Identifiability and estimation for restricted latent class models and hidden Markov models

Restricted latent class models (RLCMs) and hidden Markov models (HMMs) are widely applied in psy- chological and educational researches, of which the identifiability conditions are difficult to establish. Also, model parameters are restricted via latent structures based on identifiability conditions. This thesis devel- ops identifiability conditions of RLCMs and HMMs and proposes Bayesian framework to estimate model parameters. Diagnostic classification models (DCMs) are widely used for providing fine-grained classification of a multidimensional collection of discrete attributes. The application of DCMs requires the specification of the latent structure in what is known as the Q matrix. Expert-specified Q matrices might be biased and result in incorrect diagnostic classifications, so a critical issue is developing methods to estimate Q in order to infer the relationship between latent attributes and items. Existing exploratory methods for estimating Q must pre-specify the number of attributes, K. In this chapter, a Bayesian framework is shown to jointly infer the number of attributes K and the elements of Q, and a crimp sampling algorithm is proposed to transit between different dimensions of K and estimate the underlying Q and model parameters while enforcing model identifiability constraints. My contribution for this project focuses on adapting the Indian buffet process and reversible-jump Markov chain Monte Carlo methods to estimate Q. We conduct Monte Carlo simulation and apply the developed methodology to two datasets. Also, the hidden Markov model (HMM) have been applied in various of domains, which makes the identifiability issue of HMMs become popular among researchers. Classical identifiability conditions shown in previous studies are too strong for practical analysis, in this chapter we propose generic identifiability conditions for discrete time HMMs with finite state space. Also, recent studies about cognitive diagnosis models (CDMs) applied first-order hidden Markov model (HMM) to track changes in attributes. However, the application of CDMs requires a known Q matrix to infer the underlying structure between latent attributes and items, and the identifiability constraints of the model parameters should also be specified. We propose generic identifiability constraints for our restricted hidden Markov model and then estimate the model parameters including the Q matrix through a Bayesian framework. We present Monte Carlo simulation results to support our conclusion and apply the developed model to a real dataset. The last project is related to the analysis of multivariate responses in educational measurement. Restricted latent class models (RLCMs) provide an important framework for diagnosing and classifying respondents on a collection of multivariate binary responses. Recent research made significant advances in theory for establishing identifiability conditions for RLCMs with binary and polytomous response data. Multiclass data, which are unordered nominal response data, are also widely collected in the social sciences and psychometrics via forced-choice inventories and multiple choice tests. We establish new identifiability conditions for parameters of RLCMs for multiclass data and discuss the implications for substantive applications. The new identifiability conditions are applicable to a wealth of RLCMs for polytomous and nominal response data. We propose a Bayesian framework for inferring model parameters, assess parameter recovery in a Monte Carlo simulation study, and present an application of the model to a real dataset.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Statistics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2024

Author dc:creator

• Liu, Ying

Contributors dc:contributor

• Chen, Yuguo
• Culpepper, Steven Andrew
• Douglas, Jeffrey A
• Zhang, Susu

• Identifiability
• Dina
• Hidden Markov Model
• Restricted Latent Class Model

Statement dc:rights

• Copyright 2024 Ying Liu

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/125512

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F125512