# 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 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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