Goodness-of-Fit Testing for Latent Class Models

Area of Science:

• Statistics
• Psychometrics
• Data Analysis

Background:

• Latent class models (LCMs) are valuable for analyzing categorical data, but their performance in model comparison and selection can be compromised by sparse contingency tables.
• When data are sparse, the theoretical distributions of goodness-of-fit (GoF) indices are often unknown, leading to inaccuracies in hypothesis testing and model comparisons.
• Existing GoF indices may not accurately reflect model fit under sparsity conditions, necessitating investigation into their distributional properties.

Purpose of the Study:

• To investigate the distributional properties of common goodness-of-fit statistics (G(2), Pearson's χ(2)) and a proposed index by Read and Cressie (1988) under conditions of sparse contingency tables in latent class models.
• To assess the extent of deviations between the empirical distributions of these statistics and their theoretical chi-squared distributions.
• To evaluate the impact of factors like average cell expectation, number of indicator items, and measurement parameter extremity on these deviations.

Main Methods:

• A simulation study was conducted to examine the distributions of the likelihood ratio statistic G(2), Pearson's χ(2), and the Read and Cressie (1988) index.
• The simulation varied conditions related to data sparsity, including average cell expectation, number of indicator items, and latent class measurement parameters.
• Empirical distributions from the simulations were compared to theoretical chi-squared distributions.

Main Results:

• Substantial deviations were observed between the expected chi-squared distribution and the actual means of the G(2) and Read and Cressie distributions.
• The mean of Pearson's χ(2) distribution was generally closer to the chi-squared expectation under conditions of larger average cell expectation, fewer items, and less extreme parameters.
• While Pearson's χ(2) mean was closer to theoretical expectations, its standard deviation was considerably larger than that of G(2) and the Read and Cressie index, and also larger than the chi-squared distribution's standard deviation.

Conclusions:

• Standard goodness-of-fit indices in latent class analysis can be unreliable with sparse data due to unknown or inaccurate distributional properties.
• Reliance on theoretical chi-squared distributions for GoF statistics is problematic under sparsity, leading to potential errors in model assessment.
• Monte Carlo sampling offers a viable solution by generating empirical distributions for GoF statistics, providing more accurate expectations and quantiles for hypothesis testing and model selection.