Was the Rasch model Almost the Zermelo model?

Here is Rasch's version of the familiar Rasch model: (Rasch, 1980, p. 168):

Rasch model


Here is Zermelo's (1929) model:

Zermelo model


Both are multiplicative forms of the Rasch model. What is the difference? Rasch is contrasting persons with items. Zermelo is modeling paired comparisons. ur and us are the strengths of two chess players. Zermelo notes that this model converts the strengths of the chess-players into probabilities, urs. These strengths are estimated from their tournaments scores. A draw is accommodated by adding 0.5 to the player's raw score. Zermelo omits to mention how he would use his model to predict draws. In current practice, the prediction of draws is facilitated by using a rating-scale model (RMT 11:3, p. 584). Zermelo, however, perceives that his model is robust against unplayed games (missing data) and able to accommodate linked tournaments! (Probably the first time these attributes of a Rasch model are noticed.)

Why has Zermelo's model been ignored? His estimation technique is arduous, but worse, his explanation of the inferential meaning of the parameters is deficient. He fails to emphasize that ur /us are the odds of success of player r relative to player s. Further, his paper lacks the fundamental insight that loge(ur) linearizes "Player strength", permitting the production of a useful picture. The Figure shows such a picture, plotted from Zermelo's own example analysis, which, we can see, lacks the drama which was present at the Tournament (Soltis, 1975). Zermelo's results are the same as those obtained using Facets with a dichotomous model and half-weighting of draws. A modern analyst, however, is more likely to use a rating scale model, so that draws can be predicted explicitly. For comparison, the player measures for a paired rating-scale model are shown in the Figure and are seen to differ little from Zermelo's estimates.

New York Chess Tournament


When Bradley and Terry (1952) independently formulated Zermelo's model, they observed that: "If the estimates are converted to logarithms, the values loge [ur] occur on a linear scale and permit overall comparisons of the experimental treatments [or chess players]. Any considerations of differences among treatments [or players] should be based on the values of the log [ur]'s." (p. 326)

Bradley, R.A., and Terry, M.E. Rank analysis of incomplete block designs. I. The method of paired comparisons, Biometrika. 1952, 39, 324-45.

Soltis, A. (1975) The Great Chess Tournaments and Their Stories. Radnor, Pa.: Chilton.

Zermelo, E. (1929) The calculation of tournament results as a maximum-likelihood problem [German]. Mathematische Zeitschrift, 29, 436-460.

Was the Rasch model Almost the Zermelo model? Zermelo, E. … Rasch Measurement Transactions, 2000, 14:2 p. 754.




Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse Rasch Measurement Theory Analysis in R, Wind, Hua Applying the Rasch Model in Social Sciences Using R, Lamprianou El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
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
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias Diseño de Mejores Pruebas - free, Spanish Best Test Design A Course in Rasch Measurement Theory, Andrich, Marais Rasch Models in Health, Christensen, Kreiner, Mesba Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang 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
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
Facets Tutorials - free
Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

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
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/rmt142k.htm

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