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

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.

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.

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

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

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