Not Positive Definite Correlation Matrices in Exploratory Item Factor Analysis: Causes, Consequences and a Proposed Solution

Least-squares exploratory factor analysis based on tetrachoric/polychoric correlations is a robust, defensible and widely used approach for performing item analysis, especially in the first stages of scale development. A relatively common problem in this scenario, however, is that the inter-item correlation matrix fails to be positive definite. This paper, which is largely intended for practitioners, aims to provide a didactic discussion about the causes, consequences and remedies of this problem. The discussion is more applied than statistical and based on the factor analysis model, and the problem is linked to that of improper solutions. Solutions for preventing the problem from occurring, and the smoothing corrections available at present are described and discussed. A new smoothing algorithm is also proposed.

Automated summary

This paper provides a didactic discussion on the causes, consequences, and remedies of the issue of non-positive definite inter-item correlation matrices in least-squares exploratory factor analysis based on tetrachoric/polychoric correlations. The discussion is more applied than statistical, focusing on the factor analysis model and the connection to improper solutions. Solutions for preventing the problem and available smoothing corrections are described and discussed, along with a new smoothing algorithm proposed.