Implications of Thresholds in Discrete Choice Modelling

Abstract

Individual choices are affected by complex factors and the challenge consists of how to incorporate these factors in order to improve the realism of the modelling work. The presence of limits, cut‐offs or thresholds in the perception and appraisal of both attributes and alternatives is part of the complexity inherent to choice‐making behaviour. The paper considers the existence of thresholds in three contexts: inertia (habit or reluctance to change), minimum perceptible changes in attribute values, and as a mechanism for accepting or rejecting alternatives. It discusses the more relevant approaches in modelling these types of thresholds and analyses their implications in model estimation and forecasting using both synthetic and real databanks. It is clear from the analysis that if thresholds exist but are not considered, the estimated models will be biased and may produce significant errors in prediction. Fortunately, there are practical methods to attack this problem and some are demonstrated.

Acknowledgements

The authors are grateful to Elisabetta Cherchi, Ben Heydecker, Luis Ignacio Rizzi and Huw Williams for help in developing these ideas. The useful comments of two anonymous referees are also appreciated. Notwithstanding, and as usual, all errors are the sole responsibility of the authors. Thanks are also due to the Chilean Fund for the Development of Scientific and Technological Research (FONDECYT) for having supported this research through several projects (1970117, 1000616 and 1020981).

Notes

1. Decision strategies that use trade‐offs among attributes are called compensatory strategies (Payne et al., 1993).

2. A similar approach to consider an agent effect includes capturing correlation among responses for a given individual, as proposed by Morikawa et al. (1995).

3. An interesting topic is the identification problem. If one has a panel (at least two responses per person), for J = 2, only one parameter in ν can be identified, but if J ≥ 3, one can estimate all the parameters.

4. The LR statistic −2{l∗(θ_r ) – l∗(θ)},where l∗(θ) and l∗(θ_r ) are the log‐likelihood at convergence for the non‐restricted and the restricted model, respectively, is asymptotically distributed χ ² with r degrees of freedom, where r is the number of linear restrictions (Ortúzar and Willumsen, 2001).

5. Very large sample sizes are necessary to recover the parameters of even much simpler models with simulated data (Williams and Ortúzar, 1982; Munizaga et al., 2000)

6. The authors are grateful to a referee for pointing this out; for a detailed explanation of this effect, see Sillano and Ortúzar (2005).

7. The user was asked to provide a simple judgement of the comfort experienced during the journey described. The variable was pre‐coded into three levels: poor, sufficient and good; and two dummy variables were used: Comf1 = 1 if the level of comfort was poor, and Comf2 = 1 if comfort was sufficient. As the ‘good’ level was left as a reference, it was implicit that car had high comfort.

8. Cherchi and Ortúzar (2002) also tried an inertia variable following the approach of Bradley and Daly (1997), but it was clearly not significantly different from zero.

9. This is not exactly a threshold approach as defined in this section, but a representation of framing using a reference price that introduces asymmetry. The authors are grateful to a referee for pointing this out.

10. The model assumes that the error term is invariant in time.

11. As rightly pointed out by a referee, according to this notation, the vector of thresholds for individual q does not vary across alternatives. However, it is possible to define alternative specific thresholds (e.g. the thresholds on access or travel time for public transport and car might be different).

12. Assuming independence among thresholds.

13. To generate the synthetic data, no alternative specific constants were included, but there is no identification problem for the alternative specific constants due to the presence of thresholds.