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 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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