Normalisations in ALOGIT Tree Models

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The characteristic of the ‘tree’ logit model is that it offers the modeller the ability to escape from the assumption of complete symmetry among the alternatives embodied in the linear multinomial logit model (MNL).  The symmetry of the MNL implies that all the cross-elasticities of demand with respect to the price of one alternative are equal.  However, tree logit models contain ‘structural’ coefficients which allow the cross-elasticities of demand within groups of alternatives to be different (larger than) the cross-elasticities between groups.  The structural coefficient is, in the simplest case, simply the ratio of the cross-elasticities between groups to those within groups.  This property is generally well understood by tree logit modellers, indeed the ability to get differing cross-elasticities is an important reason for using the tree logit form.

However, it is not always realised that there is more than one way to define structural coefficients in a logit model.  Theoretically, quite a large variety of definitions could be specified.  Two alternative definitions are to be found in the text books and academic papers on the subject.  For practical purposes each of the alternatives has both advantages and disadvantages.

The purpose of this note is to elaborate on these issues and to explain why we have chosen the method of calculating structural coefficients that is used in ALOGIT.

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The Need to Constrain Coefficients

It has been known since the mid-1970's that tree logit models can be  consistent with the principle of individual utility maximisation (abbreviated as umax in the remainder of the note).  Consistency with umax implies consistency with the principle that each individual chooses the alternative that appears best to him or her.  There are of course other possible bases for defining models, with which logit models may or may not be consistent.  The choice of umax is a simple and intuitively appealing basis which has been used in the majority of theoretical modelling studies.  Adhering to the umax principle and keeping all models within a system consistent with it may at first seem of mere theoretical value, however it has clear practical advantages as well.

In particular, if a model can be guaranteed to be consistent with the umax principle, we can be certain that its behaviour will be consistent with the intuitive notions embodied in that principle.  For example, if the price of an alternative is increased (and we assume that cost has a negative impact on utility!), then the demand for that alternative will decrease and the demand for all other alternatives will increase, or at least will not decrease.  The same reasoning is appropriate for longer travel times resulting in lower utilities and decreased demand.  Other considerations, i.e. increased comfort and higher frequency (assuming these affect the utility positively), can counter the loss of utility due to higher cost or longer travel times.

When estimating and forecasting with more complex models like nested or tree logit models, it is especially essential to keep all models as far as possible consistent with umax.  Since the mid-1970 is has been known that for the models always to be consistent with umax the structural (or tree) coefficients must lie within specific ranges.  Specifically, when the model is defined as it is in ALOGIT, the structural parameters must lie between 0 and 1 (for other model specifications other ranges may apply).  The practical effect of a failure of this condition is that the change in the price of one alternative can cause the demand for other alternatives within the same 'nest' to change in an inappropriate direction.  For instance, if a structural coefficient is outside the appropriate range, a price increase for an alternative would result in a decreased demand for that alternative, but it could also result in a decreased demand for other alternatives within the same ‘nest’.  This counter intuitive effect would not always occur, so that it would not be sufficient to make a series of tests on a model to ensure its consistency within intuition.  The advantage of keeping models consistent with umax, is that it can be guaranteed that counter-intuitive results like this will not occur.

In the utility theory on which the models can be based, the concept of utility itself has no specific units (and no specific zero point).  There is not necessarily a requirement that the scale units of utility for different alternatives be the same.  One could imagine that the utilities for different alternatives were measured in sterling, francs, guilders etc., with the exchange rates between them not being directly unobservable and obtainable only as a result of the estimation process.  While such a conceptual framework can be internally consistent, we would normally believe we have more insight into the choice process.  It seems reasonable to require that equal changes in the utility of two alternatives would leave the relative demand for those two alternatives unchanged.  This requirement clearly means that the coefficients for cost and (the same ‘type’ of) time must be the same across all the alternatives.  However, another condition is also needed, affecting the scaling of the alternative utilities: specifically that a change in the utility of any alternative has an equal impact at the ‘root’ of the tree.

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Consequences for ALOGIT

The logit model is set up for ALOGIT with the specification of the utility of each of the elementary alternatives (those that can actually be chosen by the traveller) in the linear form

Vk = b1.d1k + b2.d2k + .... + bn.dnk

giving the (measured part of the) utility for alternative k in terms of observed data items d and coefficients b which are to be estimated. These utilities are used directly for calculating the conditional choice probabilities within each of the nests.

The composite alternative utilities, i.e. the utilities of the ‘nests’ are given by (note 1)

Vc = qc . log { Stk=c expVk }

where qc is the structural or tree coefficient appropriate to nest c and the sum is taken over all the elementary alternatives k for which the nesting indicator tk brings them into nest c.  The composite utility component is called the logsum, for obvious reasons.

This formulation is the most simple and straightforward known for defining the tree logit model and has advantages in terms of the programming and its flexibility.  However, some of the theoretical literature on tree logit models uses an alternative formulation, defining the composite utility by

Vc = qc . log { Stk=c exp ( Vk / qc )  }

This change of formulation has no immediately apparent impact: the values of the b coefficients are simply changed by a factor of qc and the model is the same as before.  But when the tree is more complicated and different branches have different values of qc then the two formulations are really different.

The key difference is that in the second formulation the higher-level utility is measured in the same units as the lower-level utilities. The scaling of cancels out. Clearly this consistency of units is an advantage in terms of the achievement of consistency of the definition of utility in different parts of the tree. Indeed, all that has to be done in such a model to achieve consistency is to ensure that the various utility components (cost, time in specific activities, etc.) have the same coefficients wherever they are used. However, the alternative ALOGIT formulation also has advantages.

A major advantage of the ALOGIT formulation is its simplicity in programming.  However, the more complicated programming necessary for the second form would be undertaken, were it not for the further advantage of the ALOGIT form that it gives a very easy procedure to allow the scale to be different where that is required.  For example, when SP and RP data is to be mixed it is often required that the scale of the utility functions should vary between the two data sources to allow for differences in the variance of the error term.  This difference does not cause a problem in terms of the consistency of utility because any observation refers to one data type only.  Further, consistency of utility scale can be achieved in the ALOGIT formulation by an appropriate adjustment to the model.

This adjustment is illustrated in the following diagrams.

Figure 1

Figure 2

Figure 1 shows the simplest of all possible tree structures.  Alternatives 1 and 2 form a nest, at which a ‘logsum’ is calculated and multiplied by the tree parameter q.  But because of the way in which the tree coefficient is calculated within ALOGIT, if the cost of all three alternatives is increased by a constant value, the cost of the nest will be increased only by q times that constant and (assuming q < 1) the demand for the nest alternatives will increase at the expense of alternative 3.  Obviously, the same considerations apply to any variable in the utility function, not just the cost. To counter this effect a correction is needed.

In Figure 2 the necessary correction is introduced in which alternative 3 is also included in a nest, this time containing only one alternative, but that nest is also assigned the same structural parameter q1.  Now if the cost of all the alternatives is increased by the same amount, the cost of the nests is increased by q1 times that amount and the demand remains constant.  Thus the consistency of the scale of utilities can be reinstated across all three of the alternatives.  This correction, introduced by the modeller within the model, would not be necessary if the alternative formula for calculating the logsum had been used.  But a correction to make the scale parameters different when needed would not be possible.

This type of correcting dummy alternative can be introduced into more complicated trees also to maintain the required consistency.  The only limitation is the complexity of the models thus created.

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Conclusion

There are two standard ways of defining the structural coefficients in a theoretical presentation of the tree logit model.

The procedure in which the scale of the logsum variable is equal to the scale of the lower-level utilities automatically fixes the utility scale throughout the tree.  When it is required to maintain consistency with utility maximisation this is an advantage, but it prevents the specification of different scales when that is needed, e.g. for RP / SP joint analyses.

The procedure used in ALOGIT gives a scale to the logsum variable which is equal to the scale of the lower level utility multiplied by the structural coefficient.  This means that adjustment procedures with dummy nodes must be used to achieve consistency in asymmetrical trees.  However, ALOGIT allows different scales to be adopted when this is needed.

Note 1: Here we neglect the possibility of attaching utility components to the composite alternatives or of choosing composite alternatives; both of these are possible in theory and in ALOGIT.  However these complications are distracting for the present discussion.

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