Assessing the Size of Model Misfit in Structural Equation Models

Area of Science:

• Statistics
• Quantitative Psychology
• Econometrics

Background:

• Statistical significance of mean differences is often supplemented with effect sizes (e.g., Cohen's d).
• In structural equation modeling (SEM), significant model fit tests are frequently overlooked, with inferences based on "close" models derived from goodness-of-fit indices.
• Inferences from SEM are unreliable if the model cannot be retained via an exact fit test, necessitating measurement of model misfit size.

Purpose of the Study:

• To introduce and provide statistical theory for standardized effect sizes of model misfit in SEM.
• To enable the construction of confidence intervals and tests of close fit using these effect sizes.
• To reconcile SEM practices with broader statistical conventions regarding effect size interpretation.

Main Methods:

• Utilizing standardized residual covariances and correlations as effect sizes for SEM misfit.
• Summarizing these residuals into overall effect sizes: Standardized Root Mean Squared Residual (SRMSR) and Correlation Root Mean Squared Residual (CRMSR).
• Developing statistical theory for confidence intervals and close fit tests based on SRMSR and CRMSR.

Main Results:

• Standardized residual covariances and correlations quantify the magnitude of SEM model misfit.
• SRMSR and CRMSR serve as comprehensive effect sizes for overall model misfit.
• The proposed statistical theory facilitates robust confidence intervals and close fit testing for SEM.

Conclusions:

• Standardized effect sizes (SRMSR, CRMSR) offer a crucial measure for SEM model misfit.
• Implementing these effect sizes can enhance the interpretability and reliability of SEM findings.
• This approach bridges the gap between SEM practices and established statistical principles for effect size interpretation.