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
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