Remarks on the Theory of the Measurement and Its Relation to Psychology

A theory of fundamental measurement consists of a n ordering of entities according to the attribute to he measured; some further mathematical structure over the set of these entities; a set of axioms (or empirical laws) that interrelates the ordering and the structure; a numerical representation (isomorphism) of the empirical system which is constructed just by counting and computing limits; and a uniqueness theorem that describes the relations among different representations of the same type. The two examples described in some detail are extensive and conjoint measurement. Both involve additivity the former over an operation of combining two entities to form a third, and the latter over the coordinates of the entities. When both theories apply to the same set of entities, as is often true in classical physics and occasionally may be in psychology, then an added qualitative axiom leads to a very simple relation between the two numerical systems. The construction of scales that are additive over coordinates is compared with analysis-of-variance tests for no interaction. Among other things, this involves the interplay between particular statistical statements, e.g, tests of hypotheses, and the uniqueness of the scales of measurement. The position advocated is that any monotonic increasing transformation in particular, those that lead to (approximate) normality of distributions may be applied to the measurements provided that the question asked is not about the nature of the scale. An example is whether two samples were drawn from the same distribution. However, when the question concerns the scale e.g., is there a scale that is (approximately) additive over coordinates ? then only those transformations that are admissible within the measurement theory may be used without altering the question. In general, this means that the equi-normality assumptions of classical statistics cannot be satisfied when the measurement theory is sufficiently strong to lead to interval or ratio scales.