Exploratory Factor Analysis
Introduction
Exploratory factor analysis (EFA) searches for a small number of latent factors that explain the correlations among a larger set of observed variables. It is the cornerstone of psychometric scale development, the standard tool for validating questionnaire structures, and a useful exploratory technique in any context where many correlated indicators are believed to reflect a smaller number of latent constructs. The output — rotated factor loadings together with factor correlations — directly informs item-selection, scale-construction, and CFA model-specification decisions.
Prerequisites
A working understanding of correlation matrices, principal components analysis, and the conceptual difference between observed indicators and latent constructs.
Theory
The common factor model is
\[X = \Lambda F + U,\]
where \(X\) is the observed item matrix, \(\Lambda\) is the loading matrix, \(F\) are the common factors, and \(U\) is unique (specific) variance. Unlike PCA, which lumps common and specific variance together, EFA explicitly partitions the variance.
The standard EFA workflow has five steps:
- Check factorability: Kaiser-Meyer-Olkin (KMO > 0.6 acceptable, > 0.8 good) and Bartlett’s sphericity test confirm that the correlation matrix has enough structure to factor.
- Choose number of factors: parallel analysis is the modern gold standard; the obsolete Kaiser eigenvalue-greater-than-1 rule and the subjective scree plot are still seen but should be supplemented.
- Extract: maximum likelihood (ML) for inferential purposes, principal-axis factoring (PAF) for robustness, principal-components for quick exploration.
- Rotate: varimax (orthogonal) or oblimin / promax (oblique) for interpretable loadings.
- Interpret: examine rotated loadings, factor correlations, and item communalities.
Assumptions
Multivariate Normal continuous data for ML extraction; linearity of item-factor relationships; sufficient sample size (rule of thumb: at least 5–10 cases per item, with \(n \ge 200\) as a floor).
R Implementation
library(psych); library(GPArotation)
d <- bfi[, 1:25] # 25-item big-five inventory
# Factorability
KMO(d)
cortest.bartlett(cor(d, use = "pairwise"), n = nrow(d))
# Number of factors
fa.parallel(d, fa = "fa")
# EFA with 5 factors, oblique rotation
efa <- fa(d, nfactors = 5, rotate = "oblimin", fm = "ml")
print(efa, cut = 0.3)Output & Results
KMO() returns overall and per-item factorability indices; cortest.bartlett() returns the sphericity test. fa.parallel() overlays observed eigenvalues on simulated null eigenvalues to determine the number of factors. fa() returns rotated loadings, communalities, factor correlations (under oblique rotation), and explained variance per factor.
Interpretation
A reporting sentence: “EFA on 25 personality items (n = 2,800; KMO = 0.85, Bartlett’s \(\chi^2 = 18{,}245\), \(p < 0.001\)) supported a five-factor solution by parallel analysis; oblimin-rotated loadings aligned with the Big Five structure (extraversion, neuroticism, conscientiousness, agreeableness, openness), explaining 45 % of common variance, and factor correlations ranged from 0.15 to 0.42.” Always report KMO, Bartlett, the number-of-factors method, the extraction method, and the rotation.
Practical Tips
- Use parallel analysis (
fa.parallel) for the number-of-factors decision; avoid the obsolete “eigenvalue > 1” Kaiser rule which over-extracts in most settings. - Prefer oblique rotation (oblimin, promax) by default; uncorrelated factors are rare in real psychological or biological data.
- Watch for Heywood cases — loadings above 1 or communalities above 1 — which indicate over-extraction or model misspecification.
- For ordinal Likert items, use polychoric correlations (
fa(..., cor = "poly")); Pearson correlations on Likert items underestimate the true factor structure. - For confirmatory tests of the resulting structure, run a separate CFA (
lavaan::cfa) on a held-out sample; never confirm on the data used to extract. - Report item communalities alongside loadings; communalities below 0.20 indicate items that contribute little to any factor and may need removal.
R Packages Used
psych for fa(), KMO(), fa.parallel(), and the comprehensive EFA workflow; GPArotation for the underlying rotation algorithms; EFAtools for parallel-analysis-aware extraction with bootstrap confidence intervals; lavaan for confirmatory factor analysis as the natural follow-up; MBESS for additional reliability and structural-equation utilities.