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:

likelihood curves with guessing
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

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
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
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
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