Identifiability and Estimation of Hidden Markov Models for Learning Trajectories in Cognitive Diagnosis

• In Event: Advancing Diagnostic Psychometric Models to Enhance Student Learning: Statistical Theory, Methods, and Applications

Abstract

Hidden Markov models (HMMs) have been applied in various domains, which makes the identifiability issue of HMMs popular among researchers. Classical identifiability conditions shown in previous studies are too strong for practical analysis. In this paper we propose generic identifiability conditions for discrete time HMMs with finite state space. Also, a central concern in education is to diagnose student skill profiles and to track learning. Recent studies in education and psychometrics described how to deploy cognitive diagnosis models (CDMs) to track learning in longitudinal data. Specifically, researchers used CDMs along with a first-order HMM to track changes in attributes related to learning. However, the application of CDMs requires a known Q matrix to describe the underlying structure between latent attributes and items, and the identifiability constraints of the model parameters should also be specified. To date, there is no research that has derived identifiability conditions for applying CDMs in longitudinal studies with HMMs.
Consequently, the contribution of our study is that we consider both conventional and CDM-based HMMs, which we refer to as restricted HMMs, with a time-invariant emission matrix, which governs the probability of observed variables given hidden states. We propose both strict and generic identifiability conditions for conventional HMMs with finite observable and hidden states by generalizing existing research. For restricted HMMs, we extend identifiability conditions for static RLCMs for a single time point by proposing a restricted HMM framework, which allow us to monitor the learning trajectory and uncover the underlying structure between items and skills. We propose strict and generic identifiability conditions for model parameters using both a deterministic inputs, noisy “and” gate model and the general restricted latent class measurement model. Most importantly, the new identifiability conditions contribute to the literature and provide researchers with clear guidelines for designing longitudinal formative assessment systems that aim to diagnose student skills and formulate interventions to accelerate learning. In addition to establishing new identification conditions, we develop new algorithms for inferring the Q matrix from longitudinal data. In particular, we develop fully Bayesian methods for estimating the parameters of the restricted-HMM and we present Monte Carlo simulation results to demonstrate accuracy in parameter estimation. Furthermore, we report results of an application of our methods to an educational pre- and post-test study dedicated to students learning of elementary probability theory.

Authors

• Steven Andrew Culpepper, University of Illinois at Urbana-Champaign
  Non-Presenting Author

• Ying Liu, University of Illinois at Urbana-Champaign
  Presenting Author

• Yuguo Chen, University of Illinois at Urbana-Champaign
  Non-Presenting Author