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Exploratory Factor Analysis (EFA)

Uncovers the latent factors underlying a set of items — the standard first step in developing a new measurement scale.

Reduce DimensionsMultivariate also known as: Common factor analysis, Principal axis factoring

✓ When to use

Scale development: which items cluster into which constructs?
No strong prior structure — the factor pattern is genuinely open.
Checking dimensionality of an item pool before CFA on a fresh sample.

✗ When NOT to use

A clear hypothesized structure exists — go straight to CFA.
Pure data compression without latent-variable claims — PCA.
Sample too small or correlations too weak (KMO < .60).
Running EFA and CFA on the SAME sample as 'validation' — that is double-dipping; split the sample or collect anew.

Data requirements

Dependent / outcome variable	A set of items (ideally 4+ per expected factor), interval-like; use polychoric correlations for strongly ordinal items.
Independent / grouping variable	—
Design	One sample; items measured on comparable scales.
Sample size guidance	Common guidance: n ≥ 200 or 10 cases per item, whichever is larger; stronger loadings tolerate smaller n.

Assumptions

Sufficient intercorrelation (Bartlett significant; KMO ≥ .60, ideally ≥ .80).
Linear relations among items; multivariate normality for ML extraction (PAF is less demanding).
No extreme multicollinearity (items near-duplicates).
Factors assumed to CAUSE item responses (reflective measurement logic).

Hypotheses

H₀ — — (exploratory; formal tests appear in ML extraction's fit test, but retention is the real decision).

H₁ — —

The concept

EFA models each item as loading on common factors plus a unique term, analyzing only the shared variance among items (communalities on the diagonal) — the key difference from PCA. Extraction (principal axis factoring or maximum likelihood) finds the factors; rotation then makes them interpretable: varimax forces uncorrelated factors, while oblique rotations (promax, oblimin) allow correlated factors — the realistic default in social science, where constructs correlate.

Retention should rest on parallel analysis plus interpretability, not the eigenvalue > 1 default. Read the pattern matrix: items load saliently (≥ .40) on their factor, cross-load weakly (< .30) elsewhere; violators are dropped one at a time with re-analysis. The factor correlation matrix from an oblique rotation previews discriminant validity questions that CFA will formally test.

Worked example

A researcher pilots 18 new items intended to measure workplace wellbeing on n = 260 employees. KMO = .87; parallel analysis suggests three factors.

PAF with promax rotation yields clean factors: physical (5 items, loadings .58–.81), psychological (6 items), social (4 items); three cross-loading items are dropped. Factors correlate .38–.52; total variance explained 54%.

How to run it

library(psych)
items <- df[, paste0("wb", 1:18)]

KMO(items); cortest.bartlett(cor(items), n = nrow(items))
fa.parallel(items, fa = "fa")                     # retention

efa <- fa(items, nfactors = 3, fm = "pa", rotate = "promax")
print(efa, cut = .30, sort = TRUE)                # pattern matrix
efa$Phi                                            # factor correlations

from factor_analyzer import FactorAnalyzer, calculate_kmo

kmo_all, kmo_model = calculate_kmo(df[items])
print(kmo_model)

fa = FactorAnalyzer(n_factors=3, method="principal", rotation="promax")
fa.fit(df[items])
print(fa.loadings_.round(2))
print(fa.get_factor_variance())   # SS loadings, prop., cumulative

Interpreting the output

KMO/Bartlett first: is the matrix factorable?
Number of factors defended with parallel analysis + theory, not software defaults.
Pattern matrix: salient loadings ≥ .40 on the home factor, cross-loadings < .30; communalities ≥ .30–.40.
Factor correlations (oblique) — if two factors correlate > .80, question their distinctness.
Name factors by item content, and confirm the structure with CFA on new data.

APA-style reporting

Exploratory factor analysis (principal axis factoring, promax rotation) on the 18 wellbeing items (KMO = .87; Bartlett's χ²(153) = 2,114, p < .001) supported a three-factor solution based on parallel analysis, explaining 54.3% of common variance. All retained items loaded ≥ .58 on their intended factor with cross-loadings < .30; inter-factor correlations ranged from .38 to .52.

Common mistakes

Using PCA extraction and calling it factor analysis.
Eigenvalue > 1 as the only retention rule (typically over-extracts).
Varimax by default when constructs plainly correlate.
Dropping several items simultaneously instead of iteratively.
EFA and CFA on the same sample.
Interpreting a factor from 2 salient items — under-identified and unstable.

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