Understanding the unit in the Rasch model

The purpose of this paper is to explain the role of the unit implicit in the dichotomous Rasch model in determining the multiplicative factor of separation between measurements in a specified frame of reference. The explanation is provided at two complementary levels: first, in terms of the algebra of the model in which the role of an implicit, multiplicative constant is made explicit; and second, at a more fundamental level, in terms of the classical definition of measurement in the physical sciences. The Rasch model is characterized by statistical sufficiency, which arises from the requirement of invariant comparisons within a specified frame of reference. A frame of reference is defined by a class of persons responding to a class of items in a well-defined response context. The paper shows that two or more frames of reference may have different implicit units without destroying sufficiency. Understanding the role of the unit permits explication of the relationship between the Rasch model and the two parameter logistic model. The paper also summarises an approach that can be used in practice to express measurements across different frames of reference in the same unit.