Random Effects Rasch Model

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, S_i is the number of correct answers to item i, and D_i 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

and

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 Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	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
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	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	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
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
Other 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

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