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
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