On the likelihood ratio test in structural equation modeling when parameters are subject to boundary constraints

Area of Science:

• Statistics
• Psychometrics
• Econometrics

Background:

• Inequality constraints are often implicitly used in structural equation models (SEMs).
• Common SEMs like factor, autoregressive, and latent growth models utilize these constraints.
• The impact of these boundary constraints on statistical test distributions is frequently overlooked.

Purpose of the Study:

• To investigate how inequality constraints affect the likelihood ratio test (LRT) distribution in SEMs.
• To identify the correct asymptotic distribution for LRTs when null hypotheses involve boundary constraints.
• To provide practical guidance for applying the correct distribution to improve statistical power.

Main Methods:

• Theoretical analysis of the likelihood ratio test under inequality constraints.
• Derivation of the correct asymptotic distribution for specific SEMs (common factor, autoregressive, latent growth).
• Illustrative examples demonstrating the application and benefits of the derived distribution.

Main Results:

• The asymptotic distribution of the chi-square difference test is not a standard central chi-square distribution when inequality constraints are present.
• The correct distribution depends on the specific model and the number of boundary constraints.
• The derived distributions for common factor, autoregressive, and latent growth models are presented.

Conclusions:

• Standard LRT assumptions are violated when parameters are constrained to the boundary.
• Employing the correct, non-standard distribution is crucial for accurate hypothesis testing in these SEMs.
• Using the appropriate distribution can lead to significantly increased statistical power.