| 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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