated by uncorrelated distributions, through the hierarchical logit model (HI.,), to the generalised probit function (GP) with arbitrary correlation, expressed in terms of a variance - covariance matrix. Until recently application of the generalised probit model has been restricted to a small number (3 or 4) of choice options (Haussman

.

and Wise, 1978) However by invoking the Clark approximation (Clark, 19611, Daganzo et al (1977)have extended its practical range. In spite of the advances in its applicability there appear to be many practical cases in which (a) the model cannot cope (~aganzo,1979), or (b) there is a need for a compromise between the generality it can afford m d the greater ease with which less consistent structures may

, be implemented. One such compromise is the cross-correlated logit (CCL) function (Williams, 1977). which is a closed-form model containing alternative HL functions as special cases. In this paper we wish to examine some general themes such as the relationship between certain utility functions and the structure of travel choice models; the possibility of invoking transformations in order to simplify models and derive conceptual links between them; the theoretical accuracy of particular choice models, and the problems of misspecification associated with model structures and utility functions. More specifically, we wish to address the following questions:

i)

Is it possible to apply transformations in 'utility space' in order to simplify 'symmetric' probit models and enable conceptual links to be forged with the logit family?

ii) How serious is the absence of 'similarity effects' in the multinomial logit model? In other words, how much is the well known 'independence from irrelevant alternatives' (IIA) property of the model an impediment in choice modelling? iii) How good an approximation to a general function is the crosscorrelated logit model? iv) What is the effect of misspecification of choice models with respect to model structure and their utility components? v)

Can any of the logit models display pathological response properties, and is it possible to recognise their symptons at the -. calibration stage?

vi) Can we discriminate between contending model structures on the basis of goodness of statistical fit, and the character of their inherent elasticity parameters? In particular, does a good agreement to bese year data necessarily imply good response characteristics? It should be stressed at the outset that any reference to the accuracy of a model will refer to its consistency with the underlying theoretical rationale, and not necessarily to its appropriateness in choice modelling. -

In Section 2, the basic principles of generating random utility models are reviewed, a geometric interpretation of the theory is presented, and the Monte Carlo method as a means for numerical evaluation of choice models is outlined. The existence and implications of correlation between the utility functions associated with different alternatives are then examined in Section 3 and the various approaches to its incorporation in choice models noted. In Section 4 we investigate the possibility of invoking transfomations in utility space as a means of simplifying the general probit model. Although conceptionally appealing in terms of its links with MNL, HL and CCL structures, the potential for implementing such transfomations does not, in general, appear practicable. The numerical tests to determine the theoretical accuracy of the alternative logit structures in a general choice context are then described in detail in Section 5.

2.

THE GENERATION OF M D O M UTILITY MODELS Formally, we can express the model generator equations of rando~

utility theory as follows:

Pn = Prob (Un > Unl,V Ant EA)

in which f(U) is the joint distribution function of (U1, Rn is that region of utility space defined by

(2.1)

..., UN) and

Rn:

Un L Unt

vAnl EA

Un L 0

(2.3)

(2.4)

In this paper we shall be concerned only with those cases in which a trip is actually made. The non-negativity restriction (2.4) will thus be considered inoperative. For the distribution functions considered later this will involve a negligible inconsistency, which does not affect the argument to be presented. To derive an explicit probabilistic choice model we need to know both the form of f(1) and an expression for the utility functions in terms of the attributes of alternatives in the set A. We shall take the components Un to be of the following form: Un

=

un (g.$)

+ E,

(2.5)

in which n is the so-called 'representative' utility of the population Q confronted by the choice, and E~ is a stochastic residual. Un is normally taken to be linear in terms of the attributes 'Zn characterising A n

.

That is:

The vector of parameters

g

is estimated from observed choices. It

remains to specify the distribution function f(g) or equivalently that of the stochastic residuals g. A geometric interpretation of the theory may readily be derived from expression (2.2). In the utility space%& bounded by the components (U1, , UN ) , the probability Pn is, for normalised f(g),the total density of points in the region Rn bounded by the hyperplanes defined by:

...

and U =U n n'

V AneA

This can be more easily seen in the convenient cartesian space. In Figure l(a) we illustrate the fundamental utility distributions associated with binary choice (Williams, 1977). It is important to

distinguish those distributions g (U ) and g2(U ) ' w h i c are associated 1 1 2 with the population Q confronted by the choice between alternatives A1 and A2, from

gl(ul)

and

g2(u2)

which are the "choice specific"

distributions of utility, for those members of Q who have selected options A1 and A2 respectively. The sum of these last two distributions is termed the distribution of maximum utility g,(U), functions are formally defined as follows:

and the three

We shall also write the distribution of maximum utilities in the form

-

and we note here that the mean value of this distribution, U,, has great significance in the evaluation problem (Williams, 1977). The geometric interpretation of this simple choice process, which is an extension of that provided by Robertson (1977), is given in Figure l(b). For identical and independent distributions (IID), f(g) has a - The line OZ divides circularly symmetric shape centred on and U2. 1 the positive quadrant into the regions R1 and R2, and P1 and P2 comprise of those corresponding portions of the distribution in these regions. The distinction between g(U1) and the choice specific distribution g(U ) 1 can readily be seen in terms of the respective projections onto the U 1 axis of the density function f(g), and that portion of f(g) bounded by

OZ and the U1 axis.

An important class of random utility models includes those generated by IID utility distributions for which we can decompose f(E)as follows: N

(1) Because they are identical in the figure, we have labelled them g(U1) and g(U2), respectively.

I

- 6 Here g(Un) is t h e d i s t r i b u t i o n of t h e u t i l i t y component associated w i t h A n

.

The expression f o r Pn can now b e w r i t t e n

h i s s i o n of t h e c o n s t r a i n t (2.4) allows t h e lower l i m i t s of i n t e g r a t i o n t o be extended t o minus i n f i n i t y .

It i s by now widely known t h a t t h e much favoured multinomial l o g i t model (MNL)

i s an I I D model generated from Weibull (Gnedenko) p r o b a b i l i t y d i s t r i b u t i o n s ( ~ h a r l e sRivers Associates, 1972) f o r which

This i s a skewed unimodal d i s t r i b u t i o n , i n which t h e dispersion parameter A

i s inversely r e l a t e d t o t h e standard deviation, a , a s follows

( Cochrane , 1975) :

Similarly simple probit models a r e generated from I I D Normal d i s t r i b u t i o n s . For a number of s p e c i a l d i s t r i b u t i o n s , it i s possible t o evaluate t h e i n t e g r a l (2.2) t o produce a n a l y t i c a l expressions f o r Pn, such a s t h e

MNL i n equation (2.15). form of numerical method. simulation.

I n general, however, we have t o r e s o r t t o some One such approach involves Monte Carlo

A s f a r a s we a r e aware t h e first application of t h i s method

t o t h e solution of random u t i l i t y models i s t h a t of Albright, Lerman and Manski (1977). i n t h e development of an estimation program f o r t h e general probit model.

However, and i n most cases independently, t h e power of t h e

approach has a t t r a c t e d numerous applications recently (Bonsall, 1979; Chicago Area Transportation Study, 1979;

Horowitz, 1978;

Kreibich, 1979;

Manski and Lerman, 1978;

Robertson, 1977;

Robertson and

Kennedy, 1979;

Ortuzar, 1978;

W i l l i a m s and &uzar,

1979) but c l e a r l y its roots can be

t r a c e d back t o c e r t a i n s t o c h a s t i c assignment methods ( B u r r e l l , 1968).

I n t h i s approach we follow t r a d i t i o n (~ammersleyand Handscomb, 1965); a sample of s i z e S is created, and each 'individual' member t ,

..., %.

of S, i s confronted by t h e choice between A1, number generator a set of u t i l i t y values (U1,

Using a random

..., u ~ is) drawn from

f (-~ ) and , t h e member t i s assigned t o t h a t option with t h e maximum associated u t i l i t y .

For l a r g e S, t h e proportion Sn of 'individuals'

assigned t o option An w i l l approximate t o Pn, which i s given by

I n t h e simple Cartesian u t i l i t y plane examined before, t h e method involves randm sampling of p o i n t s from f(U1,

t observation (U1.

U2).

For a given sampled

t U 2 ) , t h e corresponding 'individual' w i l l be assigned

t o A1 o r A2 according t o t h e region i n which t h e

t t if U1 > U2, i . e .

t

t

(U1,U2)

sR1,

'point' may be found.

assign t o A1 (2.19)

t t t t if U < U2, i . e . (U1,u2) cR2, assign t o A2 1 To t e s t t h e accuracy of t h e method with sample s i z e , t h e numerical s o l u t i o n of a b o p t i o n l o g i t model was compared with t h e a n a l y t i c solution. For a sample of s i z e S, t h e choice p r o b a b i l i t i e s P'

n

were determined by

drawing random values from four I I D Weibull functions, with given means

-

-

-

(ul, U2, U3,

-

Uq) and standard deviation

0.

Tnese numerically derived

p r o b a b i l i t i e s were then f i t t e d by a l o g i t function

i n which t h e parameter A

S

was estimated by t h e usual maximum l i k e l i h o o d

method (Domencich and McFadden, 1975).

I n Figure 2 , we show t h e

empirically derived r e l a t i o n s h i p between t h e variance of A~ with t h e s i z e of t h e sample S.

I n order t o examine t h e accuracy of t h e numerical

solution under d i f f e r e n t conditions, we repeated t h i s procedure f o r a binary l o g i t model and d i f f e r e n t values of t h e difference between mean utilities.

A s it can be seen, t h e c l o s e r t h e options (smaller difference

i n mean u t i l i t i e s ) , t h e l e s s s t a b l e t h e simulation becomes.

In the

numerical t e s t s described i n later s e c t i o n s t h e sample s i z e was fixed a t S = 30,000. (2

We now proceed t o consider more complex choice contexts i n which t h e presence of c o r r e l a t i o n between u t i l i t y functions i s c e n t r a l t o t h e s t r u c t u r a l develoment of t h e models.

3.

ATTRIBUTE CORRELATION AM) MODEL -

STRUCTURES

For t h e u t i l i t y d i s t r i b u t i o n s Un; n=l,

..., N we can define

a

variance-covariance matrix & w i t h elements C given by: nn'

= E (en

E

n'

)

VAn,Ant

EA

i n which E ( . ) denotes an expectation value.

(3.1) I n t h e case of I I D u t i l i t y

components t h e matrix has, by construction, a simple diagonal form

I - i s t h e u n i t matrix of dinension N , and u t h e common standard

where

deviation of t h e d i s t r i b u t i o n s g(U), t h a t i s

It is one o f t h e i n t e n t i o n s of t h i s work t o determine t h e extent t o which t h i s very simple s t r u c t u r e c o n s t i t u t e s a r e a l r e s t r i c t i o n t o choice modelling. The multinomial l o g i t ( Y ~ I L )model (2.15) generated from I I D Weibull d i s t r i b u t i o n s , which i s t h e r e f o r e characterised by a matrix with t h e diagonal s t r u c t u r e (3.2), has been very widely applied i n mode choice, and nore r e c e n t l y d e s t i n a t i o n choice modelling ( f o r a review, see Spear 1977).

It i s now well known, however, t h a t t h e model s u f f e r s a

r e s t r i c t i v e property o f cross-substitution, t h e 'independence from

(2)

Manski and Lerman (1976)-have examined t h e simulation approach carefully and have proposed l e s s naive stopping r u l e s f o r more e f f i c i e n t programs.

irrelevant alternatives' ( I I A ) property, whereby the ratio

is independent of the utility values associated with other options. The IIA property, once seen as a positive advantage to be exploited in 'new option' situations, is now recognised to be a potential hazard when certain alternatives are more 'similar' than others in the set A. In random utility theory this notion of 'similarity' is interpreted in tens of the presence of off-diagonal elements in the matrix 2. In certain applications, specific forms for the utility functions tend to suggest themselves. Consider 'two dimensional' choice contexts involving, for example, combinations of destination (D) and mode ( ! I ) . Alternatives in each dimension will be denoted by (Dl,

(y,...

..

.

,D

,

...,DN)

and ,Xm y ..,:$*), respectively, and the combination of dimensions produces the 1P.J discrete choice options (Dl D M n m' ....Dd$l), which comprise the set A. The general element An is now D M which might be a specific destination-mode combination for the nm purpose of performi~gan activity.

5,...,

For such choice contexts we shall be particularly interested in utility functions of the form U(n,m)

= U + U + Urn n m

V Dn!ImcA

(3.5)

here U and U may, for example, correspond to destination and mode n m specific utilities, respectively, while U might be the travel disutility nm associated with D M combination. This form was used in the shopping nm model developed by Ben Akiva (1974),and in a number of other applications in the United States since that time. Writing U(n,m) in terms of a 'representative' term b(n,m) and a stochastic residual ~(n,m)we have u(n,m)

= E(n,m) + ~(n.m)

(3.6)

i n which

and

-

We s h a l l now assume t h a t t h e r e s i d u a l s

E ~ E~ ,

and

a r e separatelx

E~

I I D , with

i n which 6 is t h e Kronecker d e l t a .

The elements of

-

now become

and t h e matrix i s expressed i n Figure 3, together with those corresponding t o t h e residual structures

~(n,m) =

cm +

(3.13)

which a r e c l e a r l y s p e c i a l cases of t h a t defined i n Equation ( 3 . 8 ) .

It

i s r e a d i l y seen t h a t t h e source of c o r r e l a t i o n i n 'multiple dimension' cases i s t h e existence of a ccamnon term o r 'dimension s p e c i f i c ' element (Un o r Urn) i n t h e u t i l i t y function.

For t h e four cases (3.81, (3.11)

- (3.13)

we have developed i n Figure 3, a p i c t o r i a l representation of t h e s t r u c t u r e of t h e

z- matrix

with c o r r e l a t i o n between a l t e r n a t i v e s incorporated through

common bonds as shown.

This i s t h e b a s i s f o r a representation of t h e

choice model i t s e l f (Williams, 1977).

I n t h e f i r s t case both oD and oM

a r e zero and a diagonal C - matrix r e s u l t s .

This case which is consistent

with Equation (3.11) w i l l correspond t o t h e W L model (2.15) i f t h e u t i l i t y functions a r e drawn from I I D Weibull d i s t r i b u t i o n s .

It i s c l e a r

t h a t t h e use of t h e u t i l i t y 'function (3.5) i n a KNL model of t h e form (2.15) w i l l be i n c o n s i s t e n t because t h e appropriate

z- matrix,

corresponding

t o t h a t u t i l i t y function, i s not of t h e diagonal form involved i n t h e generation of t h e model. Before t r e a t i n g t h e more general case ( 3 . 8 ) , which i s c o n s i s t e n t with t h e u t i l i t y function (3.5) and which corresponds t o t h e fourth

2 -

matrix of Figure 3, we s h a l l consider t h e derivation of a h i e r a r c h i c a l o r nested model f r o m a function consistent with t h e r e s i d u a l s t r u c t u r e

and which corresponds t o t h e second representation i n Figure 3.

In t h i s

case t h e component o

M vanishes and t h e two parameters oD and oDMallow

d i f f e r e n t degrees of cross-substitution between

intra and inter- branch

a l t e r n a t i v e s i n t h e ' t r e e ' form shown i n Figure 3 ( b ) ; t h a t is, between latter.

, in

i n the t h e former case, and between D ?I and D ,M n m' n m It may be shown (Williams, 1977) t h a t P(n,m), t h e p r o b a b i l i t y

Dn Mm and DnMm,

of s e l e c t i n g D M can be w r i t t e n n m P(n,m)

=

Pn.Pm

(3.15)

i n which

Prim

= Prob (Unm > Urn,,

Wml EM)

and Pn = Prob (Un + Unw > Unl + Unl+, Y Dd ED)

(3.17)

with

If the components Urn are Weibull distributed variables w(u.~~~,A) with mean if + y/A (where y is Euler's constant), and standard deviation nm K / ( ~ A), then it is readily shown (Cochrane, 1975) that Un, is also Weibull distributed, with mean

and standard deviation given by

The marginal distribution P is then derived from the sum of n Weibull distributed variables Un+ and variables Un, derived from some distribution r(~,> oM, then the HL specification MID corresponding to oM > oD will involve pathological behaviour. The condition, however,

(11) Results are shown for the first set of test only, i.e. when it was assumed complete knowledge of the mean utilities. (12) The performance measures used were average relative errors (ARE) defined as: N ARE = c { psim - ?dl /?dl i=l 1 xnd a izeasure, defined as :

$

1

Td= modelled share of option i, i = 1,. ..,N

I

I

appears also to depend on the underlying representative utility values. (13) If we further examine the performance of the models as reported in Figure 7(c), we note that the CCL appears to be a good approximation to the general fuuction (3.5).

Its superiority to the other logit forms was especially apparent when the three coordinates oD, uM and

uDM were different from zero and from each other. The most surprising and welcome outcome, however, was the remarkably robust performance of the MNL, even at interior points of the triangle and especially when uD uM. AS expected the IIA property was a considerable impediment near the sides of the triangle, except in the immediate region of point C. The last point to note in this section is, that out of the estimated models (MNL and two HL forms), that with the best base fit consistently provided a good estimate of the response to change: this would seem to lend some theoretical/numerical support for the suggestion of Ben Akiva (1977)that results of alternative HL and

MNL models could be compared and the appropriate model selected according to the estimated value of the 'similarity' parameter, which in our notation is 6 / A . We offer no apologies for the fact that the simulation tests were confined to a 2 x 2 example (4 options). We believe that the results and conclusions in next section, would not be qualitatively modified when the number of alternatives are increased, because the structure of the variance-covariance matrix itself was the focus of the mis-specification tests. The dependence of the results on the number of options could, of course, be tested. (14)

... ... ... ... ... ... ... ... ... ... ... ... (13) In the second series of tests the effect of omitting a particular utility ccanponent - by putting im = 0, for examplewas determined. All results were inferior to their counterparts in the first series, as would be expected because the number of 'degrees of freedom' of the model specifications had been reduced. The performance of HL (asymmetric in nature) was particularly suspect and pathological behaviour became more prevalant. (14) More important, in exercises of this kind, is to make sure that the process converges. We found it was necessary to smple 30,000 observations to get consistent results. The reader is referred to Section 2.

1 1

6.

CONCLUSIONS In this paper we have presented the essential features of random

utility theory, both formally and in a geometric framework which provides a clear interpretation of the nature of choice. We have studied the role of correlation and how the form of the utility function uniquely determines the structure of models, their complexity being dictated precisely by the structure of similarity or correlaticnbetween the alternatives. In this way, we have been able to present the basic assumptions of the most popular models, multinomial logit,

general

probit and hierarchical logit, and

discuss their implications. The general

probit model is the more conceptually appealing

of model forms, although unfortunately the least tractable of them for more than small problems, and even then with some yet unknown properties. We have investigated the possibility of invoking transformations in utility space as a means of simplifying it. We have shown that it is indeed possible to define suitable transformations that allow one to restore the simplicity of the integrand of independent 'equal variance' models to any more general model, although in the case of models incorporating correlation among many alternatives, what is gained on the joundabouts is lost on the swings because of the non-separability of the multiple integral (4.27) which in turn is due to the unhospitable form of the region of integration (4.28). Although transformations certainly give more insight into the problem, the potential for implementing them, even in the case of models with symmetric variance-covariance matrices, does not appear practicable. The results of the simulation tests are consistent with the following conclusions: i)

The cross-correlatedlogit model is a good theoretical approximation to the three parameter utility lunction (3.5). The superiority to other logit forms is especially apparent when the standard deviations oD, oM and oDMare rather different from zero and from each other. However its estimation is very complex (Williams, 1977) and for this reason it does not commend itself.

ii) The multinomial l o g i t model performs reasonably well at i n t e r i o r p o i n t s of t h e t r i a n g l e (indeed it i s considerably

-

more robust than we had a n t i c i p a t e d ) and t h i s i s p a r t i c u l a r l y t r u e when oD

oM. Its maximum e r r o r occurs near t h e s i d e s of

t h e t r i a n g l e (except i n t h e immediate region of point C ) where t h e 'independence from i r r e l e v a n t a l t e r n a t i v e s ' property i s a considerable impediment.

iii) A good base f i t does not necessarily imply a good response model, after all every m o d e l k a l i b r a t e s ' .

However, out of

t h e t h r e e a l t e r n a t i v e l o g i t s t r u c t u r e s (multinomial l o g i t and two h i e r a r c h i c a l l o g i t forms), t h e model which provides t h e b e s t base f i t provides a good estimate of t h e response t o change. iv)

A mis-specified model ( t y p i c a l case is t h e inappropriate

h i e r a r c h i c a l l o g i t form) w i l l tend t o display pathological behaviour i n a response context.

However it i s possible t o

recognise t h e symptoms a t t h e c a l i b r a t i o n s t a g e , by examining t h e consistency conditions (eg. r u l e (3.23)). I n these t e s t s we examined t h e c a p a b i l i t y of extended members of t h e l o g i t family t o accomodate t h e s t r u c t u r e of s i m i l a r i t y between a l t e r n a t i v e s embodied i n t h e u t i l i t y function ( 3 . 5 ) .

The general

probit function would be appropriate i n t h i s ' case, and indeed t o more general u t i l i t y forms.

Horowitz (1978) has, i n f a c t , examineQthe

p o t e n t i a l mis-specification problems of t h e multinomial l o g i t model when compared with a 3- a l t e r n a t i v e probit model.

H i s results are

complementary and c o n s i s t e n t with ours, i n t h e sense t h a t i n moat p r a c t i c a l cases, t h e g r e a t e r ease with which l e s s consistent s t r u c t u r e s

-

provided they a r e robust enough t o cope r e l a t i v e l y well with not serious mis-specification -may

be implemented, w i l l win t h e day.

F i n a l l y t h e powerful t o o l employed i n our a n a l y s i s , Monte Carlo simulation, deserves an e s p e c i a l word of p r a i s e .

We noted t h e increasing

a p p l i c a t i o n of t h e method i n t h e t r a n s p o r t f i e l d , but we b e l i e v e t h a t our p a r t i c u l a r use h e r e , c r e a t i n g a r t i f i c i a l d a t a s e t s on which t h e e f f e c t s of s p e c i f i c model mis-specifications can be t e s t e d i n a c o n t r o l l e d manner, i n d i c a t e s one way ahead t o a t t a c k t h e problem of model evaluation. Indeed, it has already been ue:d

i n a more ambitious p r o j e c t t o t e s t f o r

t h e o r e t i c a l misrepresentation and t o a s s e s s t h e v a l i d i t y of cross-sectional models i n general ( W i l l i a m s and Ortuzar,1979).

REFERENCES ALBRIGHT, R.L., LEREW, S.R. and MANSKI, C.F. (1977) Report on t h e developnent of an estimation program f o r t h e multinomial p r o b i t model , prepared f o r t h e Federal Highway Administration, US Department of Transportation, by Cambridge Systematics Inc. BEN-AKIVA, :-I.E. (1974) S t r u c t u r e of passenger t r a v e l demand models. Transportation Research Record, 526, 26-42. BEN-AKIVA, M.E. (1977) Passenger t r a v e l demand f o r e c a s t i n g : applications of disaggregate models and d i r e c t i o n s f o r research. Proceedings, World Conference on Transportation Research, Rotterdam.

BONSALL, P.W. (1979) ?licrosimulation of mode choice - a model of organised c a r sharing. FTRC Summer Annual Xeeting, University of Warwick. BOUTHELIER, F.J. (1978) An e f f i c i e n t methodology t o estimate and p r e d i c t with multinomial p r o b i t models: applications t o t r a n s p o r t a t i o n problems. PhD Thesis, Department of Ocean Engineering, MIT. BURRELL (1968) Multipath route assignment and i t s a p p l i c a t i o n t o capacity r e s t r a i n t . Proceedings, Fourth I n t e r n a t i o n a l Symposium on t h e Theory o f T r a f f i c Flow, Karlsruhe, Germany; CARDELL, N.S. and REDDY, B.J. (1977) A multimonial l o g i t model which permits v a r i a t i o n s i n t a s t e s across individuals. Charles Rivers Associates Inc.

CHARLES RIVERS ASSOCIATES Inc.. (1972) A disaggregate behavioural model of urban t r a v e l demand. Federal Highway Administration; _U.S.Department of Transportation. CHICAGO AREA TRANSPORTATION STUDY (1979) Travel forecasting process. A s t a f f t e c h n i c a l r e p o r t , CATS 2344-01, Chicago.

CLARK, C. (1961) The g r e a t e s t of a f i n i t e s e t of random variables. Operations Research, 9,145-62 COCHRANE, R.A. (1975) A possible economic b a s i s f o r t h e g r a v i t y model. Journal of Transport Economics and Policy 9, 34-49 DAGANZO, C.F. (1979) Calibration and prediction with random u t i l i t y models: some recent advances and unresolved questions, presented a t t h e Fourth I n t e r n a t i o n a l Conference on Behavioural Travel ModelEibsee, West Germany.

m,

DAGANZO, C.F., BOUTHELIER, F. and SHEFFI, Y. (1977) Multinomial p r o b i t and q u a l i t a t i v e choice: a computationally e f f i c i e n t algorithm. Transportation Science 11, 38-58.

DAGANZO, C.F. and SCHOENFELI), S.(1978) CHOMP User's Manual. Report UCB-ITS-RR-78-7. I n s t i t u t e of Transportation Studies, University of California a t Berkeley.

DALY, A . J . and ZACHARY, S. (1978) Improved multiple choice models. I n Hensher, D.A. and Dalvi M.Q. ( e d s . ) , Determinants of Travel Choice, Saxon House, Sussex. DOMENCICH, T. and MCFADDEN, D. (1975) Urban Travel Demand a n a l y s i s , North Holland, Amsterdam.

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a behavioural

GREEN, P.E. with contributions by CARROLL, J . D . (1976) .%thematical Tools f o r Aaplied Multivariate Analysis, Academic P r e s s , Inc., London.

HAMMERSLEY, J . M . and HANDSCOMB, D.C. Chapman and Hall, London.

(1965) Monte Carlo Xethods,

HAUSMAN, J . A . and WISE, D.A. (1978) A conditional p r o b i t model f o r q u a l i t a t i v e choice: d i s c r e t e decisions recognizing interdependence and heteregeneous preferences. Econometrica, 46, 403-26. HOROdITZ, J. (1978) The accuracy of t h e multinomial l o g i t a s an approximation t o t h e multinomial probit model of t r a v e l demand, unpublished manuscript, U.S. Environmental Protection Agency, Washington D.C. KREIBICH, V. (1979) Modelling c a r a v a i l a b i l i t y , modal s p l i t and t r i p d i s t r i b u t i o n by Monte Carlo simulation: a short way t o integrated models. Transportation, 8, 153-66. LANGDON, M. (1976) Modal s p l i t models f o r more than two modes. I n Urban Traffic ?4odels. PTRC Summer Annual Meeting, University of Warwick.

MANSKI, C.F. and LERMAN, S.R. (1978) On t h e use of simulated frequencies t o approximate choice p r o b a b i l i t i e s , Cambridge Systematics Inc. MCFADDEN, D. (1979) Quantitative methods f o r analysing t r a v e l behaviour of individuals: some recent developments, i n Hensher D.A. and Stopher P.R. (Eds. ) Behavioural Travel Modelling, Croom Helm, London. ORTUZAR, J . D . (1978) Mixed-mode demand forecasting: an assessment of current p r a c t i c e . FTRC Summer Annual Meeting, University of Warwick.

ROBERTSON, D . I . (1977) A deterministic method of assigning t r a f f i c t o multiple routes of known c o s t . ,Crothorne, ROBERTSON, D . I . and KENNEDY, J . V . (1979) The choice of r o u t e , o r i g i n and d e s t i n a t i o n by c a l c u l a t i o n and simulation. TRRL Report LR 877, Crowthorne. SENIOR, M..L. and WILLIAMS, H.C .W.L. (1977) Model based t r a n s o o r t policy assessment I: The use of a l t e r n a t i v e forecasting models. Traffic Engineering and Control, 18(9 ) , 402-406.

SPEAR, B.D. (1977)Applications of new travel demand forecasting techniques to transportation planning: a study of individual choice models. Federal Highway Administration, U.S. Department of Transportation, Washington D.C. WILLILW, H.C.W.L. (1977)On the formation of travel demand models and economic evaluation measures of user benefit. Environment and Planning A, 9, 285-344. WILLIAMS, H.C.W.L. and ORTUZAR, J.D. (1979) Behavioural travel theories model specification and the response error problem. PTRC Summer Annual Meeting, University of Warwick, (Available as Working Paper 116, ITS, University of Leeds). WILLIAJVS, H.C.W.L. and SENIOR, M.L. (1977)Model based transport policy assessment 11: Removing fundamental inconsistencies from the models. Traffic Engineering and Control 18(10), 464-9.

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.. , . .

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ACKNOWLEDGESIENTS ., .

.

,,

This work owes considerably to Dr. Huw Williams. His imprint on the paper's organization,;ideas and expositiongoes well beyond that of draft cormnen$ator. He i s , however, to be held blameless for any parts'or words that did not come out quite right. .. . .

.

I would like to express my appreciation also to Margarita Greene for the skill and effort necessary to produce the art work in the paper. The ITS team of secretaries were as usual instrumental in producing the paper and also deserve my thanks.

F I G U R E S

a.) Fundamental distributions.

b) Geometric

interpretation.

FIGURE 1 : U tilit distributions for indepent mode with equal standard deviations.

Y

1 I

FIGURE 2: Sampling errors in parameter estimution.

1 STRUCTURE I I VARIANCE -

COWRIANCE MATRIX (c)/

!

j &(D,M) = & D + ~ M + & M

rlI2

WQDM%~

a-d

0

0

aM2

4

a-M2

0

0

---.-.-----.--.-.----.----.------..-----

bo?~~d&?l I

(IJD~~DM~~M

-.

FIGURE 3 : The st'ructure

ice models:

a set of special - cases.

a1 (U1 ,U2 pluns

b j (0,,02j plans

FIGURE 5 : Transformations of the general probit - model in two di mensions.

,

.

. .

.

.

.

i . .

,

~

:

.~

c.) Model performance at

different test points.

. .

APPENDIX 1 .i

,.'"

,

If U1 and U2 q e jointly distributed bivariate normal, with means and G2, standard deviation a; and a2 and correlation coefficient 1 . then the quadratic form (QF) of f(U1, U2) is given by

p,

,

QF = I where

- 4 (g - g)T

-

z-I

1=

(g ;

- 1)) (g - 1 )

(All

=

u1 - G1

the region of integration is defined by R1: U2 < U1 In this appendix a matrix treatment of the general transformation indicated in the text will be given, both to show that the process is easily generalizable to more dimensions and to show how each transformation in turn, affects the corresponding inverse variance-covariance matrix defining the QF at each stage. First a general statement about how a transformation works in matrix terms will be given.

It is worth remembering that the QF, by definition,

is a scalar, that is, its value is invariable to the transformations, only changing the components that define it. In general a transformtion can be represented by a matrix

A acting =

upon a vector such as in ( ~ 2 ) and over a matrix such as in ( ~ 3 ) .

The QF in the new space defined by

-

QF'=I-9LTf111

r has

the general form:

-

We will show now that QF' is indeed equal to QF

(A41

Replacing (A2), (A51 and (A6) i n t o

and noting t h a t

A-T (1T )- 1 -- 2-

d e f i n i t i o n any matrix o r vector

=

g

( ~ 4 we )

L--1 -A-,

get

t h e u n i t matrix, and t h a t by

when pre- o r p o s t - m u l t i p l i e d by 2

remains unaffected, we show t h a t

QF'

= QF

(A8)

I n t h e remainder of t h e appendix we w i l l present each transformation of t h e general case i n t h e t e x t and t h e form of t h e inverse of t h e variance -covariance matrix involved. a)

The f i r s t transformation: 1

0

, in 0

(4.10)

t h i s case

-LT

=

(A9)

b)

The second transformation: (4.13)

therefore

i--'

=

fi

and

c)

The last transformation: (4.15)

therefore

2-1

-

-

h-p

APPENDIX 2 The multinomial logit model has an analytic closed form given by:

where P'!'Od

=

J

-

modelled share of alternative j, j = 1, -

u.

=

mean or representative utility of alternative j, j = 1, ...% N.

A

=

parameter to be estimated

J

.. ., N .

In this Appendix we will consider the maximum likelihood estimation of A given the values of fi. and the simulated shares P sim , of each J j alternative j. We will use a standard Newton Raphson search mechanism. In this simple case, the log-likelihood function of A (k), at iteration k, is given by

and the first derivative with respect to A (k) is

Now the Newton Raphson solution involves defining a next best estimate for the parameter, A (k+l)3 as

and the maximum likelihood estimate will be that found when the process converges (which always does, Domencich and McFadden, 19751, that is when A (k+l) =

A (k)

-.

.