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
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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 |
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