# Data Variance Explained by Rasch Measures

The Rasch model predicts that each observation will contain an exactly predictable part generated by the Rasch measures, and a well-behaved random component whose variance is also predicted by the model.

Figure 1 shows that, for dichotomous observations, as the logit measure difference between the person and the item increases (x-axis), the variance explained by the measures also increases (solid line) and the unexplained variance decreases (dotted line). When an item is as difficult as a person is able (0 on the x-axis), the outcome is completely uncertain. It is like tossing a coin. None of the variance in the observation is explained by the Rasch measures.

In Figure 2, the unexplained variance has been standardized to be the same size for every dichotomous observation. Thus each observation is modeled to contribute one unit of statistical information. An effect is to down-weight the central high-unexplained-variance observations. Standardized variances are used in the computation of standardized residuals which form the basis of several indicators of fit to the Rasch model.

In Figure 3, the decomposition of the variance in the data is shown for different combinations of item and person variance and item-person targeting. The unexplained variance has been standardized across observations a in Fig. 2. It is seen that the sum of the person S.D. and item S.D. must exceed 2 logits in order that over 50% of the standardized variance in the data be explained by the Rasch measures.

In the equivalent plot for variances using raw residuals [which have now been found to be a more accurate description], the y-axis values are about half of those plotted in Figure 3. Thus the sum of the person and item S.D.s must exceed 3 logits for over 50% of the raw observational variance to be explained.

John M. Linacre

For the expected size of the unexplained variance in the data, see www.rasch.org/rmt/rmt233f.htm.

 Fig. 1.Variance decomposition of a dichotomy. Fig. 2. Decomposition with standardized variance. Fig. 3. Decompositions of item and person variance using standardized residuals Fig. 4. Decompositions of variance using raw residuals have their explained-values about halved relative to standardized residuals. This graph from RMT 22:1 p. 1164.

Data Variance Explained by Measures, Linacre J.M. … Rasch Measurement Transactions, 2006, 20:1 p. 1045

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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
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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
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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
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Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

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