Understanding Robust Corrections in Structural Equation Modeling

Abstract Robust corrections to standard errors and test statistics have wide applications in structural equation modeling (SEM). The original SEM development, due to CitationSatorra and Bentler (1988, Citation1994), was to account for the effect of nonnormality. CitationMuthén (1993) proposed corrections to accompany certain categorical data estimators, such as cat-LS or cat-DWLS. Other applications of robust corrections exist. Despite the diversity of applications, all robust corrections are constructed using the same underlying rationale: They correct for inefficiency of the chosen estimator. The goal of this article is to make the formulas behind all types of robust corrections more intuitive. This is accomplished by building an analogy with similar equations in linear regression and then by reformulating the SEM model as a nonlinear regression model. Keywords: nonnormal datarobust standard errorsSatorra-Bentler scaled chi-square Notes 1The term robust has another meaning in mainstream statistical literature, where it refers to estimators that are robust to the presence of outliers. This is not the usage here. 2A related but slightly different property is consistency, which means the parameter estimates approach their true values as the sample size increases. This property is also necessary for SEM asymptotic theory to hold. 3In the rare instances when the variables are platykyrtic (have negative kurtosis), the ML estimator can become more efficient relative to its variability with normal data, and the naïve standard errors are too large. But even in this latter case, the ML estimator is no longer the most efficient estimator with nonnormal data, but the asymptotically distribution-free (ADF) estimator of CitationBrowne (1984) is. 4The rule used here is , where A is a constant matrix. 5The usual estimate of mean squared error is . 6 Another example of a nondiagonal residual matrix is when the data are clustered. The hierarchical linear modeling (HLM) literature (CitationRaudenbush & Bryk, 2002) provides many examples of such data. It is typically handled by specialized software. 7 For instance, in WLS regression the most common approach to estimating for each subject is to set it equal to the squared difference between each observation and its predicted value from the regular LS regression (CitationNeter et al., 1996; CitationRuud, 2000). 8 Ordinary LS regression is a special case of GLS regression with V = σ2 I. Substituting into EquationEquation 4, we obtain that the LS fit function should be . This is almost EquationEquation 2, except for a constant, which does not matter for estimation. The LS fit function is typically written without such a constant. 9 This particular model can actually be reparameterized into a structural equation model that is linear in the parameters by estimating the squared loading instead, but such reparameterizations are not possible with general SEM. 10 The term correctly specified is also sometimes used in SEM to refer to the assumption that the model being fit to data is true; this is not its meaning here. In fact, throughout the article, we assume the model fit to data is correct. 11The answer is, use EquationEquation 14 setting and , the ADF estimate of the asymptotic covariance matrix. 12Two points of clarification are necessary. First, it is theoretically possible to define polychoric correlations using any multivariate distribution for the underlying data, but it is not possible to compute them under an arbitrary underlying distribution. In practice, they are always computed assuming underlying normality. Second, the assumption of underlying normality can be equivalently stated as the assumption of a probit link function connecting categorical items and latent factors.