Assessing Goodness of Fit of Hybrid Choice Models

Recent research in travel behavior has contributed numerous technical developments for the estimation of discrete choice models with latent attributes, including the hybrid choice model (HCM). However, assessment of goodness of fit, reliability, validity, and predictive capacities of the joint model remain open research questions. The HCM is a special form of structural equation modeling (SEM). Several goodness-of-fit indexes are all in standard use in psychometric SEM. In this paper, the validity of these indexes is examined for the HCM case. Behavior of SEM fit assessment tools is known in factor analysis (some controversies in this area are reviewed in this paper), but performance of these indexes in the HCM has not been studied. A Monte Carlo study, as well as empirical microdata on bicycle route choice, was used to show that standard SEM fit assessment did not work as expected for the HCM. Important differences were discovered in model fit between the HCM and the multiple indicator multiple cause (MIMIC) model with the same structural and measurement equations for the latent attributes. Sometimes the HCM was rejected when indexes failed to reject the MIMIC structure and vice versa. One of the sources of this divergence was that the measurement equation of the choice kernel did not have an error term; this assumption was nonstandard in SEM. Until a uniform method for measuring the HCM goodness of fit is found, it is recommended that the chi-square test be used for the MIMIC component of the joint model.