# Generalizability Theory and Rasch Measurement

 Generalizability Theory Rasch Measurement Purpose: Generalize, from observed raw scores or responses, "universal" raw scores or responses. Reduce unwanted variance in future studies. Construct, from observed responses, linear measures for each facet element free of other facet distributions. Assess quantitative validity of each measure. Inference: Relative decisions, ignoring decision-neutral variance. Absolute decisions, including all variance. Linear measures with standard errors (precision), fit statistics (validity). Frame of reference is criterion by items and normative by persons. Analysis stages: Generalizability: Collection and analysis of data from which to generalize. Decision: G-study results used to evaluate error minimization and resource optimization alternatives in future research. Test conceptualization: Model enables verifying rating plan validity and estimating measurement precision prior to, during and after data collection. Measurement: Construction and statistical validation of measures from data. Data: Raw scores or responses. Raw responses. Context: Universe of admissible observations that test-user accepts as interchangeable. All responses intended to manifest measures on the same variable. Data modelled as: Linear combination of context effects. Ordinal stochastic manifestation of linear parameters. Typical model: Xptr is observed datum. v.. are context effects: p for persons, vt for items, r for raters, etc vptr is residual error. Pptrk probability of observing category k. Bp, Dt, Cr are latent parameters: Bp is person measure, Dt is item difficulty, Cr is rater severity Fk quantifies rating scale non-linearity Measurement target: Object of measurement (e.g., persons) and their universe score variance. Measures for all parameters. Facet: Aspect of situation (item, rater, time, but not object of measurement), represented by its conditions (items, raters, times). Each facet, generalized to the universe from its conditions, is a source of universe score measurement error in the object of measurement. Any component of situation (person, item, rater, time) containing its elements (persons, items, raters, times). Measure, error and fit statistics are estimated for every element of every facet,as well as for every specified element group within each facet. Facet types: Random: Observed conditions are random, exchangeables sampled from the facet universe (e.g., raters represent all raters). Fixed: Observed conditions are the universe (e.g., this test has all items of interest). Measurement: Facet elements interact to cause data (persons, items, raters). Analysis: Measured elements are summarized or decomposed according to other facets (sex, ethnicity, item type). Fixed effect: Facet elements replicate a fixed effect (community attitude to new highway), individual elements are not parameterized. To work successfully: At least one facet of random error. All elements of all facets must be linked in the data or by constraints. Assumptions: Error distributions remain fixed. Items retain difficulty variance. Measurement structure remains fixed. Facets retain construct validity. Valid rating plans: Fully crossed: all conditions of each variance source observed with all conditions of all other sources. Crossed: all conditions of one facet observed with all conditions of another variance source. Nested: two or more conditions of nested variance source appear with only one condition of another facet. Partially nested: some conditions of nested variance source appear with some conditions of another facet. Implicit Links: Data network links all elements of all facets such that all measures can be estimated unambiguously in one frame of reference. Explicit Constraints: Relations between unlinked elements (New York and Oregon) are specified (New York and Oregon tests said to be equally difficult, or New York and Oregon students said to be equally able). Estimation: Analysis of variance. Restricted maximum likelihood. Minimum variance quadratic estimation. Logit-linear maximum likelihood. Least-squares. Estimates: Variance components. Measures, errors, fits. Ideal datum for measurement: Object of measurement's mean score over all acceptable observations. Sufficient responses for each element to estimate its parameter. Ideal test: Item difficulties are all equal. Item difficulties range across person abilities. Effect of widening item difficulty range: More variability. Less generalizability. Reported as item condition variance. More measure range. Slightly less measure precision. Reported as item calibration variance. Ideal raters: Identical rating machines. Self-consistent. Shared understanding of rating scale. Effect of severe rater: More variability. Less generalizability. No effect. Effect of interaction: More variability. Less generalizability. Reported separately or as residual variance. Less fit, reported by fit stats. Motivates: DIF inquiry, test modification, bad data removal, better rater instruction, etc. Effect of random and unidentified variance: More variability. Less generalizability. Reported as residual variance. Reported in measure standard errors. Uneven stochasticity reported by fit stats. Individual improbable responses reported. Departure from ideal quantified by: Dependability of generalizing from observed to universe object of measurement score. Generalizability is overall dependability. Precision measured by Standard Errors. Quantitative validity measured by Fit Stats. Utility measured by facet-element separation. Diagnosis of data-model failure: Large residual error term. Low generalizability. Large misfit or lack of construct validity indicate some data do not support measurement. Analysis results: Variance table, e.g., Shavelson & Webb, p. 102, Table 7.2 Construct map.

John Michael Linacre, AERA, 1993

Generalizability Theory and Rasch Measurement. Linacre J.M. … Rasch Measurement Transactions, 2001, 15:1 p.806-7

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
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

 Forum Rasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.

Coming Rasch-related Events
Aug. 5 - Aug. 6, 2024, Fri.-Fri. 2024 Inaugural Conference of the Society for the Study of Measurement (Berkeley, CA), Call for Proposals
Aug. 9 - Sept. 6, 2024, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 4 - Nov. 8, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 17 - Feb. 21, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 16 - June 20, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 20 - July 18, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Facets), www.statistics.com
Oct. 3 - Nov. 7, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com