A HYPERBOLIC COSINE LATENT TRAIT MODEL FOR UNFOLDING DICHOTOMOUS SINGLE-STIMULUS RESPONSES

Social-psychological variables are typically measured using either cumulative or unfolding response processes. In the former, the greater the location of a person relative to the location of a stimulus on the continuum, the greater the probability of a positive response; in the latter, the closer the location of the person to the location of the statement, irrespective of direction, the greater the probability of a positive response. Formal probability models for these processes are, respectively, monotonically increasing and single-peaked as a function of the location of the person relative to the location of the statement. In general, these models have been considered to be independent of each other. However, if statements constructed on the basis of a cumulative model have three ordered response categories, the response function within the statement for the middle category is in fact single-peaked. Using this observation, a unidimensional model for responses to statements that have an unfolding structure was constructed from the cumulative Rasch model for ordered response categories. A location and unit of measurement parameter exist for each statement. A joint maximum likelihood estimation procedure was investigated. Analysis of a small simulation study and a small real dataset showed that the model is readily applicable. Index terms: attitude measurement, item response theory, latent trait theory, Rasch models, unfolding data, unidimensional scaling.