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Bayesian inference for dynamic Q matrices and attribute trajectories in hidden Markov diagnostic classification

Chen-Wei Liu1

  • 1Department of Educational Psychology and Counseling, National Taiwan Normal University, Taipei, Taiwan.

The British Journal of Mathematical and Statistical Psychology
|January 21, 2026
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Summary
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This study presents a Bayesian algorithm for Hidden Markov Diagnostic Classification Models, enabling tracking of student learning over time. The new method accurately estimates key parameters for longitudinal educational assessments.

Keywords:
Q matrixdiagnostic classification modelshidden Markov models

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Area of Science:

  • Educational Measurement and Psychometrics
  • Cognitive Science
  • Statistical Modeling

Background:

  • Traditional diagnostic classification models (DCMs) offer insights into student knowledge but often lack longitudinal tracking capabilities.
  • Understanding the temporal dynamics of cognitive attributes is crucial for effective, adaptive instruction and assessment.
  • Existing methods may not adequately capture the evolving nature of student mastery over multiple assessment occasions.

Purpose of the Study:

  • To introduce a novel Bayesian Markov chain Monte Carlo (MCMC) algorithm for diagnostic classification models (DCMs).
  • To enable joint estimation of time-varying parameters, including Q-matrices, latent attributes, item parameters, and transition matrices.
  • To provide a robust computational tool for longitudinal diagnostic classification.

Main Methods:

  • Development of a Bayesian MCMC algorithm tailored for time-varying DCMs.
  • Implementation of the algorithm within a new R package, 'hmdcm'.
  • Validation through Monte Carlo simulations for parameter recovery and an empirical assessment for trajectory tracing.

Main Results:

  • Monte Carlo simulations demonstrated accurate recovery of model parameters under the proposed algorithm.
  • The algorithm successfully traced attribute trajectories in an empirical probability-concept assessment.
  • The 'hmdcm' R package provides a functional tool for implementing the developed methodology.

Conclusions:

  • The proposed Bayesian MCMC algorithm offers a powerful approach for longitudinal diagnostic classification.
  • Accurate parameter estimation and attribute trajectory tracing support its utility in research and educational practice.
  • This methodology enhances the ability to model and understand cognitive attribute evolution over time.