Establishing Formal Equivalence Between Nested Logit and Hierarchical Elimination-by-aspects Choice Models
R Batley, ITS, University of Leeds, UK; A Daly, ITS, University of Leeds, UK and RAND Europe, NL
In analysing the discrete choices of transport users, travel behaviour research has shown a heavy reliance on models derived from McFadden?s (1978) Generalised Extreme Value (GEV) model. GEV models, such as multinomial logit (MNL), nested logit (NL) and cross-nested logit (CNL), have been based exclusively on the micro-economic axioms of completeness, transitivity and continuity, which together imply the principle of individual utility maximisation.
The paradigm of individual utility maximisation can readily be exploited to give a model for the prediction of a population or an individual sampled from that population by incorporating a random component in the model, thus giving the Random Utility Model (RUM). The RUM concept can be used to develop a wide range of models of behaviour, including the GEV models already mentioned, and has thus formed the basis for nearly all of the models in current use in transport analysis. The rigorous basis given by this model has allowed analysts to develop tests and procedures for model development, hypothesis testing and forecasting. RUM models exhibit consistency with a broad micro-economic theory, so that, for example, consumer surplus calculations can be made from them. These advantages of RUM models ? and, in particular of GEV models ? explain the current dominance of this approach.
A parallel development to GEV modelling, however, has been the accumulation of a substantial literature in the decision sciences suggesting that individual choice behaviour may not necessarily be consistent with utility maximisation (see for example the reviews of McFadden (1999) and Camerer (1998)). This literature has identified, under experimental conditions, behavioural phenomena that appear to violate one or more of the micro-economic axioms. Such violations have led to the advancement of alternative choice models outside of RUM.
The transport community appears to have shown recurrent interest in non-RUM choice models, but relatively few studies investigating or applying such models have materialised. Indeed, Bolduc and McFadden (2001), in their report of the workshop on Methodological Developments at the 2000 Conference of the International Association for Travel Behaviour Research concluded: ?Non-RUM models deserve to be evaluated side-by-side with RUM models to determine their practicality, ability to describe behaviour, and usefulness for transportation policy. The research agenda should include tests of these models? (p326). This paper addresses the call for comparative analysis of RUM and non-RUM choice models.
The literatures of mathematical psychology and marketing research have proposed a series of discrete choice models that are not derived from RUM. This paper focuses on a particular non-RUM model, hierarchical elimination-by-aspects (HEBA1), which was proposed by Tversky and Sattath, 1979) as a special case of the better-known elimination-by-aspects model (Tversky, 1972a; 1972b). Our paper redefines, for the present context, HEBA, and considers the mathematical and behavioural properties of the model.
Although they differ fundamentally in their theoretical underpinnings, both HEBA and NL represent the choice problem as a ?preference tree?, with subsets of similar alternatives nested together. This has prompted several authors to make observations about the degree of equivalence between the two models. Tversky and Sattath (1979) asserted; ?Although the nested logit model does not coincide with PRETREE, the two models are sufficiently close that the former may be regarded as a random utility counterpart of the latter? (p567); but offered no further insight.
McFadden (1981) presented a more detailed analysis. For a preference tree consisting of three alternatives, multinomial choice probabilities were derived analytically from binary choice probabilities for three models: NL, HEBA and multinomial probit (MNP). McFadden compared the multinomial choice probabilities forecast by the three models for given binary choice probabilities, finding that the multinomial choice probabilities forecast by NL, HEBA and MNP were, not only intuitively plausible, but extremely close to each other. It was concluded that ?...at least for simple preference trees...these models are for all practical purposes indistinguishable? (p236). We now understand the relationship of MNP to GEV models better and extensive analytical and simulation testing has illuminated the comparatively small but sometimes significant differences between MNP and NL (Whelan et al., 2002).
Extending the analysis, it is insightful to also consider the degree of equivalence between EBA and CNL, which generalise HEBA and NL respectively to accommodate a ?cross-nested? structure. Again, a number of authors have made observations. Small (1987) asserted that ordered GEV (OGEV), which is a special case of CNL, is: ?...an example of A. Tversky?s ?elimination by aspect? class of choice models? (p414). Vovsha (1997) noted: ?...within the cross-nested framework...choice can be conditionally made from several marginal nests reflecting different attraction focuses...From this point of view the cross-nested structure comes close to the elimination-by-aspects theory? (p9). Unfortunately, neither Tversky nor Vovsha elaborated on these comments, but perhaps even less transparent is the contribution of Train (forthcoming), who asserted: ?With positive [log sum] parameter, the nested logit approaches the ?elimination by aspects? model of Tversky (1972) as [log sum parameter] ® 0? (p93).
The key, and original, contribution of our paper is to establish formal conditions under which NL and HEBA are exactly equivalent. In other words, if appropriately specified, NL can be made exactly equivalent to HEBA, and vice versa. The conclusion follows that McFadden?s (1981) analytical comparison of NL and HEBA is flawed, since the two models, if appropriately specified, are identical.
Having established equivalence conditions between NL and HEBA, the paper moves on to consider the implications for the understanding of behaviour. Since a discrete choice model derived from non-RUM can be RUM-consistent, and vice versa, this raises the interesting question of whether the dichotomy between RUM and non-RUM really matters. The answer to this question depends, to a large extent, on the scientific philosophy adopted. From the instrumentalist standpoint, which appears to have been adopted by most travel behaviour modellers, the answer essentially comes down to one of whether predictive accuracy is affected. In other words, a model should be judged by its predictive ability rather than the realism of its assumptions.
To this end, the paper addresses the following substantive questions. Are the parameters of NL and HEBA comparable, and can we draw inferences in the same way from the two types of model? If NL and HEBA models are equivalent, is there any way of distinguishing between them in practice, and are there any differences in the interpretation of each? Is GEV strengthened or weakened by the apparent increase in the generality of NL? Finally, are there any implications for forecasting?
1. HEBA has an alternative, but formally equivalent form, known as elimination-by-tree (EBT), and both models also go by the generic term ?PRETREE? (Tversky and Sattath, 1979).
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