Rasch Notes

Links

See rasch.org - 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).

John M. Linacre

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