Fit Indexes, Lagrange Multipliers, Constraint Changes and Incomplete Data in Structural Models

Area of Science:

• Statistics
• Psychometrics
• Structural Equation Modeling

Background:

• Existing model modification and evaluation methods, including those by Kaplan (1990), are foundational.
• Bentler-Bonett indices are recognized for their utility in assessing model fit.
• Assessing model misspecification is crucial for reliable statistical inference.

Purpose of the Study:

• To discuss and expand upon existing viewpoints on model modification and evaluation.
• To reiterate the usefulness of specific fit indices like the Bentler-Bonett indices.
• To extend incomplete data methods to a wider range of statistical applications.

Main Methods:

• Utilizing the noncentrality parameter of a chi-squared distribution to measure misspecification.
• Employing the comparative fit index as a general index of model adequacy.
• Discussing the Lagrange Multiplier chi-squared statistic's dependence on parameters and constraints.
• Developing a sensitivity theorem for model modification in structural models.
• Extending recent incomplete data methods.

Main Results:

• The comparative fit index offers a useful measure of model adequacy, independent of misspecification source identification.
• The Lagrange Multiplier statistic's behavior is clarified concerning parameter estimation and constraints.
• A sensitivity theorem is presented to guide modifications in structural models.
• Incomplete data methods are made more versatile.

Conclusions:

• The study provides refined methods for evaluating and modifying statistical models.
• Comparative fit indices and sensitivity theorems enhance the understanding of model adequacy and modification.
• Extended incomplete data methods increase the applicability of advanced statistical techniques.