Normal Versus Noncentral Chi-square Asymptotics of Misspecified Models

Area of Science:

• Statistics
• Psychometrics
• Quantitative Psychology

Background:

• The noncentral chi-square distribution is a standard approximation for likelihood ratio (LR) test statistics in structural equation modeling (SEM).
• Recent discussions question its universal applicability, suggesting normal distributions might be superior in specific contexts.
• Evaluating these distributional approximations is crucial for accurate SEM analysis.

Purpose of the Study:

• To assess the practical validity of using noncentral chi-square versus normal distributions for approximating LR test statistics in SEM.
• To determine the performance of these approximations under varying degrees of model misspecification and sample sizes.

Main Methods:

• Monte Carlo simulations were employed to generate data and evaluate the fit of different distributional approximations.
• The study examined the behavior of the LR test statistic under conditions of small, moderate, and severe model misspecification.
• The Thurstone data from a classic mental ability study was used for illustration.

Main Results:

• The noncentral chi-square distribution demonstrated robust performance, accurately describing LR test statistic behavior across a range of misspecifications and sufficient sample sizes.
• A bias-corrected normal distribution provided a marginally better approximation only under extremely severe misspecifications.
• Neither approximation offered a reasonable fit for the LR test statistic under extremely severe misspecifications, which are of limited practical relevance.

Conclusions:

• The noncentral chi-square distribution remains a reliable approximation for LR test statistics in SEM, even with moderate to severe model misspecification.
• While normal distributions offer a slight advantage in extreme cases, their practical utility is limited.
• Researchers should be cautious with extremely misspecified models, as standard approximations may fail.