Testing mixture models of transitive preference: comment on Regenwetter, Dana, and Davis-Stober (2011).

This article contrasts 2 approaches to analyzing transitivity of preference and other behavioral properties in choice data. The approach of Regenwetter, Dana, and Davis-Stober (see record 2011-00732-003) assumes that on each choice, a decision maker samples randomly from a mixture of preference orders to determine whether A is preferred to B. In contrast, Birnbaum and Gutierrez (2007) assumed that within each block of trials, the decision maker has a true set of preferences and that random errors generate variability of response. In this latter approach, preferences are allowed to differ between people; within-person, they might differ between repetition blocks. Both approaches allow mixtures of preferences, both assume a type of independence, and both yield statistical tests. They differ with respect to the locus of independence in the data. The approaches also differ in the criterion for assessing the success of the models. Regenwetter et al. fitted only marginal choice proportions and assumed that choices are independent, which means that a mixture cannot be identified from the data. Birnbaum and Gutierrez fitted choice combinations with replications; their approach allows estimation of the probabilities in the mixture. It is suggested that researchers should separate tests of the stochastic model from the test of transitivity. Evidence testing independence and stationarity assumptions is presented. Available data appear to fit the assumption that errors are independent better than they fit the assumption that choices are independent.