Simulating Data from Marginal Scores

Simulating Rasch datasets from known item difficulties and person abilities is straightforward www.rasch.org/rmt/rmt213a.htm

More challenging is simulating Rasch datasets with known marginal scores. Verhelst et al. (2007) achieve this for complete rectangular dichotomous datasets using a Markov-Chain Monte-Carlo (MCMC) algorithm. Its operates on dichotomous data by shuffling observations between rows and columns. The program is implemented as an R package, RaschSampler, which can simulate matrices of up to 1024 rows and 64 columns.

Another approach is to simulate the data using Rasch measures. The dataset can be dichotomous or polytomous, complete or incomplete, and of any size.

1. Compute the marginal scores (persons, items) and counts (polytomous categories) in the generating dataset.

2. In the simulated dataset, impute the extreme observations corresponding to all extreme marginal scores in the generating dataset. Flag those observations as missing in the generating dataset. If there were extreme marginal scores, repeat from 1.

3. Estimate the measures (persons, items, polytomies) for the generating dataset. Rough estimates are good enough. Polytomies: allow for intermediate categories with sampling zeroes.

4. Select a non-missing observation at random in the generating dataset. Simulate its value based on the estimated measures and place it in the matching location in the simulated dataset. Flag that data-point as missing in the generating dataset.

5. Repeat the procedure from 1 until there are no active observations in the original dataset. For dichotomous data, only the measures for one person and one item will need to be re-estimated for each simulated observation. For polytomous data all measures may need to be re-estimated.

When this procedure completes, the simulated dataset will have the same marginal scores and category counts as the original generating dataset.

John Michael Linacre

Verhelst N., Hatzinger R., Mair R. (2007) The Rasch Sampler. Journal of Statistical Software, 20:6. www.jstatsoft.org/v20/i04

Simulating Data from Marginal Scores … J.M. Linacre, Rasch Measurement Transactions, 2008, 22:2, 1168

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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