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
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 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|
|Forum||Rasch Measurement Forum to discuss any Rasch-related topic|
Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement
Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.
|Coming Rasch-related Events|
|Sept. 27-29, 2017, Wed.-Fri.||In-person workshop: Introductory Rasch Analysis using RUMM2030, Leeds, UK (M. Horton), Announcement|
|Oct. 13 - Nov. 10, 2017, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|Oct. 25-27, 2017, Wed.-Fri.||In-person workshop: Applying the Rasch Model hands-on introductory workshop, Melbourne, Australia (T. Bond, B&FSteps), Announcement|
|Dec. 6-8, 2017, Wed.-Fri.||In-person workshop: Introductory Rasch Analysis using RUMM2030, Leeds, UK (M. Horton), Announcement|
|Jan. 5 - Feb. 2, 2018, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|Jan. 10-16, 2018, Wed.-Tues.||In-person workshop: Advanced Course in Rasch Measurement Theory and the application of RUMM2030, Perth, Australia (D. Andrich), Announcement|
|Jan. 17-19, 2018, Wed.-Fri.||Rasch Conference: Seventh International Conference on Probabilistic Models for Measurement, Matilda Bay Club, Perth, Australia, Website|
|April 13-17, 2018, Fri.-Tues.||AERA, New York, NY, www.aera.net|
|May 25 - June 22, 2018, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|June 29 - July 27, 2018, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com|
|Aug. 10 - Sept. 7, 2018, Fri.-Fri.||On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com|
|Oct. 12 - Nov. 9, 2018, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
The URL of this page is www.rasch.org/rmt/rmt201f.htm