Ordinal Independence and Functional Equations in the Theory of Psychological Difference

Abstract

Suppose that ψ is an interval scale that represents the magnitude of differences along a psychological continuum like loudness or subjective value. This paper analyzes qualitative assumptions that constrain the specific form of ψ. For example, what assumptions would imply that ψ is an additive or multiplicative function of a multiattributed stimulus? Or a power function of a physical magnitude? Many hypotheses regarding the specific form of ψ are axiomatized by ordinal independence (OI) assumptions, i.e., assumptions that assert that an ordering of stimuli is invariant under a transformation of the stimuli. In the context of difference measurement, OI assumptions imply functional equations; specific forms for ψ can be derived as the solutions of such equations. The functional equations analysis of OI assumptions reveals striking analogies that may not be evident from the assumptions themselves. The major goal of this paper is to present these analogies, and to show that they suggest a general methodology for investigating interval-scale representations of psychological attributes.

I would like to thank David H. Krantz, Jay Lundell, and Shihfen Tu for useful discussions of difference measurement I would also like to express my gratitude to Edward Roskam, Thom Bezembinder, Peter Wakker, Ronald Jansen, and Yvonne Schouten for their efforts in creating an intellectually and culturally stimulating experience at the 20th EMPG meeting. The criticisms of two anonymous reviewers substantially improved the form of this paper.