Equivalence Set Restricted Latent Class Models (ESRLCM)

TL;DR

This work presents ESRLCM, a Bayesian latent-class approach that identifies clusters sharing item response probabilities through equivalence-set restrictions, extending beyond traditional RLAMs. It introduces a base-class matrix $B$ and a repelled Beta prior to promote well-separated class probabilities, supported by identifiability theory and a tailored MCMC sampler. Through simulations and real-data applications (e.g., Progressive Matrices and fraction subtraction tasks), ESRLCM demonstrates superior recovery of restriction structures and predictive performance relative to competitors. The framework yields interpretable, flexible clustering of respondents across $J$ items and $C$ classes, with practical implications for educational and psychological assessment.

Abstract

Latent Class Models (LCMs) are used to cluster multivariate categorical data, commonly used to interpret survey responses. We propose a novel Bayesian model called the Equivalence Set Restricted Latent Class Model (ESRLCM). This model identifies clusters who have common item response probabilities, and does so more generically than traditional restricted latent attribute models. We verify the identifiability of ESRLCMs, and demonstrate the effectiveness in both simulations and real-world applications.

Figures (2)

• Figure 1: Density of repelled beta distribution. Each row of matrix $\boldsymbol{\alpha}$ corresponds to a different component of $\underline{\boldsymbol{{\rho}}}$.
• Figure 2: Dependencies between parameters in equivalence set RLCMs.

Theorems & Definitions (63)

• Definition 1: Equivalence set restrictions
• Definition 2
• Definition 3
• Theorem 1
• example 1
• Theorem 2: Generic Identifiability of Equivalence Set RLCM
• proof
• example 2
• Corollary 1
• Definition 4