When fit indices and residuals are incompatible

Area of Science:

• Statistics
• Psychometrics
• Quantitative Psychology

Background:

• Chi-square-based fit indices are standard in factor analysis.
• These indices assess model fit by comparing observed and model-implied covariance matrices.
• A known issue is their sensitivity to sample size.

Purpose of the Study:

• To investigate a lesser-known property of chi-square fit indices related to unique variances.
• To explain why fit indices may contradict correlation residuals.
• To provide empirical and artificial examples illustrating this incompatibility.

Main Methods:

• Analysis of standard chi-square-based fit indices in factor analysis.
• Examination of the relationship between unique variances and model fit.
• Generation of empirical and artificial datasets for comparison.

Main Results:

• Chi-square fit indices are more sensitive to misfit with small unique variances.
• Excellent model fit indicated by small residuals can be contradicted by poor fit indices when unique variances are small.
• Artificial data with larger unique variances showed excellent fit indices with identical residuals.

Conclusions:

• The sensitivity of fit indices to unique variances is a critical consideration in model evaluation.
• Researchers should be aware of potential discrepancies between residuals and fit indices.
• Understanding the role of unique variances enhances the interpretation of factor analysis model fit.