Question: My referee insists that I write the Rasch model with an error term. How do I do that, and what is the error distribution?
Answer: When the Rasch dichotomous model is written with an error term it looks like this:
Xni = Pni ± √(Pni*(1-Pni))
where Xni is the scored response of person n to item i, and Pni is the Rasch-model probability of a correct response, so that Pni = exp(Bn-Di) / (1+exp(Bn-Di)),
where Bn is the ability of person n and Di is the difficulty of item i.
The distribution of each error term √(Pni*(1-Pni )) is binomial, because only two outcomes are possible for each observation, but when the error terms are accumulated across all the observations (as they are for estimation), the binomial errors approximate normality.
For a polytomous Rasch model,
Xni = Eni ± √ (Wni )
where Eni is the Rasch-model expected value of the response and Wni is the Rasch-model variance of the response around its expectation. The error distribution is multinomial, approximating normality across the dataset.
Linacre J.M. (2010) Rasch Model with an Error Term, Rasch Measurement Transactions, 2010, 23:4, 1238
|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|
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