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Page 1: FRANCESCO BARTOLUCCI

Dimensionality of the latent structure and item selection via

latent class multidimensional IRT models

FRANCESCO BARTOLUCCI

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Outline

• Introduction• Data set• Statistic Methodology• Strategy of Analysis• Application to the Dataset• Conclusion

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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.

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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

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Model

• Latent class

• Multidimensional 2PL

– Constraint:

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Latent class model

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Model

• log-likelihood

• number of free parameter– LC:

– 2PL:

– Difference between them:

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Estimate

• Expectation-Maximization (EM)• E-step: conditional expected value

• M-step: maximizing the log-likelihood where is replaced.

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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

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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

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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.

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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

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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.

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Step 4. Reduction of the Number of Items

• Standardized ability: • Transform the items parameter:

• normalized Garma:

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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.

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Application to the ULISSE Dataset

• Selection of the Number of Latent Classes

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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.

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Application to the ULISSE Dataset

• Assessment of the Number of Dimensions

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Application to the ULISSE Dataset

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Application to the ULISSE Dataset• The initial number of dimension (8) may be

excessive.

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Application to the ULISSE Dataset

• The latent classes can be interpretation of different degrees of impairment of health status.

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Application to the ULISSE Dataset

• investigate the stability of 5 dimension, compare between s=5 and s=4 model.– Cross –validated log-liklihood.

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Cross-validated log-liklihood

• 2 Randomly chosen partitions of equal size– Training data– Test data

• Training: s=4 Test: s=4Training: s=5 Test: s=5

• BIC: s=5 is a proper solution

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Reduction of the Number of Items

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Reduction of the Number of Items

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Reduction of the Number of Items

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Conclusion

• More general structure• Missing responses• Polytomous items

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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.


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