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 crosselasticities of demand with respect to the price of one alternative
are equal. However, tree logit
models contain ‘structural’ coefficients which allow the
crosselasticities of demand within groups of alternatives to be different
(larger than) the crosselasticities between groups. The structural coefficient is, in the
simplest case, simply the ratio of the crosselasticities between groups to
those within groups. This property
is generally well understood by tree logit modellers, indeed the ability to get
differing crosselasticities 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.
It has been known since the
mid1970'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 mid1970 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
counterintuitive 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.
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
V_{k}
= b_{1}.d_{1k} + b_{2}.d_{2k} + .... + b_{n}.d_{nk}
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)
V_{c}
= q_{c} . log { S_{tk=c} expV_{k} }
where q_{c} 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 t_{k} 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
V_{c}
= q_{c} . log { S_{tk=c} exp ( V_{k} / q_{c} ) }
This change of formulation
has no immediately apparent impact: the values of the b coefficients are simply changed by a factor of
q_{c} and the model is the same as before. But when the tree is more complicated and
different branches have different values of q_{c} then the two formulations are really
different.
The key difference is that
in the second formulation the higherlevel utility is measured in the same
units as the lowerlevel 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 q_{1}. Now if the cost of all the alternatives
is increased by the same amount, the cost of the nests is increased by q_{1} 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.
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 lowerlevel 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.