A theory of relative judgment

Abstract

This paper is concerned with the scaling method of "ratio estimation." The simple theory that equates reported ratio judgments to ratios of psychological magnitudes is first considered, then two close relatives of this theory are formulated, each of which places weaker constraints on the structure of the data. Structural conditions are stated that express the relations that must hold among observed ratio judgments for each of the models. The models proposed are "cumulative" in the sense that the second is a weakened version of the first, and the third a weakened version of the second. A special feature of the models is that they may be tested entirely in terms of observables, avoiding the necessity of scale construction prior to testing. Tests were carried out on data from 9 published studies. The strongest model, typically used in scale construction using ratio estimation data, was generally inadequate, showing large systematic errors. However, the weakest version generally passed the tests of internal consistency, and the model equation provided a basis for constructing ratio scales utilizing bias parameters.This paper is concerned with the scaling method of ratio estimation. In a typical task, a pair of stimuli (a, b) is presented, and the subject is instructed to respond with a number that corresponds to the sensation "ratio" of a to b relative to a defined attribute. In another variation, called "free ratio estimation" (Mashhour, 1964), subjects assign a number to each of the two stimuli such that the ratio of the numbers assigned reflects the sensation ratio. Thus the method applies to any response that can be transformed to a ratio. The method does not require a modulus, and usually all possible pairs are presented.The method of ratio estimation has a long history (Ekman, 1958;Stevens, 1958), but recently the simple theory that equates the reported sensation ratio to a ratio of psychological magnitudes has been questioned (e.g., Fagot, Stewart, & Kleinknecht, 1975;SjOberg, 1971). If the simple theory does not lead to valid ratio scales, how might the theory be modified without abandoning the basic assumption that subjects are capable of making ratio judgments that are consistent in a well-defined sense?This paper answers the question by developing two close relatives of the simple theory of ratio estimation that place weaker constraints on the structure of the data. The different versions constitute a theory of relative judgment in the obvious sense that a stimulus is judged relative to another stimulus. This relativity is in no sense trivial, since it leads to the formulation of tests of internal consistency not possible with methods such as the pure form of magnitude estimation in which stimuli are judged one at a time.Work on this paper was carried out, in part, during the author's tenure as a Fellow at the Netherlands Institute for Advanced Study, Wassenaar, Holland. 243The basic question asked is: Assuming that a ratio scale exists, what relations must hold among observed ratio estimations? The answe...

A theory of bidirectional judgments

“…The theory (Fagot, 1978(Fagot, , 1979 deals with any set of judgments Jab for which there exists a response transformation J:b that is a function of sensory ratios 'l'a/'l'b and possible bias parameters. Three specific model equations were analyzed:…”

“…(1) These and other questions are addressed by the theory of bidirectional judgments to be presented, based on a theory of relative judgment by Fagot (1978Fagot ( , 1979. First the theory will be outlined briefly, followed by further theoretical development that takes explicit account of bidirectionality.…”

“…The extensive data base for ratio estimation and similarity judgments examined by Fagot (1978Fagot ( , 1979 consisted (8) (9) We are now in a position to give a more precise definition of bidirectional judgments. Let Jab be any judgment of stimulus a relative to stimulus b.…”

“…Equations 1-6 are not restricted to the case of unidirectional judgments, as applied by Fagot (1978) to ratio estimation. If judgments are bidirectional, however, the number of test cases for the structural conditions is substantially increased, as will be shown below.…”

“…With the assumption that judgments are mediated by perceived relations of pairs of stimuli, the theory is developed for judgments of comparison stimuli in relation to standards, taking explicit account of the location of the comparison stimulus relative to the standard. The theory of bidirectional judgments, based on a theory of relative judgment by Fagot (1978Fagot ( ,1979,entails a partial nesting of models characterized by a progressive weakening of the constraints placed on the structure of the data. The weakest model, the relative bias/directional standard (RBDS) model, allows each standard to have two biasing effects, depending on the location of the standard above or below the comparison stimulus.…”