Roskam, E.E., & Jansen, P.G.W. (1984). A new derivation of the Rasch model. In E. Degreef & J. van Bruggenhaut (Eds.), Trends in mathematical psychology (pp. 293-307). Amsterdam: North-Holland.
Consistent stochastic ordering of items and subjects is analogous to the ordering derived from composite transitivity in Guttman's scalogram structure. Guttman ordering is equivalent to the deterministic condition that
iRj =: nSj & -nSi, _n
This means that if item i is harder than item j [iRj] for some subject n, a condition implied by the fact that [=:] subject n succeeds on item j [nSj] and [&] subject n does not succeed on item i [-nSi], then item order is i then j for any n [_ n], i.e., item ordering is independent of subject.
Differential item ordering iRj can only be observed for subject n who succeeds on one item and fails on the other, i.e., whose number of successes on the two items, Rn, is 1. Therefore, the equivalent stochastic ordering is
p(iRj) =: p(nSj & -nSi ! Rn =1), _n
This means that the probability that (item i is more difficult than item j) is the probability that subject n, who succeeds on only one of the items, succeeds on item j and does not succeed on item i.
Proceeding algebraically, and specifying local independence,
Reparameterize p(nSi) as fi, p(nSj) as fj, where f is some continuously differentiable and nowhere equal to zero item-dependent function of z, a subject-dependent, but item-independent parameter.
But, for subject-independent stochastic ordering, the probability that item i is more difficult than item j must be independent of the subject forming the basis of the comparison, i.e., of the value of zn for subject n. Thus,
Thus the Rasch model follows from the requirement of stochastically consistent item orders, and so is the probabilistic counterpart of Guttman's scalogram rule. Since a scalogram is symmetrical in its treatment of subjects and items, the Rasch model is also obtained by considering stochastically consistent subject ordering.
Rasch Model derived from Consistent Stochastic Guttman Ordering, E Roskam & P Jansen Rasch Measurement Transactions, 1992, 6:3 p. 232
|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|
|Apr. 14-17, 2020, Tue.-Fri.||International Objective Measurement Workshop (IOMW), University of California, Berkeley, https://www.iomw.org/|
|May 22 - June 19, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|June 26 - July 24, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com|
|June 29 - July 1, 2020, Mon.-Wed.||Measurement at the Crossroads 2020, Milan, Italy , https://convegni.unicatt.it/mac-home|
|July - November, 2020||On-line course: An Introduction to Rasch Measurement Theory and RUMM2030Plus (Andrich & Marais), http://www.education.uwa.edu.au/ppl/courses|
|July 1 - July 3, 2020, Wed.-Fri.||International Measurement Confederation (IMEKO) Joint Symposium, Warsaw, Poland, http://www.imeko-warsaw-2020.org/|
|Aug. 7 - Sept. 4, 2020, Fri.-Fri.||On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com|
|Oct. 9 - Nov. 6, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|June 25 - July 23, 2021, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com|
The URL of this page is www.rasch.org/rmt/rmt63e.htm