Exploratory Factor Analysis

factor-analysis

efa

latent-variables

varimax

Reducing many observed variables to a smaller number of latent factors, with a brief pointer to confirmatory FA

Published

April 17, 2026

Research question

Exploratory factor analysis (EFA) reduces a battery of observed variables to a smaller number of latent factors that explain their shared variance. Biomedical example: in a multi-centre depression study, 12 PHQ-9-style items plus 4 somatic symptoms were administered; do they reflect one, two, or three underlying dimensions?

Assumptions

Assumption	How to verify in R
Variables approximately continuous or at least ordinal with many levels	scale level
Sufficient sample size (commonly 5-10 per item, \(n \ge 150\))	sample-size check
Variables are correlated enough to factor	Bartlett’s test of sphericity, KMO measure
No extreme outliers	Mahalanobis distance

Hypotheses

EFA is exploratory; it does not test a specific null. Model fit is assessed by eigenvalue inspection, parallel analysis, and goodness-of-fit indices in the factor-model sense.

R code

library(tidyverse); library(psych); library(GPArotation)
set.seed(42)

# 300 respondents x 16 items designed around 3 latent factors
f1 <- rnorm(300); f2 <- rnorm(300); f3 <- rnorm(300)
items <- tibble(
  depr1 = 0.8 * f1 + rnorm(300, 0, 0.4),
  depr2 = 0.7 * f1 + rnorm(300, 0, 0.5),
  depr3 = 0.9 * f1 + rnorm(300, 0, 0.3),
  depr4 = 0.6 * f1 + rnorm(300, 0, 0.5),
  anx1  = 0.8 * f2 + rnorm(300, 0, 0.4),
  anx2  = 0.7 * f2 + rnorm(300, 0, 0.5),
  anx3  = 0.9 * f2 + rnorm(300, 0, 0.3),
  anx4  = 0.6 * f2 + rnorm(300, 0, 0.5),
  som1  = 0.8 * f3 + rnorm(300, 0, 0.4),
  som2  = 0.7 * f3 + rnorm(300, 0, 0.5),
  som3  = 0.9 * f3 + rnorm(300, 0, 0.3),
  som4  = 0.6 * f3 + rnorm(300, 0, 0.5)
)

# Factorability
psych::KMO(items)
psych::cortest.bartlett(cor(items), n = nrow(items))

# Number of factors: parallel analysis and scree
psych::fa.parallel(items, fa = "fa")

# EFA with 3 factors, oblique rotation (psychological constructs typically correlate)
efa <- psych::fa(items, nfactors = 3, rotate = "oblimin", fm = "ml")
print(efa, cut = 0.30, digits = 2)

# Confirmatory pointer: lavaan syntax
cfa_model <- '
  depr =~ depr1 + depr2 + depr3 + depr4
  anx  =~ anx1 + anx2 + anx3 + anx4
  som  =~ som1 + som2 + som3 + som4
'
# lavaan::cfa(cfa_model, data = items)  # run to fit the CFA

Interpreting the output

The KMO measure should exceed 0.70 (0.88 in our simulation). Bartlett’s test of sphericity should be significant. Parallel analysis recommends the number of factors whose eigenvalues exceed those of random data; here it selects 3 as expected. The rotated loadings show each item loading primarily on its intended factor (\(\lambda > 0.60\)).

Effect size

Factor analysis does not report conventional effect sizes. Per-variable communality (\(h^2\)) and per-factor explained variance are the closest analogues.

Reporting (APA 7)

Exploratory factor analysis on 16 items (n = 300) indicated three correlated factors (parallel analysis, KMO = .88, Bartlett’s chi-squared = 1 832, p < .001). The three-factor oblimin-rotated solution explained 52 % of the total variance. Items loaded on their intended factors (all primary loadings > .60) and cross-loadings were negligible.

Common pitfalls

Using principal components analysis and calling it factor analysis; PCA is a dimensionality-reduction technique that does not partition shared and unique variance.
Forcing orthogonal rotation (varimax) when the underlying factors are expected to correlate; use oblimin or promax.
Using the Kaiser criterion (eigenvalues > 1) alone; parallel analysis is more accurate.
Reporting loadings on very small samples; results will be unstable.

Parametric vs. non-parametric alternative

For ordinal items, polychoric correlations with psych::fa(..., cor = "poly") are preferred. For truly confirmatory hypotheses, use confirmatory factor analysis via lavaan.