Robust estimation of structural equation models with ordinal data

M. Welz${}^a$, P. Mair${}^b$ and A. Alfons${}^c$

${}^a$University of Zurich, ${}^b$Harvard University, ${}^c$Erasmus University Rotterdam

Structural Equation Models (SEMs) are typically fitted to a given correlation matrix, which is commonly estimated from a sample of Likert-type rating data. However, noisy or low-quality observations—such as (but not limited to) careless responses [1]—might be present in the data, which can introduce a sizable bias in correlation estimates [2, 3]. We demonstrate that this bias is inherited by the SEM estimate, possibly leading to worse model fit and biased estimates of factor structure. As a remedy, we propose to use a robust estimate of a polychoric correlation matrix [3]. We show through simulation studies and empirical applications that fitting a SEM to a robustly estimated polychoric correlation matrix can substantially improve SEM fit, enhance the accuracy of parameter estimates, and help identify potentially low-quality responses. In particular, we demonstrate how the fit of commonly used SEM estimators such as maximum likelihood or least-squares-based approaches like diagonally weighted least squares (DWLS) can be improved by using a robustly estimated polychoric correlation matrix. Our proposed procedure is implemented in the free open-source R package robcat, which is implemented using fast and efficient C++ code.

Keywords: Careless responding, Polychoric correlation, Partial misspecification