Comparison of Smoothing Approaches to Polychoric Correlation Matrices in CFA

Abstract

Confirmatory factor analysis (CFA) using polychoric correlations has become standard in psychometric and item analyses. Nevertheless, issues such as sparse data can lead to non-positive definite (NPD) polychoric correlation matrices, posing notable challenges. Smoothing algorithms to address this issue can play an important role in eliminating noise, enhancing signal quality, and regularizing data. In the present paper, a series of simulation studies were conducted to compare the eigenvalue substitution smoothing method with Higham’s nearest correlation approach. Both aim to transform NPD matrices into positive definite ones but differ in technique. Eigenvalue substitution adjusts eigenvalues below a set threshold and rescales, while Higham’s method employs iterative eigen decomposition, selectively choosing eigenvalues above a certain threshold and reconstructing the matrix until convergence. It was found that although Higham’s correlation approach slightly outperforms the eigenvalue substitution method in terms of parameter bias, the converse was more efficient. Neither approach was particularly favorable at assessing fit. Recommendations for empirical data analysis and potential future avenues of research are discussed.