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Goodness-of-Fit Tests for Latent Class Models with Ordinal Categorical Data

Huan Qing

Abstract

Ordinal categorical data are widely collected in psychology, education, and other social sciences, appearing commonly in questionnaires, assessments, and surveys. Latent class models provide a flexible framework for uncovering unobserved heterogeneity by grouping individuals into homogeneous classes based on their response patterns. A fundamental challenge in applying these models is determining the number of latent classes, which is unknown and must be inferred from data. In this paper, we propose one test statistic for this problem. The test statistic centers the largest singular value of a normalized residual matrix by a simple sample-size adjustment. Under the null hypothesis that the candidate number of latent classes is correct, its upper bound converges to zero in probability. Under an under-fitted alternative, the statistic itself exceeds a fixed positive constant with probability approaching one. This sharp dichotomous behavior of the test statistic yields two sequential testing algorithms that consistently estimate the true number of latent classes. Extensive experimental studies confirm the theoretical findings and demonstrate their accuracy and reliability in determining the number of latent classes.

Goodness-of-Fit Tests for Latent Class Models with Ordinal Categorical Data

Abstract

Ordinal categorical data are widely collected in psychology, education, and other social sciences, appearing commonly in questionnaires, assessments, and surveys. Latent class models provide a flexible framework for uncovering unobserved heterogeneity by grouping individuals into homogeneous classes based on their response patterns. A fundamental challenge in applying these models is determining the number of latent classes, which is unknown and must be inferred from data. In this paper, we propose one test statistic for this problem. The test statistic centers the largest singular value of a normalized residual matrix by a simple sample-size adjustment. Under the null hypothesis that the candidate number of latent classes is correct, its upper bound converges to zero in probability. Under an under-fitted alternative, the statistic itself exceeds a fixed positive constant with probability approaching one. This sharp dichotomous behavior of the test statistic yields two sequential testing algorithms that consistently estimate the true number of latent classes. Extensive experimental studies confirm the theoretical findings and demonstrate their accuracy and reliability in determining the number of latent classes.
Paper Structure (48 sections, 16 theorems, 277 equations, 3 figures, 4 tables, 3 algorithms)

This paper contains 48 sections, 16 theorems, 277 equations, 3 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Model and problem
  3. Latent class model
  4. Problem statement
  5. Test statistic
  6. Ideal normalized residual matrix
  7. Practical test statistic
  8. Algorithms
  9. GoF-LCM algorithm
  10. RGoF-LCM algorithm
  11. Numerical Studies
  12. General simulation setup
  13. Class membership matrix $Z$
  14. Item parameter matrix $\Theta$
  15. Response matrix $R$
  16. ...and 33 more sections

Key Result

Lemma 1

When Assumption ass:A1 holds, for any $\epsilon > 0$, we have

Figures (3)

  • Figure 1: Accuracy (left) and running time (right) of GoF-LCM and RGoF-LCM for $K=8$, $J=60$, $\delta=0.3$, with varying $N$.
  • Figure 2: Accuracy of GoF‑LCM and RGoF‑LCM $K=8$, $N=600$, $\delta=0.3$, with varying $J$.
  • Figure 3: Test statistic $T_{K_0}$ (left) and ratio $r_{K_0}=|T(K_0-1)/T(K_0)|$ (right) versus candidate number of latent classes $K_0$ for the BFPT dataset.

Theorems & Definitions (36)

  • Definition 1: Latent class model for ordinal categorical data
  • Lemma 1: Spectral norm of ideal residual matrix
  • Remark 1
  • Lemma 2: Perturbation control of normalized residual matrix
  • Theorem 1: Null behavior of test statistic
  • Theorem 2: Alternative behavior of the test statistic
  • Theorem 3: Consistency of GoF–LCM
  • Remark 2: Choice of $\tau_N$
  • Theorem 4: Asymptotic behaviour of the ratio statistic
  • Theorem 5: Consistency of RGoF-LCM
  • ...and 26 more