Georg Rasch was concerned to construct measurement models with "parameters that are specific to the individuals" (1960, p. 9). An early extension of the Rasch model specified that the "individuals" were to be treated as random variables drawn from some distribution. This is now done routinely using the PROX algorithm and Marginal Maximum Likelihood Estimation MMLE.
Lalitha Sanathanan in "Some Properties of the Logistic Model for Dichotomous Response" (JASA, 69, 347, 744-749, 1974!) attempts this, but immediately encounters a major hurdle, the multiplicative form of the Rasch model then in common use. After Herculaean effort, she derives a simple approximation, but is forced to present it in additive form in order to make it tractable. Rewritten, the approximation is:
where N is the sample size, Si is the number of correct answers to item i, and Di is the difficulty of item i. α and β are sample dependent, but how they relate to the mean and standard deviation of the distribution is omitted from the paper. In fact, this formulation is equivalent to the PROX equation for a sample distributed N(μ,σ²), when
Sanathanan realizes that she has "shown how the parameters in the model can be calculated in a rough ready manner" (p. 749), but the utility of her insight was lost due to her abstruse math.
Sanathanan's plot of expected score against item difficulty for different sample distributions.
Random effects Rasch model. Sanathanan, L. Rasch Measurement Transactions, 2002, 16:2 p.881
|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/rmt162i.htm