The sensitivity of structural equation modeling with ordinal data to underlying non-normality and observed distributional forms

Abstract

Structural equation modeling (SEM) of ordinal data is often performed using normal theory maximum likelihood estimation based on the Pearson correlation (cont-ML) or using least squares principles based on the polychoric correlation matrix (cat-LS). While cont-ML ignores the categorical nature of the data, cat-LS assumes underlying multivariate normality. Theoretical results are provided on the validity of treating ordinal data as continuous when the number of categories increases, leading to an adjustment to cont-ML (cont-ML-adj). Previous simulation studies have concluded that cat-LS outperforms cont-ML, and that it is quite robust to violations of underlying normality. However, this conclusion was based on a data simulation methodology equivalent to discretizing exactly normal data. The present study employs a new simulation method for ordinal data to reinvestigate whether ordinal SEM is robust to underlying non-normality. In contrast to previous studies, we include a large set of ordinal distributions, and our results indicate that ordinal SEM estimation and inference is highly sensitive to the interaction between underlying non-normality and the ordinal observed distributions. Our results show that cont-ML-adj consistently outperforms cont-ML, and that cat-LS is less biased than cont-ML-adj. The sensitivity of cat-LS to violation of underlying normality necessitates the need for a test of underlying normality. A bootstrap test is found to reliably detect underlying non-normality.