Robust Structural Equation Modeling
Also known as: Robust SEM, SEM with robust standard errors, Satorra-Bentler SEM, non-normal SEM
Robust structural equation modeling (Robust SEM) applies the full SEM framework — simultaneous estimation of measurement and structural relations among latent variables — while using corrected test statistics and sandwich standard errors that remain valid when observed data depart from multivariate normality. The Satorra-Bentler scaled chi-square is the most widely used correction.
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When to use it
Use robust SEM whenever observed indicators show notable departures from multivariate normality — for example, Mardia's kurtosis coefficient substantially above zero, or variables with obvious floor or ceiling effects. It is the appropriate default in applied social, behavioural, and health research because non-normality is the rule rather than the exception. Do not use it as a replacement for addressing ordinal data: if indicators are Likert-type with five or fewer categories, WLSMV with polychoric correlations is usually preferred. Also avoid robust SEM with very small samples (n < 100–200): the sandwich estimator requires a reasonably large n to be stable, and with small n the correction can behave erratically.
Strengths & limitations
- Corrects inflated chi-square and biased standard errors under non-normality without changing the ML point estimates.
- The Satorra-Bentler scaled chi-square is the best-studied and most widely accepted robust correction, implemented in lavaan, Mplus, EQS, and LISREL.
- Requires no distributional assumption beyond finite fourth moments — no user-specified alternative distribution is needed.
- Robust CFI, TLI, and RMSEA can be derived from the SB chi-square, enabling the full standard model-fit assessment workflow.
- Applicable to a wide range of SEM models including confirmatory factor analysis, path models with latent variables, and multi-group models.
- Requires larger samples than standard SEM because the empirical estimation of the asymptotic covariance matrix of sample moments is noisy in small samples.
- Does not fully solve the problem for ordinal data: it adjusts for continuous non-normality but ordinal indicators call for polychoric correlations and WLSMV.
- The SB correction addresses kurtosis primarily; it may underperform when data are extremely skewed or multimodal.
Frequently asked
What is the difference between robust SEM and standard SEM?
Standard ML SEM assumes multivariate normality; its chi-square test is inflated and standard errors are biased when that assumption fails. Robust SEM keeps the same model equations and ML estimates but replaces the chi-square with the Satorra-Bentler scaled statistic and replaces standard errors with sandwich (robust) standard errors, correcting both problems without requiring a different distribution.
In lavaan or Mplus, how do I request robust SEM?
In lavaan, set estimator = 'MLR' in the sem() or cfa() call; this provides the Yuan-Bentler T2* robust chi-square and robust standard errors. In Mplus, specify ESTIMATOR = MLR in the ANALYSIS section. Both yield the Satorra-Bentler family of corrections.
Can I compare nested robust SEM models with a chi-square difference test?
No, not with the ordinary chi-square difference. Because the SB scaled statistics for two nested models do not subtract to give an SB-scaled difference, you must use the special scaled chi-square difference formula (Satorra-Bentler chi-square difference test), which rescales the difference. lavaan's lavTestLRT() with method = 'satorra.bentler.2010' handles this automatically.
When should I prefer WLSMV over robust ML for non-normal data?
WLSMV (diagonally weighted least squares with mean and variance adjustment) is preferred when indicators are ordinal with five or fewer response categories; it uses polychoric correlations and is designed specifically for categorical data. Robust ML is more appropriate when indicators are treated as continuous but deviate from normality.
Is robust SEM appropriate for small samples?
The sandwich estimator underlying robust standard errors is consistent only asymptotically and can be unstable with n below 100–200. With small samples, Bayesian SEM or bootstrap confidence intervals may be more reliable alternatives.
Sources
- Satorra, A. & Bentler, P. M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In A. von Eye & C. C. Clogg (Eds.), Latent variables analysis (pp. 399–419). Sage. link ↗
- Yuan, K.-H. & Bentler, P. M. (1998). Normal theory based test statistics in structural equation modelling. British Journal of Mathematical and Statistical Psychology, 51(2), 289–309. DOI: 10.1111/j.2044-8317.1998.tb00682.x ↗
How to cite this page
ScholarGate. (2026, June 3). Robust Structural Equation Modeling. ScholarGate. https://scholargate.app/en/statistics/robust-structural-equation-modeling
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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