Headings: | Rasch Dichotomous Model | Birnbaum 3-PL Model |
---|---|---|
Item parameters | 1 item parameter | 3 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
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