Rasch with an Error Term

Question: Regression models include an explicit error term, why don't Rasch models?

Answer: The Rasch model is usually presented in a way which emphasizes its unique statistical properties, but it can be written to conform with a "general linear hypothesis" as :

X = E ± sqrt(W)

where X is the empirical observation and E is the expected value of the observation according to the relevant Rasch model. W is the error variance, specific to this observation, i.e., modeled as heteroscedastic, in contrast to the typical regression model in which the error variance is averaged across all observations, i.e., assumed to be homoscedastic.

The algebraic expressions for E and W are shown on p. 100 of Rating Scale Analysis (Wright & Masters, 1982). For dichotomous data they are

and

Wni = Pni (1-Pni)

Rasch with an error term, … Rasch Measurement Transactions, 2004, 18:3 p. 993

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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