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Structural Properties and Measurement Theory of Certain Sets Admitting a Concatenation Operation

Kristof, Walter
Publication Year:
Report Number:
ETS Research Bulletin
Document Type:
Page Count:
Subject/Key Words:
Scaling, Statistical Analysis, Test Theory


This paper contributes to the theory underlying unidimensional scaling on the basis of a concatenation operation defined on pairs of objects of certain continuous and ordered sets M. After stating the axiomatic base, main emphasis is placed on inferences about structural properties of M, such as reflexivity and commutativity of the concatenation operation through an entirely nonmetric and general treatment. It is shown that reflexivity holds either throughout or just for a single element and that commutativity of two different elements holds either generally or in no instance. These statements are independent. The study employs ad hoc defined concatenation operations which formally and necessarily satisfy first the reflexivity requirement and then, in addition, commutativity. Using this means, the existence and essential uniqueness of scales of M with linear expressions for the various concatenation operations are established. The rules by which the parameters entering into these expressions change under admissible scale transformations are derived.

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