Polytomous item scores and Guttman dependence

Some theoretical relations are established between the score on a polytomous item and the total score on a set of Guttman dependent binary items. Conditions are derived under which these two scores are identically distributed. Application of the theoretical results to three well-known models for polytomous data yields, among others, the result that the score on a graded response item (Samejima, 1969) is never distributed as the total score on a set of independent binary Rasch items (Rasch, 1969).