A general theory of polynomial conjoint measurement

The present theory generalizes conjoint measurement in five major respects and shows that a data structure D satisfies a given polynomial measurement M if and only if D satisfies an abstract irreflexivity axiom with respect to M.

Abstract

The present theory generalizes conjoint measurement in five major respects. (a) It is formulated in terms of partially rather than fully ordered data. (b) It applies to both ordinal and numerical data. (c) It is applicable to finite as well as infinite data structures. (d) It provides a necessary and sufficient condition for measurement. (e) This condition applies to any polynomial measurement model; that is, any model where each data element is expressed as a specified real-valued, order-preserving polynomial function of its components.Examples of polynomial measurement models include Savage's subjective expected utility model, Hull's and Spence's performance models, Luce's choice model, and multidimensional scaling models.It is shown that a data structure D satisfies a given polynomial measurement M if and only if D satisfies an abstract irreflexivity axiom with respect to M. The interpretation of the result and its implications to measurement theory are discussed.