# Sample Independency in the Rasch Model?

In my last Note I reminded you of the fact that the Rasch model has simple sufficient statistics for its parameters. A favorable consequence of the presence of sufficient statistics is that conditional likelihood equations can be established that only have ability or difficulty parameters. This Note is to warn you not to over-interpret this property as "sample-independent estimation in the Rasch model."

For example, let's take the case of conditional maximum likelihood estimation CMLE of the ability parameters and assume that the Rasch model holds for a certain pool of items. An overly simple interpretation would be to state that from the above property it follows that "the same difficulty parameters can be estimated from any subset of examinees." This statement holds for any estimation problem! The pertinent question is not if we can estimate parameters from different samples, but how well we can estimate them. Statements that do not refer to any criterion of optimality holding for the estimators are therefore incomplete.

Such criteria of optimality are always formulated in terms of the distributions of estimators over repeated sampling. Now what properties of the sampling distribution remain invariant if we replicate measurement of the same examinee, but include different items in the test? Is it the full distribution of the ability estimator that does not change? Or just its expected value (mean) or variance?

We know that the variance of the estimator of the ability parameter depends on the item parameters. This property is welcome, for it allows us to construct an optimal test for a population of examinees. Thus, for different items we have different variances of our estimators. And hence even different distributions! But what about the mean? Does the mean not remain invariant? For tests of finite sizes, the estimators for the ability parameters, like most maximum likelihood estimators, are known to be biased. And the bias may be dependent on the values of the parameters in the sample - as is demonstrated, for example, by the non-zero probability that some of the items have to be removed from the test because all examinees have them correct. Hence, for different tests of finite length, we are likely to have estimators of the same ability with a different mean.

So what is left? It is only the property that if test length goes to infinity, each ability estimator will have the same expected value or mean, whatever the composition of the test. This property is known as consistency in statistics. It holds uniquely for the Rasch model among all response models with incidental parameters.

It is recommended to use this well-defined term, consistency, rather than the contradictio in terminis "sample-independent estimation."

Sample Independency in the Rasch Model?, W van der Linden … Rasch Measurement Transactions, 1993, 6:4 p. 247

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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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