Rasch Model with an Error Term

# Rasch Model with an Error Term

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 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.

Coming Rasch-related Events
Oct. 9 - Nov. 6, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 22 -Feb. 19, 2021, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 21 -June 18, 2021, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 25 - July 23, 2021, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 13 - Sept. 10, 2021, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith,Facets), www.statistics.com
June 24 - July 22, 2022, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com