Assessing Dimensionality in Non-Positive Definite Tetrachoric Correlation Matrices: Does Matrix Smoothing Help?

Area of Science:

• Psychometrics
• Statistical Modeling

Background:

• Assessing dimensionality in factor analysis is crucial for model accuracy.
• Non-positive definite (NPD) tetrachoric correlation matrices pose challenges for standard analyses.
• Parallel analysis is a common method for determining the number of factors.

Purpose of the Study:

• To investigate the impact of matrix smoothing on dimensionality recovery using parallel analysis for NPD tetrachoric correlation matrices.
• To extend previous research by examining a wider range of factors, model error, and realistic item parameters.
• To evaluate the practical utility of smoothing algorithms in improving parallel analysis for binary data.

Main Methods:

• Two simulation studies were conducted, replicating and extending prior work.
• Non-positive definite (NPD) tetrachoric correlation matrices were generated and smoothed using three algorithms.
• Parallel analysis was applied to both original and smoothed matrices to assess dimensionality.
• Factor analysis was applied to an empirical dataset (Adjective Checklist data) to demonstrate the effects.

Main Results:

• Matrix smoothing provided modest improvements in the performance of parallel analysis with binary data.
• The benefits of smoothing were often small in practical terms.
• The study explored the influence of varying numbers of factors and levels of model error.

Conclusions:

• Matrix smoothing can enhance parallel analysis for NPD tetrachoric correlation matrices with binary data, though improvements are often marginal.
• The findings suggest that while smoothing is beneficial, its practical impact on dimensionality assessment may be limited.
• The study highlights the importance of considering matrix properties and smoothing techniques in factor analysis research.