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 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

Forum

Rasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.

The URL of this page is www.rasch.org/rmt/rmt201a.htm

Website: www.rasch.org/rmt/contents.htm