Robust Model Testing Research — Robust SEM and Structural Model Evaluation

Both use the Satorra-Bentler scaling idea but differ in their corrections. MLM applies a mean adjustment to the chi-square statistic only, leaving standard errors as standard ML estimates. MLR applies both mean and variance adjustments, producing robust standard errors and confidence intervals in addition to the scaled chi-square. MLR is generally preferred because it also corrects standard errors, which affects significance tests for individual parameters.

You cannot simply subtract one scaled chi-square from another. You must use the Satorra-Bentler scaled difference test, which divides the difference in scaled chi-squares by a correction factor computed from the scaling factors of each model. Mplus produces this automatically with the DIFFTEST option. Failing to use this correction commonly leads to erroneous conclusions about whether adding or removing a path improves fit.

For purely ordinal data, WLSMV (weighted least squares mean and variance adjusted, the default for categorical indicators in Mplus) is generally preferred over MLR because it treats ordinal variables as categorical, estimating polychoric correlations. MLR with ordinal data assumes the variables are continuous approximations, which can be acceptable for five or more response categories but is technically misspecified for truly ordinal scales.

Robust estimators are asymptotically valid and therefore perform better in large samples. As a practical guide, n >= 200 is commonly recommended for robust SEM, compared to n >= 100–150 for standard ML with well-behaved data. In small samples the scaling corrections themselves become unstable; bootstrapped confidence intervals or Bayesian SEM are better options below n = 150.

Good global fit does not mean the model is uniquely correct — a misspecified model can still yield acceptable fit indices while systematically mis-representing specific relationships. Inspect residuals and modification indices as diagnostics, but make re-specifications only when they are theoretically justified. Post-hoc modifications driven purely by modification indices constitute capitalisation on chance and must be disclosed and cross-validated.