1. Estimation

2. The Model Probability

3. The Model for N Items — 1 The vector probability takes this form if we assume independence

4. The Model for N Items — 2

5. Probabilities of scoring 2 for different response patterns Probability Mode is ability that makes this pattern most likely Likelihood principle: Which ability maximises the probability of what was obtained? All modes at same location

6. Graphical Display of the likelihoods for a five item test Probability All items with difficulty parameter zero

7. Estimation Methods Approximate Methods – PROX – UFORM Pair-wise Minimum Chi-square Maximum likelihood

8. Maximum Likelihood Methods–1 Joint maximum likelihood – also called unconditional maximum likelihood (UCON) – method used in Quest, WinSteps, Facets, ConQuest, TAM – Ability and difficulty estimates

9. Maximum Likelihood Methods–2 Marginal Maximum Likelihood – really a new model that invokes a population distribution assumption – Works well with more general models – ConQuest, TAM, default method

10. Joint Maximum Likelihood

11. Maximising the Joint Likelihood — 1

12. Maximising the Joint Likelihood — 2

13. Maximising the Joint Likelihood — 3 A total of N+I equations to be solved

14. At the solution — 1 First derivatives are zero (called scores) What the student scored What the model predicts It happens to be the expected score

15. We only need the marginals! Many sets of patterns will satisfy a given set of marginals Estimates, errors, reliability do not depend on the patterns Parameter estimates do not depend upon fit Implications for the order debate We do not need to use this

16. Further Implications – Student and item scores are sufficient statistics for Rasch estimation. – Students with the same score will have the same ability estimate. – One-to-one match between raw score and Rasch ability estimate (when no missing data). – Use of score equivalence table. – So why (and when) do we need Rasch scores?

17. At the solution: What must the distribution of estimates look like? First derivatives are zero (called scores)

18. The Resulting Ability Distribution Score 0 Score 1 Score 2 Score 3 Score 4 Score 5 Score 6 Proficiency on Logit Scale