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

Please help with Standard Dataset 4: Andrich Rating Scale Model

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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