Folded and unfolded scaling from preferential paired comparisons

A Comparison of Probabilistic Unfolding Theories for Paired Comparisons Data

“…Basically the same model for this limiting case of the Coombs-Zinnes-Griggs theory was presented by Bechtel (1968). He arrived at it by adding parametric assumptions to a relation between choice probabilities and interaltemative midpoints, detected by Greenberg (1965).…”

“…closer to 1 or 0) when the distance from the ideal to the midpoint increases. Specifications of models of this midpoint unfolding theory have been presented by Bechtel (1968), Sixtl (1973) and Jansen (1981). The latter two are both originally inspired by the Rasch item-response theory.…”

“…Greenberg's model specification is functionally identical to a non-parametric midpoint unfolding model. Bechtel's (1968) theory is based on a parametric specification of Greenberg's assumption. Bechtel suggested a normal distribution of the coordinates of ideals over the psychological continuum.…”

“…A model specification of this random ideal coordinate theory for a unidimensional space was presented by Bechtel (1968). However, Bechtel did not do so with the intention of defining a variation of Coombs' theory.…”

“…Several if not most expositions of new probabilistic unfolding theories include a discussion on the theory's implications in terms of assumed properties of probabilistic choice behavior (see, for example, Bechtel, 1968;De Soete, Carroll & DeSarbo, 1986, DeSarbo, De Soete & Jedidi, 1987. It seems that these properties are regarded as valuable indications on what kind of probabilistic choice behavior a probabilistic choice theory wants to describe.…”