Rasch Model with an Error Term

Question: My referee insists that I write the Rasch model with an error term. How do I do that, and what is the error distribution?

Answer: When the Rasch dichotomous model is written with an error term it looks like this:

X_ni = P_ni ± √(P_ni*(1-P_ni))

where X_ni is the scored response of person n to item i, and P_ni is the Rasch-model probability of a correct response, so that P_ni = exp(B_n-D_i) / (1+exp(B_n-D_i)),

where B_n is the ability of person n and D_i is the difficulty of item i.

The distribution of each error term √(P_ni*(1-P_ni )) is binomial, because only two outcomes are possible for each observation, but when the error terms are accumulated across all the observations (as they are for estimation), the binomial errors approximate normality.

For a polytomous Rasch model,

X_ni = E_ni ± √ (W_ni )

where E_ni is the Rasch-model expected value of the response and W_ni is the Rasch-model variance of the response around its expectation. The error distribution is multinomial, approximating normality across the dataset.

See www.rasch.org/rmt/rmt34e.htm.

Linacre J.M. (2010) Rasch Model with an Error Term, Rasch Measurement Transactions, 2010, 23:4, 1238

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