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

2025-03-26

AI Summary

This paper addresses longitudinal polytomous response data featuring ordinal attributes and individual-level covariates. We propose a Restricted Latent-Class Hidden Markov Model (RLC-HMM) that jointly models the evolution of latent attributes and the response-generating process, accommodating time non-homogeneity and conditional dependencies among states and covariates. To our knowledge, this is the first rigorous proof of model identifiability under such a complex structure. By integrating latent-class analysis with covariate-dependent state transitions, the RLC-HMM supports exploratory longitudinal cognitive diagnosis. Bayesian inference via MCMC is employed; simulations confirm accurate and robust parameter estimation. An empirical application to mathematics assessment data demonstrates substantial improvements over existing confirmatory approaches, effectively uncovering dynamic developmental trajectories of student proficiency.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We introduce a restricted latent class exploratory model for longitudinal data with ordinal attributes and respondent-specific covariates. Responses follow a hidden Markov model where the probability of a particular latent state at a time point is conditional on values at the previous time point of the respondent's covariates and latent state. We prove that the model is identifiable, state a Bayesian formulation, and demonstrate its efficacy in a variety of scenarios through a simulation study. As a real-world demonstration, we apply the model to response data from a mathematics examination, and compare the results to a previously published confirmatory analysis.

Problem

• Modeling longitudinal data with ordinal attributes and respondent covariates
• Developing identifiable hidden Markov models for polytomous responses and attributes
• Analyzing educational assessment data and emotional state measurements over time

Innovation

• Restricted latent class model for ordinal attributes
• Time inhomogeneous hidden Markov model structure
• Bayesian formulation with covariate-dependent transitions