Rasch measures never lose their unidimensionality (nor their linearity for the additive form of the Rasch model). Those properties are forced by the Rasch model. But we can lose the connection between the Rasch measures and the intended unidimensional latent variable.
Example: we want to measure "arithmetic ability". 1,000 children take our arithmetic test. 500 children respond to the items carefully. 500 children guess at random.
If we Rasch-analyze the 500 careful children, then we will obtain ability measures and item difficulties on the intended unidimensional, linear "arithmetic" latent variable with good fit of the data to the Rasch measures.
If we Rasch-analyze the 500 guessing children, then we will obtain person measures and item difficulties on a unidimensional, linear "random guessing" latent variable with good fit of the data to the Rasch measures..
If we Rasch-analyze 500 careful children + 500 guessing children, then we will obtain "ability" measures and item difficulties on a unidimensional, linear "arithmetic + random guessing" latent variable with poor fit of the data to the Rasch measures.
We might say, but "arithmetic + random guessing" is not substantively unidimensional! We know that, but the Rasch model does not. It analyzes the data as though they are unidimensional, and then the fit statistics report how well the data match the mathematically unidimensional framework that the Rasch analysis has constructed.
John M. Linacre
Rasch Measures and Unidimensionality, J.M. Linacre. ... Rasch Measurement Transactions, 2011, 24:4, 1310
Please help with Standard Dataset 4: Andrich Rating Scale Model
|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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