# 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 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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