# Meaningfulness, Sufficiency, Invariance and Conjoint Additivity

Consider the following statements from widely respected authorities in statistics and measurement:

• "If there exists a minimal sufficient statistic [i.e., one that is both necessary and sufficient] for the individual parameter Theta which is independent of the item parameters, then the raw score is the minimal sufficient statistic and the model is the Rasch model" (Andersen, 1977, p. 72).

• "The set of invariant rules based on a sufficient statistic is an essentially complete subclass of the class of invariant rules" (Arnold 1985, p. 275; citing Hall, Wijsman, & Ghosh, 1965).

• "The hallmark of a meaningless proposition is that its truth-value depends on what scale or coordinate system is employed, whereas meaningful propositions have truth-value independent of the choice of representation, within certain limits. The formal analysis of this distinction leads, in all three areas [measurement theory, geometry, and relativity], to a rather involved technical apparatus focusing upon invariance under changes of scale or changes of coordinate system" (Mundy, 1986, p. 392).

Andersen (1977) shows that summing ratings to a score is meaningful and useful only if that score is a minimally sufficient statistic, and if that statistic exists, then the Rasch model holds. Arnold (1985) and Hall, Wijsman, and Ghosh (1965) show that statistical sufficiency is effectively equivalent with measurement invariance. Mundy (1986) shows that meaningful propositions all share the property of invariance. Luce and Tukey (1964) show that conjoint additivity is another way of arriving at the invariance characteristic of fundamental measurement.

These principles of meaningfulness, sufficiency, invariance, and conjoint additivity are ubiquitous in the production of scientific knowledge, which explains why we find so many strong statements in the history of science to the effect that measurement and quantification are absolutely essential to any science worthy of the name (Michell, 1990, pp. 6-8). We have, unfortunately, confused the mere use of number with meaningful measurement, when, in fact, it is the realization of the qualitatively mathematical ideal of invariance that makes science what it is. Even as unlikely a philosopher as Heidegger (1967, pp. 75-6), who was held by some to be, at best, a poet, understood that the broad qualitative sense of the mathematical is "the fundamental presupposition of all 'academic' work" and "of the knowledge of things."

Multiple harmonious definitions of meaningful measurement are effectively embodied in Rasch models (Fischer, 1995; Fisher, 2004; Wright, 1997). It then follows that the Rasch model's "singular significance for measurement is that it is a unique (necessary and sufficient) deduction from the (fundamental) measurement requirements of joint order and additivity" (Wright, 1984).

Analytic methods implementing Rasch measurement test the hypothesis of qualitative yet mathematical meaningfulness more effectively, easily and efficiently than any other available methods. It is the norm today to presume scientific status and the achievement of measurement even when sufficiency and invariance have not been tested or established. The day may soon be coming when such hubris will be considered tantamount to fraud. When that day arrives, research employing Rasch models will be sought after as paradigmatic examples of mathematically meaningful methodology.

William P. Fisher

Reference

Andersen, E. B. (1977). Sufficient statistics and latent trait models. Psychometrika, 42(1), 69-81.

Arnold, S. F. (1985, September). Sufficiency and invariance. Statistics & Probability Letters, 3, 275-279.

Fischer, G. H. (1995). Derivations of the Rasch model. In G. Fischer & I. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 15-38). New York: Springer-Verlag.

Fisher, W. P., Jr. (2004, October). Meaning and method in the social sciences. Human Studies: A Journal for Philosophy and the Social Sciences, 27(4), 429-54.

Hall, W. J., Wijsman, R. A., & Ghosh, J. K. (1965). The relationship between sufficiency and invariance with applications in sequential analysis. Annals of Mathematical Statistics, 36, 575-614.

Heidegger, M. (1967). What is a thing? (W. B. Barton, Jr. & V. Deutsch, Trans.). South Bend, Indiana: Regnery/Gateway.

Michell, J. (1990). An introduction to the logic of psychological measurement. Hillsdale, New Jersey: Lawrence Erlbaum Associates.

Mundy, B. (1986). On the general theory of meaningful representation. Synthese, 67, 391-437.

Wright, B. D. (1984). Despair and hope for educational measurement. Contemporary Education Review, 3(1), 281-288 www.rasch.org/memo41.htm

Wright, B. D. (1997, June). Fundamental measurement for outcome evaluation. Physical Medicine & Rehabilitation State of the Art Reviews, 11(2), 261-88.

Meaningfulness, Sufficiency, Invariance and Conjoint Additivity, Fisher W.P. Linacre J.M. … Rasch Measurement Transactions, 2006, 20:1 p. 1053

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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