Hyperbolic cosine latent trait models for unfolding direct responses and pairwise preferences

The hyperbolic cosine unfolding model for direct responses of persons to individual stimuli is elaborated in three ways. First, the parameter of the stimulus, which reflects a region within which people located there are more likely to respond positively than negatively, is shown to be a property of the data and not arbitrary as first supposed. Second, the model is used to construct a related model for pairwise preferences. This model, for which joint maximum likelihood estimates are derived, satisfies strong stochastic transitivity. Third, the role of substantive theory in evaluating the fit between the data and the models, in which unique solutions for the estimates are not guaranteed, is explored by analyzing responses of one group of persons to a single set of stimuli obtained both as direct responses and pairwise preferences. Index terms: direct responses, hyberbolic cosine model, item response theory, latent trait models, pair comparisons, pairwise preferences, unfolding models