Exploratory factor analysis decomposes variance into shared (communalities) + unique components. Latent-variable model; distinct from PCA in framing.
PCA assumes all variance is shared — variables are exact linear combinations of components. Factor analysis adds a unique-variance term per variable (uniqueness): each item is a linear combination of factors plus item-specific noise. This latent-variable framing matches the questionnaire / scale-development use case where each item measures the latent construct with its own error.
Three extraction methods: maximum likelihood (ML — gives goodness-of-fit χ² and modern criteria like RMSEA / TLI), principal axis (iterative communality estimation), minimum residual (least-squares fit). ML is the default for confirmatory / inferential framings.
Rotation matters for interpretation. Varimax (orthogonal, default) maximises high vs low loadings — gives 'simple structure'. Promax / Oblimin (oblique) let factors correlate — more realistic when factors are conceptually related (depression and anxiety, for example).
Two sample-adequacy checks: KMO (Kaiser-Meyer-Olkin) measures how well the data supports factor analysis (> 0.6 marginal, > 0.8 great). Bartlett's test of sphericity checks the correlation matrix isn't the identity — needs to be significant for FA to be meaningful.
Numeric features + # factors + extraction (ML / PAF / minimum residual) + rotation (none / Varimax / Quartimax / Promax / Oblimin).
Loadings + communalities + uniquenesses + SS loadings + % variance + ML goodness-of-fit χ² + RMSEA + TLI + KMO + Bartlett. Loadings ≥ |0.4| highlighted.
Pick # factors via scree plot + parallel analysis. Inspect rotated loadings: items loading > 0.4 on one factor and < 0.3 on others = clean structure.