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.

1. PCA: Data Variance: Explained, Modeled and Empirical
2. Critical Eigenvalue Sizes (Variances) in Standardized Residual Principal Components Analysis (PCA)
3. More about Critical Eigenvalue Sizes (Variances) in Standardized-Residual Principal Components Analysis (PCA)
4. Data Variance Explained by Rasch Measures
5. PCA: Variance in Data Explained by Rasch Measures

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

Rasch Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	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
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	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	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
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
Other 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

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