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Bayesian Inference for an Unknown Number of Attributes in Restricted Latent Class Models

Published online by Cambridge University Press:  01 January 2025

Yinghan Chen Open the ORCID record for Yinghan Chen [Opens in a new window] ,
Steven Andrew Culpepper Open the ORCID record for Steven Andrew Culpepper [Opens in a new window]  and
Yuguo Chen Open the ORCID record for Yuguo Chen [Opens in a new window]
Yinghan Chen*
Affiliation:
University of Nevada, Reno
Steven Andrew Culpepper
Affiliation:
University of Illinois at Urbana-Champaign
Yuguo Chen
Affiliation:
University of Illinois at Urbana-Champaign
*
Correspondence should bemade to Yinghan Chen, Department of Mathematics and Statistics, University of Nevada,Reno, 1664 North Virginia Street, Reno, NV, 89557, USA. Email: [email protected]
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Abstract

The specification of the Q\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\varvec{Q}}$$\end{document} matrix in cognitive diagnosis models is important for correct classification of attribute profiles. Researchers have proposed many methods for estimation and validation of the data-driven Q\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\varvec{Q}}$$\end{document} matrices. However, inference of the number of attributes in the general restricted latent class model remains an open question. We propose a Bayesian framework for general restricted latent class models and use the spike-and-slab prior to avoid the computation issues caused by the varying dimensions of model parameters associated with the number of attributes, K. We develop an efficient Metropolis-within-Gibbs algorithm to estimate K and the corresponding Q\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\varvec{Q}}$$\end{document} matrix simultaneously. The proposed algorithm uses the stick-breaking construction to mimic an Indian buffet process and employs a novel Metropolis–Hastings transition step to encourage exploring the sample space associated with different values of K. We evaluate the performance of the proposed method through a simulation study under different model specifications and apply the method to a real data set related to a fluid intelligence matrix reasoning test.

Keywords

Bayesian analysiscognitive diagnosis modelIndian buffet processlatent class modelspike-and-slab prior
Type
Theory and Methods
Information
Psychometrika , Volume 88 , Issue 2 , June 2023 , pp. 613 - 635
Copyright
Copyright © 2023 The Author(s) under exclusive licence to The Psychometric Society

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