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A Restricted Latent Class Hidden Markov Model for Polytomous Responses, Polytomous Attributes, and Covariates: Identifiability and Application

Eric Alan Wayman, Steven Andrew Culpepper, Jeff Douglas, Jesse Bowers

Abstract

We introduce a restricted latent class exploratory model for longitudinal data with ordinal attributes and respondent-specific covariates. Responses follow a time inhomogeneous hidden Markov model where the probability of a respondent's latent state at the current time point is conditional on the respondent's latent state at the previous time point as well as the respondent's covariates at the current time point. We prove that the model is identifiable, state a Bayesian formulation, and demonstrate its efficacy in a variety of scenarios through two simulation studies. We apply the model to response data from a mathematics examination, comparing the results to a previously published confirmatory analysis, and also apply it to emotional state response data which was measured over a several-day period.

A Restricted Latent Class Hidden Markov Model for Polytomous Responses, Polytomous Attributes, and Covariates: Identifiability and Application

Abstract

We introduce a restricted latent class exploratory model for longitudinal data with ordinal attributes and respondent-specific covariates. Responses follow a time inhomogeneous hidden Markov model where the probability of a respondent's latent state at the current time point is conditional on the respondent's latent state at the previous time point as well as the respondent's covariates at the current time point. We prove that the model is identifiable, state a Bayesian formulation, and demonstrate its efficacy in a variety of scenarios through two simulation studies. We apply the model to response data from a mathematics examination, comparing the results to a previously published confirmatory analysis, and also apply it to emotional state response data which was measured over a several-day period.

Paper Structure

This paper contains 18 sections, 10 theorems, 104 equations, 3 figures, 34 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Measurement Model
  4. Monotonicity Condition
  5. Structural Model
  6. Bayesian Model
  7. Likelihood and Data Augmentation Prior for Observed Data
  8. Priors for Measurement Model Coefficients and Related Parameters
  9. Priors for Structural Model Parameters
  10. Integrability of the Model with Respect to the Variance Parameter
  11. Model Identifiability
  12. Parameter Expansion and Algorithm
  13. Software
  14. Simulation Studies
  15. Applications
  16. ...and 3 more sections

Key Result

Theorem 1

Define the following conditions: In addition, define two conditions on the $\delta$ matrix: If conditions (C1) through (C6) and (D1) hold, then $\{p(Y \mid \theta)\}$ is generically identifiable up to label swapping, and if conditions (C1) through (C6) and (D1) and (D2) hold, then $\{p(Y \mid \theta)\}$ is strictly identifiable up to label swapping.

Figures (3)

  • Figure 1: Simplified version of model in directed graphical model form (for alt text, see Supplementary Material A)
  • Figure 2: Bayesian model in directed graphical model form, part one (for alt text, see Supplementary Material A)
  • Figure 3: Bayesian model in directed graphical model form, part two (for alt text, see Supplementary Material A)

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • ...and 12 more