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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Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
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Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
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