Rasch Dichotomous Model vs Birnbaum 3-PL Three-Parameter Logistic Model

Headings:Rasch Dichotomous ModelBirnbaum 3-PL Model
Item parameters1 item parameter3 item parameters
A-1 Binomial probability of person n succeeding on dichotomous item i: loge [P/(1-P)] = Bn - Di loge [(P-ci)/(1-P)] = 1.7 ai (t - bi)
A-2 Item characteristic curve (ICC): Monotonic ogive with slope and lower asymptote to be estimated Logistic ogive with specified slope and asymptotes
A-3 Person ability: Bn measures person n B-distribution estimated from data Subjects assumed to be sampled from N(0,1) or other arbitrary distribution
A-4 Item difficulty: Di calibrates item i bi estimates ICC inflection point
A-5 Item discrimination: Specified at constant or unity. Misfit detects variation ai estimates ICC slope at bi. Sample dependent
A-6 Guessing success on item by low ability persons: Preset at constant or zero. Person fit detects lucky guessers ci estimates ICC lower asymptote. Sample dependent
B-1 Motivation: Measurement construction Data description
B-2 Ruled by: Theory and intention Data and chance
B-3 Substance of latent variable: Definitive. Items uniquely ordered Ambiguous. Item order varies with ability level because ai and ci variation causes ICCs to cross
B-4 Unidimensionality: Specified by model Assumed
B-5 Local independence: Verified by fit analysis Not evaluated
B-6 Sufficient statistics: Unweighted raw scores Weighted raw scores if and only if weights known a priori
B-7 Unit of calibration: Log-odds unit (logit) Normit-scaled logits (logit/1.7)
C-1 Estimation Raw scores are sufficient. No arbitrary constraints needed No sufficient statistics. Arbitrary constraints required to control parameter interactions
C-2 Standard errors: Well defined Skewed by arbitrary constraints
C-3 Fit statistics: Based on asymptotic distributions of responses Clouded by parameter interactions
C-4 Gross misfit between model and data: Fit statistics identify invalid data and guide diagnosis and remediation Hidden by over-parameterization and arbitrary constraints required for estimation
C-5 Person diagnosis and quality control: Guided by individual person fits and specific item response residuals Person estimates defined to be random events!?
C-6 Item diagnosis and quality control: Guided by individual item fits and specific person response residuals Hidden by over-parameterization and arbitrary constraints required for estimation
C-7 Random guessing: Response set: Scanning error: Identified by misfit which cues remediation or elimination of error Increases ci and decreases ai of whatever item encounters unexpected successes
C-8 Item miskeying: Many correct options: No correct options: Identified by misfits which cue remediation or elimination of errors Decreases unestimated upper asymptote. Decreases ai
C-9 Duplicate test item: Detected by model overfit Increases ai. Seems to improve test!?
C-10 Item bias: Different item function: Size and significance estimated from person group residuals Not detectable. Requires additional analysis
D-1 Missing data: No problem Biases estimates
D-2 Minimum useful data: 4 items by 10 persons Said to be at least 1000 persons
D-3 Typical stable data: 20 items by 200 persons Does not exist
D-4 Common-item equating: Item Banking: Computer-adaptive tests: Straightforward Impossible, unless bi assumed to dominate (i.e. Rasch model approximated)
D-5 Common-person equating: Straightforward Only if person distributions match
D-6 Weighting to combine items of differing discriminations Model holds when weights decided rather than estimated. Weight validity assessed by fit When ai pre-set, then approximates weighted Rasch analysis
D-7 Polytomous data: Solved by Rating Scale model Not addressed
D-8 Judge intermediation: Solved by Facets model Not addressed

Rasch Dichotomous Model vs Birnbaum 3-PL Three-Parameter Logistic Model. Wright BD. … Rasch Measurement Transactions, 1992, 5:4 p.178




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

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