The standard dichotomous Rasch model does not incorporate guessing. Instead, guessing is detected as off-dimensional behavior by means of quality-control fit statistics. But for "minimum competency" tests guessing may need to be incorporated into the Rasch model as a lower asymptote to the item characteristic curve (ICC). In these circumstances, the guessability of an item is not a parameter to be estimated, but a constant to be specified. (In practice, the lower asymptote is specified as a constant in many supposedly 3-PL analyses.)
Here is a quasi-Rasch model (Keats' generalization) for guessing:
where ci is the probability of guessing the item, the lower asymptote to the ICC. This can be rewritten:
It is seen that when ci=0, this is the standard dichotomous model.
Estimation Equations
The slopes of the complementary ICCs are given by:
where Pnix is the probability that x = Xni = {0,1} is observed when person n encounters item i.
The likelihood of the data is:
The log-likelihood is:
Looking for the maximum-likelihood of the data across all values of the parameters, here for Bn:
This does not have convenient sufficient statistics, except when ci=c, so that guessability is constant across items. But this is how many MCQ tests are intended to function.
When ci=c, then the maximum likelihood condition for Bn is:
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Maximum Likelihood Curves with Guessing |
where Rn is the score for person n. There is a paradox here (and consequently also in 3-PL analyses). It is seen that, for a given raw score, success on easy items yields a higher estimated measure than success on hard items. The Figure shows this for a score of 5 right on a test of 10 items, uniformly distributed .1 logits apart, with guessability probability of .25.
The second derivative is, in general,:
with the Newton-Raphson iteration equation:
John Michael Linacre
Colonius, H. (1977). On Keats' generalization of the Rasch model. Psychometrika, 42, 443-445.
Dichotomous Quasi-Rasch Model with Guessing. Linacre J.M. 15:4 p. 856
Dichotomous Quasi-Rasch Model with Guessing Linacre J.M. Rasch Measurement Transactions, 2002, 15:4 p. 856
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