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 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

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