1. Dimensionality of the latent structure and item selection via latent class multidimensional IRT models FRANCESCO BARTOLUCCI

2. Outline Introduction Data set Statistic Methodology Strategy of Analysis Application to the Dataset Conclusion

3. Introduction Dimensionality issue of health conditions: Subjects show a degenerative health status to a specific pathology, but have overall good health status. Assume the population is divided into a certain number of latent classes. Address the issue of item selection.

4. The ULISSE Dataset A network on health care services for older people Longitudinal survey Filled out by the nursing assistant Since 2004 through the repeated administration every 6months 79 items – 1: presence of a specific health problem – 0: its absence

5. Model Latent class Multidimensional 2PL – Constraint:

6. Latent class model

7. Model log-likelihood number of free parameter – LC: – 2PL: – Difference between them:

8. Estimate Expectation-Maximization (EM) E-step: conditional expected value M-step: maximizing the log-likelihood where is replaced.

9. Strategy of Analysis Selection of the number of latent class Validation of the multidimensional 2PL model Assessment of the number of dimensions Reduction of the number of items

10. Step 1. Selection of the number of latent classes BIC: – LC or 2PL #par: penalization term – Number of classes increasing, #par rising AIC tends to overestimate the number of classes

11. Step 2. Validation of the Multidimensional 2PL model Compare the LC and 2PL model by BIC. For validate the structure of the questionnaire. LC, which is completely unconstrained, allows each item to measure a separate dimension. If 2PL proves preferable in BIC, the evidence of item structure is found.

12. Step 3. Assessment of the Number of Dimensions Chi-square, df=k-2 – is the probability under the s dimensions – is the probability under the s-1 dimensions When sample size is large, the criterion is too severe, it may lead to overestimating the number of dimensions. Adopt BIC

13. Step 4. Reduction of the Number of Items Discrimination index between 0 and 1 (constraint) Minimum number of items is 5 retained for each dimension. However, indices are not comparable across dimensions, so latent trait standardized for each dimension is required.

14. Step 4. Reduction of the Number of Items Standardized ability: Transform the items parameter: normalized Garma:

15. Step 4. Reduction of the Number of Items Item reduction changes the classification of the subjects. – Posterior on the full set items, and then on the subset. – Use the same parameter obtained with the initial set.

16. Application to the ULISSE Dataset Selection of the Number of Latent Classes

17. Application to the ULISSE Dataset Validation of the 2PL Model – 2PL: BIC=68,653.32 < – LC: BIC=69845.39 2PL proves preferable The structure is also validated, and the assumption (each section measures each dimension) is supported.

18. Application to the ULISSE Dataset Assessment of the Number of Dimensions

19. Application to the ULISSE Dataset

20. The initial number of dimension (8) may be excessive.

21. Application to the ULISSE Dataset The latent classes can be interpretation of different degrees of impairment of health status.

22. Application to the ULISSE Dataset investigate the stability of 5 dimension, compare between s=5 and s=4 model. – Cross –validated log-liklihood.

23. Cross-validated log-liklihood 2 Randomly chosen partitions of equal size – Training data – Test data Training: s=4  Test: s=4 Training: s=5  Test: s=5 BIC: s=5 is a proper solution

24. Reduction of the Number of Items

27. Conclusion More general structure Missing responses Polytomous items

28. Question why not studying the number of latent classes and dimensionality simultaneously? MNSQ item-fit statistic used to reducing items could be tried in this process. Simulation studies should be conducted to confirm its efficiency and accuracy of the proposed approach.