A Rasch Unidimensionality coefficient

[ Note: later work has indicated that this is not an effective indicator of unidimensionality. ]

If you are so disinterested in your data that you are willing to reduce its state of dimensionality to a single index, (rather than studying where and how it departs from your intended dimension), then here is a Rasch-based procedure of assessing unidimensionality.

For all persons, on all items,

1. compute R(model), the person separation reliability using model (asymptotic) standard errors. This treats the data as unidimensional. All fluctuations away from stochastic unidimensionality are regarded as due to expected local stochastic variation in stochasticity.

2. compute R(real), the person separation reliability using real (misfit-inflated) standard errors. This treats the data as though it might be multidimensional. All fluctuations away from stochastic unidimensionality are regarded as multidimensionalities of whatever cause. Since the misfit-inflated standard errors are larger than the model errors, R(real) is always less than R(model).

3. compute R(unidimensional), a "reliability" of item unidimensionality:

This reliability coefficient can be interpreted in the same way as a conventional reliability coefficient. Values above 0.9 indicate a clearly unidimensional variable. Values below 0.5 might be cause for alarm.

Ben Wright

Unidimensionality coefficient. Wright BD. … Rasch Measurement Transactions, 1994, 8:3 p.385

Rasch Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene	Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver	Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone	Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind	Rasch Measurement: Applications, Khine	Winsteps Tutorials - free Facets Tutorials - free	Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre	Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan
Other Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free	An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse	Rasch Measurement Theory Analysis in R, Wind, Hua	Applying the Rasch Model in Social Sciences Using R, Lamprianou	El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias	Diseño de Mejores Pruebas - free, Spanish Best Test Design	A Course in Rasch Measurement Theory, Andrich, Marais	Rasch Models in Health, Christensen, Kreiner, Mesba	Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen

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