Under Rasch model conditions, there is some probability that a person will succeed or fail on any item, no matter how easy or hard. This means that there is some probability that any person could produce any response string. Even the most able person could fail on every item.
The measure estimated for a person is usually that for which the observed response string is most likely, or that for which the response string best fits a Rasch model. We may, however, have some rough idea about a person's ability measure (or an item's difficulty) prior to the current data collection and wish to incorporate this idea into the newly estimated measure. To do this, we calibrate the test items in the usual way. Then we combine the item calibrations, our prior rough idea, and the observed responses to obtain an improved, a posteriori, person measure. Mislevy and Stocking (1989) recommend this approach for IRT models. John Uebersax (1993 and on his website) outlines a general procedure for this.
The technique capitalizes on an insight of Thomas Bayes:
Prior Probability x Data Probability => Posterior Probability
which implies that
Prob (B' given {X}) =
Prob (B' ) x Prob ({X} given B' ) / Sum over all B [ Prob (B) x
Prob ({X} given B) ]
where B' is a particular value of the person measure, and the
sum is over all possible values of our rough idea, B. {X} is the
person's response string. The EAP estimate of the person measure is
the expected value of this:
EAP estimate = Sum over all B [B x Prob (B given {X})].
Thus, suppose that our rough idea, the prior distribution of B, φ(B), is a convenient distribution, such as N(μ,σ²). The test consists i=1,L items. P_{Xni} is the probability of person n of ability B scoring X_{ni} on item i.
EAP estimates may be more central or more diverse than MLE estimates depending on the choice of prior distribution.
Then
This can be evaluated using numeric quadrature to approximate the integrals.
John M. Linacre
Mislevy RJ & Stocking ML (1989) A consumer's guide to LOGIST and BILOG. Applied Psychological Measurement, 13, 57-75.
Uebersax JS (1993) Statistical modeling of expert ratings on medical treatment appropriateness. Journal of the American Statistical Association, 88, 421-427.
Expected A Posteriori (EAP) Measures. Uebersax JS. … 16:3 p.891
Expected A Posteriori (EAP) Measures. Uebersax JS. … Rasch Measurement Transactions, 2002, 16:3 p.891
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 |
in Spanish: | Análisis de Rasch para todos, Agustín Tristán | Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez |
Forum | Rasch Measurement Forum to discuss any Rasch-related topic |
Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement
Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.
Coming Rasch-related Events | |
---|---|
June 23 - July 21, 2023, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com |
Aug. 11 - Sept. 8, 2023, Fri.-Fri. | On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com |
The URL of this page is www.rasch.org/rmt/rmt163i.htm
Website: www.rasch.org/rmt/contents.htm