As each of you knows, the Rasch model is the only model for dichotomous responses that has (non-trivial) "sufficient" statistics for its parameters. All other fine statistical features of the model, such as the existence of consistent Conditional Maximum Likelihood (CMLE) estimators and the presence of feasible statistical tests for goodness of fit, follow from this property.
Statisticians define sufficiency by the concept of data reduction. Obviously, for any response model, a sample of response vectors contains all the information about the parameters at hand and is trivially sufficient for these parameters. The property of sufficiency becomes interesting only if we are able to reduce the number of response vectors, combining them into a statistic without losing any information about the parameters. This process of data summary or reduction may go on and on, until we reach a point where any further reduction would create loss of information. For the Rasch model, if we start with response vectors (X1,X2,...,XN) for a fixed examinee on an N-item test, then a possible representation of this process of data summary is: (X1,...,XN), (X1+X2,X3,...,XN), (X1+X2+X3,X4,...,XN),..., (X1+X2+...+XN). Each of these statistics is sufficient for the ability parameter of the examinee!
The other day, in a statistical textbook by Casella and Berger (1990) that is now my latest favorite, I found a reference to a paper by Dynkin (1951) that gives an answer to the question: "Are there any necessary statistics?" Dynkin defines a statistic as "necessary", if it is a function of every sufficient statistic. In the above representation, the endpoint of the process, which is the simple sum of the item responses, is a necessary statistic since it is a function of the statistics earlier in the series, as well as of the statistics in any other series that can be defined.
Dynkin's terminology has not become popular; it has been beaten by the more familiar concept of a "minimal sufficient" statistic, which is precisely a statistic that is both sufficient and necessary. But it may be fun to keep this older terminology in mind. Next time you get involved in a discussion about properties of the Rasch model, just casually remark that it is the only response model for which the number of correct response is a "necessary" statistic!
Casella G, Berger RL. 1990. Statistical Inference. Pacific Grove, CA: Wadsworth.
Dynkin EB. 1951. Necessary and sufficient statistics for a family of probability distributions. English translation in Selected Translations in Mathematical Statistics and Probability, 1961, 1, 23- 41
Sufficient and Necessary Statistics, W van der Linden Rasch Measurement Transactions, 1992, 6:3 p. 231
|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|
|Jan. 30-31, 2020, Thu.-Fri.||A Course on Rasch Measurement Theory - Part 1, Sydney, Australia, course flyer|
|Feb. 3-7, 2020, Mon.-Fri.||A Course on Rasch Measurement Theory - Part 2, Sydney, Australia, course flyer|
|Jan. 24 - Feb. 21, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|Apr. 14-17, 2020, Tue.-Fri.||International Objective Measurement Workshop (IOMW), University of California, Berkeley, https://www.iomw.org/|
|May 22 - June 19, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|June 26 - July 24, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com|
|June 29 - July 1, 2020, Mon.-Wed.||Measurement at the Crossroads 2020, Milan, Italy , https://convegni.unicatt.it/mac-home|
|July 1 - July 3, 2020, Wed.-Fri.||International Measurement Confederation (IMEKO) Joint Symposium, Warsaw, Poland, http://www.imeko-warsaw-2020.org/|
|Aug. 7 - Sept. 4, 2020, Fri.-Fri.||On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com|
|Oct. 9 - Nov. 6, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|June 25 - July 23, 2021, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com|
The URL of this page is www.rasch.org/rmt/rmt63d.htm