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 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 |
Forum | Rasch Measurement Forum to discuss any Rasch-related topic |
Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement
Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.
Coming Rasch-related Events | |
---|---|
Apr. 14-17, 2020, Tue.-Fri. | International Objective Measurement Workshop (IOMW), University of California, Berkeley, https://www.iomw.org/ |
May 22 - June 19, 2020, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
June 26 - July 24, 2020, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com |
June 29 - July 1, 2020, Mon.-Wed. | Measurement at the Crossroads 2020, Milan, Italy , https://convegni.unicatt.it/mac-home |
July - November, 2020 | On-line course: An Introduction to Rasch Measurement Theory and RUMM2030Plus (Andrich & Marais), http://www.education.uwa.edu.au/ppl/courses |
July 1 - July 3, 2020, Wed.-Fri. | International Measurement Confederation (IMEKO) Joint Symposium, Warsaw, Poland, http://www.imeko-warsaw-2020.org/ |
Aug. 7 - Sept. 4, 2020, Fri.-Fri. | On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com |
Oct. 9 - Nov. 6, 2020, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
June 25 - July 23, 2021, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com |
The URL of this page is www.rasch.org/rmt/rmt213a.htm
Website: www.rasch.org/rmt/contents.htm