Reflecting on the "Robust" Standard Errors for Two-Stage SEM Estimation With Categorical Data: Mistakes and Correction

Estimating structural equation models with categorical data often proceeds in two stages. Stage 1 obtains polychoric correlations among categorical variables. Stage 2 fits the researcher's model to the polychoric correlations, often with diagonally-weighted least squares (DWLS) or unweighted least squares (ULS). For DWLS/ULS model parameter estimates, the current literature contains two standard error (SE) methods: the classic SE and robust SE. Although the robust SE is generally believed to be satisfactory, in this paper we show it is reliable only given perfect models. When a model is misspecified, the robust SE can sometimes be egregiously biased, and the bias is not directly related to the model's overall misfit. We also propose a new SE method for DWLS and ULS. The new SE is consistent regardless of the model fit, and substantially outperforms the robust SE in our simulation studies. The classic SE is the worst among the three SE methods.