GENERALIZED THURSTONE MODELS FOR RANKING - EQUIVALENCE AND REVERSIBILITY

Previous work has determined the conditions under which generalized versions of Thurstone’s theory of comparative judgment are formally equivalent (i.e., empirically indistinguishable) for choice experiments. This note solves the analogous problem for ranking experiments: It is shown that if two “Generalized Thurstone Models” are equivalent for choice experiments with n alternatives they are also equivalent for ranking experiments with n alternatives, despite the fact that ranking generates many more preference probabilities. This result in turn allows one to determine which Generalized Thurstone Models are “reversible,” i.e., satisfy the requirement that regardless of whether the subject ranks from best to worst or from worst to best, rankings that express the same preference order will occur with the same probability. Several recent articles in this journal have dealt with the equivalence properties of a family of random utility models for choice experiments that can be regarded as generalizations of Thurstone’s (1927) Theory of Comparative Judgment-generalized in the sense that the utility random variables (Thurstone’s “discriminal processes”) are no longer required to have normal distributions. (“Equivalence” here means experimental indistinguishability. Two models-that is, theories-are said to be equivalent for a given class of experiments if results that satisfy one always satisfy the other-so that no experimental decision can be made between them. Yellott (1977, 1978), Moszner (1978), and Rockwell and Yellott (1979), all deal with a generalized version of Thurstone’s Case V in which the utility random variables are independent and identically distributed except for shifts: Such models were referred to there simply as “Thurstone models”; here they will be called “Generalized Case V Thurstone (GT-V) Models.” Strauss (1979), analyzes a broader class of “Generalized Thurstone (GT) Models” which includes the GT-V models and also nonindependent cases. All this work is reviewed here in Section 2.) This paper deals with the equivalence properties of these same models when they are applied to ranking experiments-that is, experiments in which the subject does not simply choose a single best alternative from some set, but instead rank orders all the alternatives from best to