Advances in Polytomous Log-Linear Rasch Models

The ever-increasing complexity of Rasch datasets, combined with a desire to use standard statistical software for estimation has motivated advances in log-linear Rasch models. The relationship between log-linear Rasch models and logit-linear (or exponential) Rasch models is shown at www.rasch.org/rmt/rmt113r.htm

Log-linear models simplify estimation by eliminating nuisance parameters (usually those of the subjects), but often add complexity through the need for design matrices. They are also awkward to implement when data-points are missing.

Hatzinger & Katzenbeisser (2008) derive dichotomous and partial-credit log-linear Rasch models incorporating multiple time-points. This can be estimated using conditional maximum-likelihood estimation (CMLE) with standard statistical software, such as R. It models dependency across time-points, and also allows different subjects to be observed at different numbers of time-points. The example dataset has dichotomous data, 3 items, 45 subjects observed at up to 11 time-points.

The estimation of the parameters of multidimensional polytomous Rasch models presents an even greater technical challenge. Anderson et al. (2007) achieve it by formulating the Rasch model as a log-linear-by-association (LLLA) model. The multidimensional structure renders conventional CMLE impossible in general, so a pseudo-likelihood technique is employed. The overall likelihood of the data is decomposed into a set of parallel regression models which are maximized simultaneously. This is implemented in the plRasch package for R, and the SAS plgRasch macro. Example datasets have up to 30 items, 1000 subjects, 3 response categories and 2 dimensions.

Anderson, C.J., Li, Z., & Vermunt, J.K. (2007). Estimation of models in the Rasch family for polytomous items and multiple latent variables. Journal of Statistical Software, 20:4. www.jstatsoft.org/v20/i06

Hatzinger R. & Katzenbeisser W. (2008). Log-linear Rasch-type Models for Repeated Categorical Data with a Psychobiological Application. Department of Statistics and Mathematics, Wirtschaftsuniversitaet Wien. Research Report 69.

http://epub.wu-wien.ac.at/dyn/virlib/wp/eng/showentry?ID=epub-wu-01_dab

Advances in Polytomous Log-Linear Rasch Models … Rasch Measurement Transactions, 2008, 22:2 p. 1167

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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