Computation of OUTFIT and INFIT Statistics

Reproduced from Rating Scale Analysis (Wright BD & Masters GN, Chicago, MESA Press, 1982, p.100) with permission

Page 100 of Rating Scale Analysis (Wright and Masters 1982), which summarizes the calculation of outfit and infit statistics, is displayed in a revised form adjacent.

The Rasch residuals are the differences between the observations and their expected values according to the Rasch model. Outfit is based on a sum of squared standardized residuals. Standardized residuals are modeled to approximate a unit normal distribution. Their sum of squares approximates a χ² distribution. Dividing this sum by its degrees of freedom yields a mean-square value, OUTFIT MEANSQ, with expectation 1.0 and range 0 to infinity. Values larger than 1.0 indicate unmodeled noise. Values are on a ratio scale, so that 1.2 indicates 20% excess noise. Values less than 1.0 indicate overfit of the data to the model, i.e., the observations are too predictable.

A Wilson-Hilferty transformation standardizes the mean-square into its OUTFIT ZSTD value. This approximates a unit-normal distribution corresponding to a t-statistic with infinite degrees of freedom. It is a test of the hypothesis "these data fit the Rasch model (exactly)."

Infit is an information-weighted form of outfit. The weighting reduces the influence of less informative, low variance, off-target responses. It is also computed in INFIT MEANSQ and INFIT ZSTD forms.

This computation is the same for persons and items with appropriate adjustment of subscripts and summations.

Note: Practical considerations in the computation of the ZSTD values.
1. the lower limit of the degrees of freedom of the mean-square statistic is set at 1. q_i is not allowed to be more than √2 for both
2. the contribution to the model variance of the OUTFIT MEANSQ by one observation is not allowed to overwhelm the contributions of the other observations. W²_ni is not allowed to be less than 0.00001

Computation of OUTFIT and INFIT Statistics. Wright BD, Masters GN. … Rasch Measurement Transactions, 1990, 3:4 p.84-5

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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
Rasch Books and Publications: Winsteps and Facets
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	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang	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
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

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