Measures are ideals with precise mathematical properties - much more than the assignment of numbers to qualities. Practical measurement requires the application of the ideal to the empirical. These are themes from Joel Michell's "An Introduction to the Logic of Psychological Measurement", Hillsdale, NJ:Lawrence Erlbaum Associates, 1990.
Requirements for measurement:
Michell censures S.S. Stevens (1951) for the confusion caused by his definition of measurement as the assignment of numbers according to any rule, and by his misguided classification of scales as nominal, ordinal, interval, and ratio. Michell argues that the scaling methods of Thurstone (1927), Coombs (1964) and Guttman (1950) have not received due attention because Stevens' shallow definitions have encouraged the acceptance of poorly constructed data. On the other hand, Michell criticizes physicist N.R. Campbell (1920) for reducing additive relations to empirical concatenation, and praises Luce and Tukey (1964) for showing that additivity is an ideal that does not depend upon empirical operations.
The misconstrual of nominal and ordinal data as "measures" together with the empirical problem of constructing data good enough for fundamental measurement have, according to Michell, led psychologists to find refuge in "quantitative methods that, because they assume more, demand less foundational research as the basis for their application. Methods that always yield a scaling solution, like the method of summated ratings, are almost universally preferred to methods which, like Coombs', do not produce a scaling solution when they are falsified by the data. Surprisingly, vulnerability to falsification is commonly deemed by psychologists to be a fault rather than a virtue" (p. 130). Methods that do not specify conditions for validity leave researchers at the mercy of unsupported "assumptions" as to whether a valid scale has been constructed.
Rasch measurement, though almost unnoticed by Michell, is a rigorous measurement technique. By asserting that measurement must be a function of only the relevant person and item characteristics, Rasch measurement does not assume that such is the case, as when ratings are simply summed; rather, it tests the hypothesis that empirical person- item interactions are, in fact, dominated by the variable of interest. Rasch measurement, sometimes criticized for assuming too much, actually liberates researchers from unsupportable assumptions about scale validity.
Is Quantitative Science Platonic?:
Michell contends that psychological measurement has foundered on the myth that scientific research must be quantitative. He traces the quantitative imperative from Pythagoras, claiming that Plato furthered it, but that Aristotle countered it with his emphasis on qualities. According to Michell, the Platonic view dominating psychology has resulted in psychologists "accepting a definition [of measurement] so inflated as to rule out none of their methods" (p. 9).
Michell's perception of Plato accords with that of Galileo, but is refuted by recent studies (Fisher 1991). Gadamer's reading of Plato contradicts Michell's, but accords with Michell's claim that "measurement" is a qualitative-quantitative continuum with its qualitative end no less rigorous or desirable than its quantitative end. The most Platonic aspect of Galileo's physics may not be his sense of mathematics as the language of nature, but his phrase "mente concipio" (I conceive in my mind, I imagine), which he used when describing the ideal circumstances under which his theory of gravity would hold. Just as Plato founded geometry with ideal definitions (a point is an indivisible line segment), so did Galileo found physics (in a frictionless vacuum, particular relations of force, mass, and acceleration will hold).
Michell's message is that psychological measurement must emphasize the ideal. Like the additive conjoint models Michell presents, Rasch requires us to think of person ability and item difficulty as ideals, and then to test to what extent these ideals can have empirical support.
Re-examining Thurstone Measurement:
Measurement is an ideal. Unfortunately, Michell insists that the empirical reflect the ideal in a way that abandons the useful and verges on the impossible. He examines Thurstone's (1925) example of conjoint measurement constructed from data on the seriousness of crimes, and requires ordered statistically significant differences everywhere, even between items which appear to be of the same difficulty (crimes of the same seriousness). This leads Michell to the futile conclusion that "either seriousness of crimes is not a quantitative variable or else some other part of Thurstone's theory of comparative judgment is false" (p.107).
But Michell tests the wrong hypothesis. His hypothesis is that every measurement criteria must be completely met for measurement to exist. But then not even physical measurement is possible when two objects happen to be of equal length. A better hypothesis is that measurement exists unless a measurement criterion is clearly failed. Then empirical disordering of objects of nearly equal length is due not to failure of the measurement process, but to its inescapable imprecision.
Nevertheless, Michell's book is a dynamic and readable addition to the literature on rigorous measurement. His criticisms of what have passed for measurement theory in psychology are cogent, and the alternatives he offers are well formulated and documented. Now that Joel Michell has spoken at IOMW, perhaps the next edition of his soon- to-be-classic book will include a chapter on Rasch's Separability Theorem!
Coombs CH 1964. A theory of data. New York: Wiley
Fisher WP Jr 1991. Chapter 3 of Objective Measurement: Theory into Practice, Mark Wilson (Ed.). Norwood NJ: Ablex
Guttman L 1950. The basis for Scalogram analysis. In Stouffer et al. Measurement & Prediction, The American Soldier, Vol IV. New York: Wiley
Campbell NR 1920. Physics: the elements. Cambridge: Cambridge University Press
Luce RD & Tukey JW 1964. Simultaneous conjoint measurement. Journal of Mathematical Psychology 1 1-27
Stevens SS (Ed) 1951. Handbook of experimental psychology. New York: Wiley
Thurstone LL 1927. The unit of measurement in educational scales. Journal of Educational Psychology 18 505-24
Scientific Measurement: the Supremacy of Ideals, W Fisher Jr. Rasch Measurement Transactions, 1991, 5:2 p. 139-140
Please help with Standard Dataset 4: Andrich Rating Scale Model
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