## Dichotomous Quasi-Rasch Model with Guessing

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:

 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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Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
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