1. How Should We Assess the Fit of Rasch-Type Models? Approximating the Power of Goodness-of-fit Statistics in Categorical Data Analysis Alberto Maydeu-Olivares Rosa Montano

2. Outline Introduction Rasch-Type Models for Binary Data Rationale of Goodness-of-Fit Statistics ◦ Full Picture ◦ M2, R1 and R2 Estimating the Power Empirical Comparison of R1, R2 and M2 Numerical Examples Discussion and Conclusion

3. Introduction Two properties of Rasch-Type models ◦ Sufficient statistics ◦ Specific objectivity Estimation methods ◦ Specific for Rasch-Type models (CML) ◦ General procedures (MML via EM) Goodness-of-fit testing procedures ◦ Specific to Rasch-Type models ◦ General to IRT or multivariate discrete data models

4. Introduction Compare the performance of certain goodness- of-fit statistics to test Rasch-Type models in MML via EM ◦ Binary data ◦ 1PL (random effects) R1 and R2 for 1PL M2 for multivariate discrete data

5. Rasch model and 1PL Fixed effects ◦ The distribution of ability is not specified Random effects ◦ Specify a standard normal distribution for ability ◦ The less restrictive definition of specific objectivity still hold

6. Rationale (000)(100)(010)(001)(110)(101)(011)(111) 101000000 200000100 300001000 Marginal Total for each cell > 5 1. High-dimensional contingency table C = 2^n cells which n is the number of items. For example, 20 items test C = 2^20 = 1048576 cells To fulfill the rule of thumb >5, at least 1048576*5 sample size is needed.

7. (000)(100)(010)(001)(110)(101)(011)(111) 101000000 200000100 … Marginal Total101721321581219134 Observed proportion 0.07 Probability Under Model 0.11 2.

8. When order r = 2, Mr -> M2 M2 used the univariate and bivariate information The degree of freedom is It is statistics of choice for testing IRT models 3. Limited information approach (M2) Pooling cells of the contingency table

9. Degree of freedom is n(n-2) Specific to the monotone increasing and parallel item response functions assumptions 3. Limited information approach (R1 and R2) Degree of freedom is (n(n-2)+2)/2 Specific to the unidimensionality assumption

10. Estimating the Asymptotic Power Rate Under the sequence of local alternatives ◦ The noncentrality parameter of a chi-square distribution can be calculated given the df for M2, R1 and R2 The Kullback-Leibler discrepancy function can be used ◦ The minimizer of DKL is the same as the maximizer of the maximum likelihood function between a “true” model and a null model

11. Study 1: Accuracy of p-values under correct model df = Mean; df = ½ Var Another Study by Montano (2009), M2 is better than R1 and the discrepancies between the empirical and asymptotic rate were not large. Group the sum scores ->

12. The degree of freedom is also adjust An iterative procedure When appropriate score ranges are used, the empirical rejection rate of R1 should be closely match the theoretical rejection rates. This should be also done in R2

13. Study 2: Asymptotic Power to reject a 2PL

14. Study 3: Empirical Power to reject a 2PL

15. Study 4: Asymptotic Power to reject a 3PL

16. Study 5: Asymptotic Power to reject a multidimensional model

17. Empirical Example 1: LSAT 7 Data The agreement in ordering between value/df ratio and power

18. Empirical Example 2: Chilean Mathematical Proficiency Data

19. Discussion and Conclusions Generally, M2 is more powerful than R1, R2. That is, the R1 and R2 which developed specific to Rasch-type models is not superior than the general M2