The measurement of latent traits by proximity items

Abstract

A probabilistic parallelogram model for the measurement of latent traits by proximity items (the PARELLA model) is introduced. This model assumes that the responses of persons to items result from proximity relations: the smaller the distance between person and item, the larger the probability that the person will agree with the content of the item. The model is unidimensional and assigns locations to items and persons on the latent trait. The parameters of the PARELLA model are estimated by marginal maximum likelihood and expectation maximization. The efficiency of the estimation procedure is illustrated, a diagnostic for the fit of items to the model is presented, and the PARELLA model is used for the analysis of three empirical datasets. Index terms: expectation maximization, latent trait theory, marginal maximum likelihood, nonmonotone trace lines, single-peaked preference functions, unfolding.