Confirmatory Factor Analysis (CFA)

EFA is data-driven: it discovers how many factors exist and which items cluster together without prior specification. CFA is theory-driven: the researcher specifies the full factor pattern before seeing the fit statistics and then tests whether the data are consistent with that pattern. A standard workflow runs EFA first on a calibration sample and then validates the resulting structure with CFA on a hold-out sample.

Report at minimum chi-square (with df and p-value), CFI, TLI, RMSEA (with 90% CI), and SRMR. Adequate fit is commonly indicated by CFI/TLI ≥ .95, RMSEA < .06, and SRMR < .08, though these are guidelines rather than hard thresholds. Always inspect modification indices for large, theoretically interpretable sources of local misfit.

Yes, but the standard ML estimator assumes continuous, normally distributed indicators. For ordinal items — typically those with fewer than five response categories or with skewed distributions — use polychoric correlations as input and choose a robust estimator such as DWLS (also called WLSMV in lavaan/Mplus) or robust ML (MLR).

Measurement invariance means that the scale measures the same construct in the same way across groups (e.g., men vs. women, or cultures). Multi-group CFA tests this by fitting a sequence of increasingly constrained models — configural, metric, scalar — and comparing their fit. Scalar invariance is required before latent mean differences across groups can be validly interpreted.

A common minimum is 200 cases, but requirements depend on model complexity, number of factors, and indicator reliability. Simpler models with strong loadings can work adequately with fewer cases, while complex models or weak indicators need substantially more. Monte Carlo simulation studies suggest that at least 5–10 observations per freely estimated parameter is a useful starting heuristic.