Rasch Model derived from Consistent Stochastic Guttman Ordering

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


  1. The Rasch Model derived from E. L. Thorndike's 1904 Criteria, Thorndike, E.L.; Linacre, J.M. … 2000, 14:3 p.763
  2. Rasch model derived from consistent stochastic Guttman ordering, Roskam EE, Jansen PGW. … 6:3 p.232
  3. Rasch model derived from Counts of Right and Wrong Answers, Wright BD. … 6:2 p.219
  4. Rasch model derived from counting right answers: raw Scores as sufficient statistics, Wright BD. … 1989, 3:2 p.62
  5. Rasch model derived from Thurstone's scaling requirements, Wright B.D. … 1988, 2:1 p. 13-4.
  6. Rasch model derived from Campbell concatenation: additivity, interval scaling, Wright B.D. … 1988, 2:1 p. 16.
  7. Dichotomous Rasch model derived from specific objectivity, Wright BD, Linacre JM. … 1987, 1:1 p.5-6



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

To be emailed about new material on www.rasch.org
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from Rasch.org

www.rasch.org welcomes your comments:

Your email address (if you want us to reply):

 

ForumRasch 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
May 17 - June 21, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 12 - 14, 2024, Wed.-Fri. 1st Scandinavian Applied Measurement Conference, Kristianstad University, Kristianstad, Sweden http://www.hkr.se/samc2024
June 21 - July 19, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 5 - Aug. 6, 2024, Fri.-Fri. 2024 Inaugural Conference of the Society for the Study of Measurement (Berkeley, CA), Call for Proposals
Aug. 9 - Sept. 6, 2024, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 4 - Nov. 8, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 17 - Feb. 21, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 16 - June 20, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 20 - July 18, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Facets), www.statistics.com
Oct. 3 - Nov. 7, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com

 

The URL of this page is www.rasch.org/rmt/rmt63e.htm

Website: www.rasch.org/rmt/contents.htm