The Logit-Linear Model

Statistical texts, which focus on descriptive, variance-explaining, models rather than on linear measurement attributes, view the Rasch model as a "linear logistic" or "logit-linear" models. Nevertheless, those texts can provide insight into technical aspects of Rasch analysis.

Fox (1984) chapter 5 mentions the true-score model (his "linear probability model") only to discard it as statistically untenable. He does not refer explicitly to Rasch, but his discussion of the logit- linear model is reassuring to those familiar with Wright and Stone (1978). His estimation method and fit diagnosis is similar to theirs, though expressed in matrix arithmetic. His discussion of polytomous data parallels Rasch's multi-dimensional approach and Andrich's ordered categories.

Fox adds two aspects not usually mentioned in Rasch texts. He investigates the concept of the influence of an observation. This relates to how much parameter estimates would change were that observation left out. Influence increases as the size of the residual (observed - expected) increases, and also as the number of other observations made under similar circumstances decreases. Examples of influential observations are the one and only correct response given by any examinee to the hardest item of a math test, and also any of the responses made by an examinee who was only administered three items.

Fox also shows that which pieces of the data are modelled as rating scale categories and which as parameters depends on our intention. In data about Canadian women's participation in the labor force, he reports for each woman, her employment (1=full time, 2= part time, 3=not working), whether she has children (1=yes, 0=no), her husband's income (in $000's), and the region where she lives (Atlantic, Quebec, Ontario, Prairie, BC). Here, any of these 4 pieces of information could be the rating scale, while the other three pieces are parameterized as three facets!

Fox J 1984. Linear statistical models and related methods. New York: John Wiley & Sons.

Wright BD, Stone MH 1979. Best Test Design. Chicago: MESA Press.

Logit-linear model. Fox J, Linacre JM. … Rasch Measurement Transactions, 1992, 5:4 p.187

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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