How to Simulate Rasch Data

Dichotomous data:

1. Decide about the items. They are usually uniformly distributed. How many items? How wide the interval? The item mean is usually set at 0 logits. Simulate the item difficulties.

2. Decide about the person sample. This is usually normally distributed. How big a sample? What is the mean? What is the standard deviation? Simulate the person abilities.

4. For each response by a person to an item:

4A. Generate a random number U = uniform [0,1]

4B. Probability of failure = 1/(1 + exp(ability - difficulty))

4C. If U > Probability of failure, then X=1 else X=0.

4D. X is the simulated observation.

5. Check this by simulating data for a very high ability person (logit = 10): the data should all be "1".
Simulate data for a very low ability person (logit = -10): the data should all be "0"

Polytomous (rating scale or partial credit) data:

1. Decide about the items. They are usually uniformly distributed. How many items? How wide the interval? The item mean is usually set at 0 logits. Simulate the item difficulties.

2. Decide about the person sample. This is usually normally distributed. How big a sample? What is the mean? What is the standard deviation? Simulate the person abilities.

3. Decide about the number of categories, m. The higher categories, 2 to m, have Rasch-Andrich threshold values that are usually ascending and sum to zero across all the categories. Simulate the threshold values.

4. For each response by a person to an item:

4A. Generate a random number U = uniform [0,1]

4B. Compute the cumulative exponential of observing each category:
  measure = 0
  cumexp(1) = 1
  Compute for category j = 2 to m
    measure = measure + ability - difficulty - threshold(j)
    cumexp(j) = cumexp(j-1) + exponential(measure)
  Next category

4C. Identify the simulated observation:
  U = U * cumexp(m)
  For category j = 1 to m
    if U <= cumexp(j) then X = j: exit
  Next category

4D. X is the simulated observation.

5. Check this by simulating data for a very high ability person (logit = 10): the data should all be "m" (the top category).
Simulate data for a very low ability person (logit = -10): the data should all be "1" (the bottom category).

Unobserved Categories: sampling zeroes

Unobserved categories have a very low probability of being observed, so set the threshold values:
very low for an unobserved bottom category: example: 5 categories, bottom category unobserved: -40, -1, 0, 1
very high for an unobserved top category: example: 5 categories top category unobserved: -1, 0, 1, 40
very high then very low for the an unobserved intermediate category: example: 5 categories middle category unobserved: -1, 40, -40, 1

Many-Facets data:

As Polytomous data with the addition of:
1A. Decide about the other facets (tasks, demographics, etc.). Choose logit values for their elements.

4B. is amended:
    measure = measure + ability - difficulty + {measures of elements of other facets that apply to this observation} - threshold(j)

John M. Linacre

Linacre J.M. (2007) How to Simulate Rasch Data … Rasch Measurement Transactions 21:3 p. 1125

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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