On the boundary problems in diagnostic classification models

In diagnostic classification models, parameter estimation sometimes provides estimates that stick to the boundaries of the parameter space, which is called the boundary problem and may lead to extreme values of standard errors. However, the relationship between the boundary problem and irregular sta...

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Bibliographic Details
Published in:Behaviormetrika 2023, Vol.50 (1), p.399-429
Main Author: Yamaguchi, Kazuhiro
Format: Article
Language:English
Subjects:
Achievement tests
Chemistry and Earth Sciences
Computer Science
Economics
Estimates
Finance
Insurance
Management
Mathematics and Statistics
Original Paper
Parameter estimation
Physics
Probability
Statistical Theory and Methods
Statistics
Statistics for Business
Statistics for Engineering
Students
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Summary:In diagnostic classification models, parameter estimation sometimes provides estimates that stick to the boundaries of the parameter space, which is called the boundary problem and may lead to extreme values of standard errors. However, the relationship between the boundary problem and irregular standard errors has not been analytically explored. In addition, prior research has not shown how maximum-a-posteriori estimates avoid the boundary problem and affect the standard errors of estimates. To analyze these relationships, the expectation–maximization algorithm for maximum-a-posteriori estimates and a complete data Fisher information matrix are explicitly derived for a mixture formulation of saturated diagnostic classification models. Theoretical considerations show that the emptiness of attribute mastery patterns causes both the boundary problem and the inaccurate standard error estimates. Furthermore, unfortunate boundary problem without emptiness causes shorter standard errors. A simulation study shows that the maximum-a-posteriori method prevents boundary problems. Moreover, this method with monotonicity constraint estimation improves standard error estimates more than unconstrained maximum likelihood estimates do.
ISSN:0385-7417
1349-6964
DOI:10.1007/s41237-022-00187-7