## Random Scale Heterogeneity in Discrete Choice Models

### Authors

S Hess, ITS, University of Leeds, UK; S Bain, J M Rose, ITLS Sydney, AU

### Description

This paper looks at the modelling of random scale heterogeneity in discrete choice data

### Abstract

The concept of scale is one of the main notions of the theory of random utility model. It describes the relative weight of the deterministic and unobserved utility components, with higher scale meaning a larger weight for the modelled utility, i.e. the choices become more sensitive to the attributes included in the utility function.

Over the years, the topic of scale differences has received attention in different contexts. A large body of work has been dedicated to allowing for scale differences between different subsets of the estimation data (see e.g., Cherchi and Ortuzar 2002; Caussade et al. 2005; Hensher and Rose 2007), for example when estimating models jointly on revealed preference (RP) and stated preference (SP) data, or when combining data from different SP surveys, e.g. mode choice and route choice. Additionally, some analysts have allowed for scale differences between different population segments, e.g. commuters and non-commuters.

Focussing on a simple linear-in-attributes specification of the utility function, scale differences between two respondents would manifest themselves in that the various coefficients will be higher for one of the respondents. In the most extreme case, the relative values for the coefficients would stay identical (e.g. willingness-to-pay indicators would be the same), and the coefficients would all differ by the same percentage. In general, the situation will however not be as simple, and the differences in scale will co-exist with differences in relative sensitivities. An illustration would be the case where one respondent will have higher absolute sensitivities (and in turn elasticities) than another respondent, while at the same time also having different relative sensitivities (with no preconditions on the direction of these differences).

In recent years, it has become increasingly popular for analysts to allow for random variations in sensitivities across respondents, for example with the use of the Mixed Multinomial Logit model. A fact that has been recognised by some, but not by others, is that the differences picked up in this way may in fact not be purely differences in relative sensitivities but also differences in absolute sensitivities, i.e. the presence of random scale heterogeneity (e.g., Louviere 2001; Amador and Cherchi 2008; Louviere et al. 2008). In the most extreme case, there may be homogeneity in relative sensitivities (e.g. value of time), with all heterogeneity being in the absolute sensitivities.

In this paper, we discuss the development of a modelling methodology that allows for these random differences in scale across respondents in a simple extension of standard GEV methodology. We discuss the properties of this model and illustrate its performance on two separate datasets. Here, we show that a large share of the heterogeneity in coefficients across respondents is in fact scale heterogeneity, where our model significantly reduces the amount of residual heterogeneity in relative preferences. Additionally, we show how accounting for random scale heterogeneity can lead to important corrections in the retrieved correlation structures. As an example, one application using a simple Mixed Logit model showed positive correlation between the time and cost coefficients, where our model showed that this was due to confounding between taste and scale heterogeneity, and the new results showed the expected negative correlation. Throughout, we observe major improvements in model performance.

The developments in this paper fall into a small but growing stream of research in this area. For example, Fosgerau and Bierlaire (in press) recently developed a discrete choice model with multiplicative errors terms (in contrast to the typically assumed linear additive error terms typically assumed) that also allows for a random scale parameter. Whilst conceptually analogous to the model we propose here, we believe that the model developed within this paper is somewhat more flexible and requires a less restrictive functional form than that implemented in the multiplicative approach. The proposed model also has some similarities with recent developments on models estimated in willingness-to-pay space (e.g., Train and Weeks, 2005), but once again is more general.

#### Publisher

Association for European Transport