Robust Confirmatory Factor Analysis
Also known as: Robust CFA, CFA with robust standard errors, Satorra-Bentler CFA, non-normal CFA
Robust confirmatory factor analysis fits a pre-specified factor structure to observed data while correcting standard errors and goodness-of-fit statistics for violations of multivariate normality. It is the preferred variant of CFA whenever Likert-type, skewed, or kurtotic indicators make the classical normal-theory estimator unreliable.
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When to use it
Use robust CFA whenever the observed indicators show meaningful non-normality — in practice, whenever absolute skewness exceeds about 2 or kurtosis exceeds about 7, or whenever items are Likert-type ordinal variables. It is also warranted when confirmatory fit under standard ML looks poor and non-normality is the likely culprit. Do not use robust CFA when the hypothesised factor structure itself has not been grounded in prior theory or a previous EFA; in that situation, EFA should come first. Also avoid robust CFA as the default when sample sizes are very small (n < 100), because the asymptotic correction is itself unreliable in small samples — bootstrap SEs or Bayesian CFA may be preferable then.
Strengths & limitations
- Maintains the confirmatory, theory-driven logic of CFA while removing sensitivity to the normality assumption.
- The Satorra-Bentler scaled chi-square and associated fit indices are well-calibrated across a wide range of non-normal conditions in large samples.
- Robust standard errors provide accurate inference for individual loadings even when the distributional assumption fails.
- Widely available in standard software (lavaan in R, Mplus, LISREL) with simple option switches, making implementation straightforward.
- Compatible with ordinal indicators via WLSMV, which extends the robustness principle to Likert and binary items.
- Robust corrections are asymptotic and may not work well in small samples (n < 100), where standard errors can still be biased.
- The method adjusts inference for non-normality but does not change the factor structure — a misspecified model remains misspecified.
- Computing robust corrections requires estimating the full asymptotic covariance matrix of the observed moments, which can be unstable when p (number of indicators) is large relative to n.
- Different software implementations of robust CFA can produce slightly different scaled statistics and SEs, complicating direct comparison across studies.
Frequently asked
What is the difference between robust CFA and standard CFA?
Both fit a pre-specified factor model. Standard CFA uses normal-theory maximum likelihood, whose chi-square and standard errors are correct only if data are multivariate normal. Robust CFA applies the Satorra-Bentler correction to the chi-square and uses sandwich standard errors, keeping inference valid when data are non-normal.
When should I use WLSMV instead of MLR?
Use WLSMV (or WLSM) when indicators are genuinely ordinal (Likert scales with few categories, binary items). WLSMV fits the model to polychoric correlations and applies mean-and-variance adjustments. Use MLR when indicators are treated as continuous and non-normality is the primary concern.
Can I compare nested robust CFA models with a chi-square difference test?
Not with ordinary chi-square differences. When models are estimated with MLR or Satorra-Bentler corrections, chi-square differences must be computed using the scaled difference chi-square formula (implemented in lavTestLRT in lavaan and DIFFTEST in Mplus) because the scaled statistics do not subtract like standard chi-squares.
How large a sample do I need for robust CFA?
The asymptotic corrections work well from roughly n = 200 upwards, depending on the number of indicators and degree of non-normality. With fewer than 100 cases the corrections may be unreliable; bootstrapped standard errors or Bayesian CFA are safer alternatives in that range.
Does robust CFA fix a poorly fitting model?
No. Robust corrections adjust the test statistic and standard errors for non-normality — they do not change the factor structure itself. If the hypothesised structure does not match the data, the robust chi-square will still indicate poor fit. Robust CFA corrects inference; it does not rescue misspecification.
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: Applications for developmental research (pp. 399–419). Sage. link ↗
- Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37(1), 62–83. DOI: 10.1111/j.2044-8317.1984.tb00789.x ↗
How to cite this page
ScholarGate. (2026, June 3). Robust Confirmatory Factor Analysis. ScholarGate. https://scholargate.app/en/statistics/robust-confirmatory-factor-analysis
Which method?
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