Sufficient and Necessary Statistics

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-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free	An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse	Rasch Measurement Theory Analysis in R, Wind, Hua	Applying the Rasch Model in Social Sciences Using R, Lamprianou	El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias	Diseño de Mejores Pruebas - free, Spanish Best Test Design	A Course in Rasch Measurement Theory, Andrich, Marais	Rasch Models in Health, Christensen, Kreiner, Mesba	Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene	Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver	Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone	Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind	Rasch Measurement: Applications, Khine	Winsteps Tutorials - free Facets Tutorials - free	Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre	Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

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