efa() performs Exploratory Factor Analysis (EFA) to discover underlying
patterns in your survey items. Supports both Principal Component Analysis (PCA)
and Maximum Likelihood (ML) extraction. This is the R equivalent of SPSS's
FACTOR procedure.
For example, if you have 6 items measuring different attitudes, EFA can reveal whether they group into 2-3 underlying dimensions (factors or components).
Usage
efa(
data,
...,
n_factors = NULL,
rotation = "varimax",
extraction = "pca",
weights = NULL,
use = "pairwise",
sort = TRUE,
blank = 0.4,
na.rm = TRUE
)Arguments
- data
Your survey data (a data frame or tibble)
- ...
The items to analyze. Use bare column names separated by commas, or tidyselect helpers like
starts_with("trust").- n_factors
Number of components to extract (a whole number). Default
NULLuses the Kaiser criterion (eigenvalue > 1). A single component cannot be rotated; it is shown unrotated with SPSS's note.- rotation
Rotation method:
"varimax"(default, orthogonal),"oblimin"(oblique direct oblimin, delta = 0, allows correlated factors),"promax"(oblique, power 4), or"none". All rotations use Kaiser normalization and SPSS FACTOR's own algorithms and stopping rules (at most 25 iterations, as SPSS/CRITERIA ITERATE(25)), so they reproduce SPSS's rotated matrices; they differ slightly fromstats::varimax(),stats::promax()andGPArotation::oblimin().- extraction
Extraction method:
"pca"(default, Principal Component Analysis) or"ml"(Maximum Likelihood, enables goodness-of-fit testing, assumes multivariate normality). ML uses SPSS's starting values, bounds a Heywood variable's communality at .999 as SPSS does and keeps SPSS's factor order.- weights
Optional survey weights for population-representative results. Give a column name (unquoted or as a string), an expression such as
sampling_weight * 2, or a numeric vector with one weight per row.- use
How to handle missing data for correlation computation:
"pairwise"(default, matches SPSS) or"complete"(listwise;"listwise"is accepted as an alias).- sort
Logical. Sort loadings by size within each component? Default
TRUE.- blank
Numeric. Suppress (hide) loadings with absolute value below this threshold in the print output. Default
0.40(matches SPSS BLANK(.40)). Set to0to show all loadings.- na.rm
Logical. Remove missing values? Default
TRUE.
Value
An efa result object containing:
- loadings
Component loading matrix (rotated if rotation applied)
- unrotated_loadings
Unrotated component matrix
- eigenvalues
All eigenvalues from the correlation matrix
- variance_explained
Tibble with the initial eigenvalues: Total, % of Variance, Cumulative % (one row per variable)
- extraction_variance
Tibble with the extraction sums of squared loadings of the extracted components/factors (SPSS "Extraction Sums of Squared Loadings"). For ML this is the variance the common factors explain - the figure the compact print reports.
- rotation_variance
Tibble with rotation sums of squared loadings (for the oblique rotations the column sums of squares of the structure matrix, as SPSS reports them)
- communalities
Extraction communalities for each variable
- kmo
List with overall KMO and per-item MSA values
- bartlett
List with chi_sq, df, and p_value
- rotation
Rotation method used
- extraction
Extraction method used
- n_factors
Number of components extracted
- correlation_matrix
Correlation matrix used for analysis
- initial_communalities
Initial communalities (1.0 for PCA, SMC for ML)
- goodness_of_fit
Goodness-of-fit test (ML only): chi_sq, df, p_value. NULL for PCA.
- uniquenesses
Unique variances per variable (ML only, NULL for PCA)
- pattern_matrix
Pattern matrix (oblimin/promax only, NULL otherwise)
- structure_matrix
Structure matrix (oblimin/promax only, NULL otherwise)
- factor_correlations
Factor correlation matrix (oblimin/promax only, NULL otherwise)
- rotation_iterations, rotation_converged
Iterations of the rotation as SPSS counts them ("Rotation converged in 4 iterations"; for promax those of its varimax step) and whether it converged; NULL without rotation
- variables
Character vector of variable names
- variable_labels
Named character vector with the variable labels (
NAwhere a variable has none);summary()shows them next to the names, shortened to the console width in the tables- weights
Weights variable name or NULL
- item_statistics
Tibble with mean, SD, analysis N and missing N per item. With
use = "pairwise"each item uses its own valid cases; withuse = "complete"all items use the complete cases.- n
Sample size: the smallest pairwise N (
use = "pairwise", the N of Bartlett's test as in SPSS) or the number of complete cases (use = "complete"); the sum of weights when weighted- use
The missing-data handling used (
"pairwise"or"complete")- col_prefix
Column name prefix:
"PC"for PCA,"Factor"for ML- sort
Whether loadings are sorted
- blank
Suppression threshold
Use summary() for the full SPSS-style output with toggleable sections.
Details
Understanding the Results
KMO (Kaiser-Meyer-Olkin) measures sampling adequacy:
KMO > 0.90: Marvelous
KMO 0.80 - 0.90: Meritorious
KMO 0.70 - 0.80: Middling
KMO 0.60 - 0.70: Mediocre
KMO 0.50 - 0.60: Miserable
KMO < 0.50: Unacceptable - don't use factor analysis
Bartlett's Test of Sphericity tests whether correlations are significantly different from zero. A significant result (p < .05) means the correlation matrix is suitable for factor analysis.
Eigenvalues indicate how much variance each component explains. The Kaiser criterion retains components with eigenvalue > 1.
Factor Loadings show how strongly each item relates to each component:
|loading| > 0.70: Strong association
|loading| 0.40 - 0.70: Moderate association
|loading| < 0.40: Weak (suppressed by default)
Communalities show how much of each item's variance is explained by the extracted components. Low communalities (< 0.40) suggest the item doesn't fit well with the others.
Choosing an Extraction Method
PCA (default): Extracts components explaining maximum total variance. Simple and robust. Does not assume normality.
ML: Extracts factors explaining shared variance only. Assumes multivariate normality. Provides a goodness-of-fit test to evaluate model fit. A non-significant chi-square (p > .05) suggests adequate fit.
Choosing a Rotation
Varimax (default): Assumes factors are uncorrelated. Produces simpler, easier-to-interpret results.
Oblimin: Allows factors to be correlated. More realistic for social science data. Produces both a Pattern Matrix (unique contributions) and Structure Matrix (total correlations).
Promax: Oblique rotation based on a power transformation of Varimax results. Like Oblimin, produces Pattern and Structure matrices. Common alternative to Oblimin in SPSS.
None: No rotation. Rarely useful for interpretation.
See also
reliability for checking scale reliability before creating indices.
row_means for creating mean indices after identifying factors.
summary.efa for detailed output with toggleable sections.
Other scale:
pomps(),
reliability(),
row_count(),
row_means(),
row_sums()
Examples
library(dplyr)
data(survey_data)
# Basic EFA with Varimax rotation
efa(survey_data,
political_orientation, environmental_concern, life_satisfaction,
trust_government, trust_media, trust_science)
#> Exploratory Factor Analysis: 6 items, 3 components (PCA/Varimax)
#> KMO = 0.505 (Miserable), Variance explained: 61.0%, N = 2168 (smallest pairwise)
#> Use summary() for detailed output.
# With Oblimin rotation
efa(survey_data,
political_orientation, environmental_concern, life_satisfaction,
trust_government, trust_media, trust_science,
rotation = "oblimin")
#> Exploratory Factor Analysis: 6 items, 3 components (PCA/Oblimin)
#> KMO = 0.505 (Miserable), Variance explained: 61.0%, N = 2168 (smallest pairwise)
#> Use summary() for detailed output.
# Maximum Likelihood extraction
efa(survey_data,
political_orientation, environmental_concern, life_satisfaction,
trust_government, trust_media, trust_science,
extraction = "ml")
#> Exploratory Factor Analysis: 6 items, 3 factors (ML/Varimax)
#> KMO = 0.505 (Miserable), Variance explained: 24.1%, N = 2168 (smallest pairwise)
#> Use summary() for detailed output.
# Promax rotation (oblique)
efa(survey_data,
political_orientation, environmental_concern, life_satisfaction,
trust_government, trust_media, trust_science,
rotation = "promax")
#> Exploratory Factor Analysis: 6 items, 3 components (PCA/Promax)
#> KMO = 0.505 (Miserable), Variance explained: 61.0%, N = 2168 (smallest pairwise)
#> Use summary() for detailed output.
# Fix number of factors
efa(survey_data,
political_orientation, environmental_concern, life_satisfaction,
trust_government, trust_media, trust_science,
n_factors = 2)
#> Exploratory Factor Analysis: 6 items, 2 components (PCA/Varimax)
#> KMO = 0.505 (Miserable), Variance explained: 44.0%, N = 2168 (smallest pairwise)
#> Use summary() for detailed output.
# With survey weights
efa(survey_data,
political_orientation, environmental_concern, life_satisfaction,
trust_government, trust_media, trust_science,
weights = sampling_weight)
#> Exploratory Factor Analysis: 6 items, 3 components (PCA/Varimax) [Weighted]
#> KMO = 0.505 (Miserable), Variance explained: 61.0%, N = 2182 (smallest pairwise)
#> Use summary() for detailed output.
# Grouped by region
survey_data %>%
group_by(region) %>%
efa(political_orientation, environmental_concern, life_satisfaction,
trust_government, trust_media, trust_science)
#> [region = East]
#> Exploratory Factor Analysis: 6 items, 3 components (PCA/Varimax)
#> KMO = 0.475 (Unacceptable), Variance explained: 62.1%, N = 419 (smallest pairwise)
#> [region = West]
#> Exploratory Factor Analysis: 6 items, 3 components (PCA/Varimax)
#> KMO = 0.505 (Miserable), Variance explained: 61.3%, N = 1749 (smallest pairwise)
#> Use summary() for detailed output.
# --- Three-layer output ---
result <- efa(survey_data, political_orientation, environmental_concern,
life_satisfaction, trust_government, trust_media, trust_science)
result # compact overview
#> Exploratory Factor Analysis: 6 items, 3 components (PCA/Varimax)
#> KMO = 0.505 (Miserable), Variance explained: 61.0%, N = 2168 (smallest pairwise)
#> Use summary() for detailed output.
summary(result) # full detailed output with all sections
#>
#> Exploratory Factor Analysis (PCA, Varimax) Results
#> --------------------------------------------------
#> - Variables:
#> political_orientation Political orientation (1=left, 5=right)
#> environmental_concern Environmental concern (1=low, 5=high)
#> life_satisfaction Life satisfaction (1=dissatisfied, 5=satisfied)
#> trust_government Trust in government (1=none, 5=complete)
#> trust_media Trust in media (1=none, 5=complete)
#> trust_science Trust in science (1=none, 5=complete)
#> - Extraction: Principal Component Analysis
#> - Rotation: Varimax with Kaiser Normalization
#> - N of Components: 3
#> - N (smallest pairwise): 2168
#>
#> Descriptive Statistics
#> -------------------------------------------------------------------
#> Variable Mean Std. Deviation Analysis N Missing N
#> -------------------------------------------------------------------
#> political_orientation 2.722 1.086 2299 201
#> environmental_concern 3.573 1.194 2400 100
#> life_satisfaction 3.628 1.153 2421 79
#> trust_government 2.621 1.163 2354 146
#> trust_media 2.452 1.163 2367 133
#> trust_science 3.641 1.028 2398 102
#> -------------------------------------------------------------------
#>
#> KMO and Bartlett's Test
#> ----------------------------------------
#> Kaiser-Meyer-Olkin Measure: 0.505
#> Bartlett's Chi-Square: 932.068
#> df: 15
#> Sig.: <.001
#>
#> Communalities
#> ---------------------------------------------------------------------
#> Variable Initial Extraction
#> ---------------------------------------------------------------------
#> political_orientation Political orientation ... 1.000 0.786
#> environmental_concern Environmental concern ... 1.000 0.783
#> life_satisfaction Life satisfaction (1=d... 1.000 0.668
#> trust_government Trust in government (1... 1.000 0.347
#> trust_media Trust in media (1=none... 1.000 0.475
#> trust_science Trust in science (1=no... 1.000 0.598
#> ---------------------------------------------------------------------
#> Extraction Method: Principal Component Analysis.
#>
#> Total Variance Explained
#> -------------------------------------------------------------------------
#> Initial Eigenvalues Extraction Sums Rotation Sums
#> Component Total % Var. Cum. % Total % Var. Cum. % Total % Var. Cum. %
#> -------------------------------------------------------------------------
#> 1 1.600 26.666 26.666 1.600 26.666 26.666 1.598 26.635 26.635
#> 2 1.041 17.358 44.024 1.041 17.358 44.024 1.039 17.324 43.959
#> 3 1.017 16.955 60.979 1.017 16.955 60.979 1.021 17.020 60.979
#> 4 0.980 16.334 77.313
#> 5 0.949 15.814 93.127
#> 6 0.412 6.873 100.000
#> -------------------------------------------------------------------------
#> Sums = sums of squared loadings.
#> Extraction Method: Principal Component Analysis.
#>
#> Component Matrix (unrotated)
#> ----------------------------------------
#> PC1 PC2 PC3
#> political_orientation Political orientation (1=lef... -0.885
#> environmental_concern Environmental concern (1=low... 0.885
#> trust_science Trust in science (1=none, 5=... 0.672
#> trust_government Trust in government (1=none,... 0.547
#> trust_media Trust in media (1=none, 5=co... 0.524 0.448
#> life_satisfaction Life satisfaction (1=dissati... 0.809
#> Extraction Method: Principal Component Analysis.
#>
#> Rotated Component Matrix
#> ----------------------------------------
#> PC1 PC2 PC3
#> political_orientation Political orientation (1=lef... -0.887
#> environmental_concern Environmental concern (1=low... 0.884
#> trust_science Trust in science (1=none, 5=... 0.762
#> trust_government Trust in government (1=none,... 0.566
#> life_satisfaction Life satisfaction (1=dissati... 0.789
#> trust_media Trust in media (1=none, 5=co... 0.620
#> Extraction Method: Principal Component Analysis.
#> Rotation Method: Varimax with Kaiser Normalization.
#> Rotation converged in 4 iterations.
summary(result, communalities = FALSE) # hide communalities table
#>
#> Exploratory Factor Analysis (PCA, Varimax) Results
#> --------------------------------------------------
#> - Variables:
#> political_orientation Political orientation (1=left, 5=right)
#> environmental_concern Environmental concern (1=low, 5=high)
#> life_satisfaction Life satisfaction (1=dissatisfied, 5=satisfied)
#> trust_government Trust in government (1=none, 5=complete)
#> trust_media Trust in media (1=none, 5=complete)
#> trust_science Trust in science (1=none, 5=complete)
#> - Extraction: Principal Component Analysis
#> - Rotation: Varimax with Kaiser Normalization
#> - N of Components: 3
#> - N (smallest pairwise): 2168
#>
#> Descriptive Statistics
#> -------------------------------------------------------------------
#> Variable Mean Std. Deviation Analysis N Missing N
#> -------------------------------------------------------------------
#> political_orientation 2.722 1.086 2299 201
#> environmental_concern 3.573 1.194 2400 100
#> life_satisfaction 3.628 1.153 2421 79
#> trust_government 2.621 1.163 2354 146
#> trust_media 2.452 1.163 2367 133
#> trust_science 3.641 1.028 2398 102
#> -------------------------------------------------------------------
#>
#> KMO and Bartlett's Test
#> ----------------------------------------
#> Kaiser-Meyer-Olkin Measure: 0.505
#> Bartlett's Chi-Square: 932.068
#> df: 15
#> Sig.: <.001
#>
#> Total Variance Explained
#> -------------------------------------------------------------------------
#> Initial Eigenvalues Extraction Sums Rotation Sums
#> Component Total % Var. Cum. % Total % Var. Cum. % Total % Var. Cum. %
#> -------------------------------------------------------------------------
#> 1 1.600 26.666 26.666 1.600 26.666 26.666 1.598 26.635 26.635
#> 2 1.041 17.358 44.024 1.041 17.358 44.024 1.039 17.324 43.959
#> 3 1.017 16.955 60.979 1.017 16.955 60.979 1.021 17.020 60.979
#> 4 0.980 16.334 77.313
#> 5 0.949 15.814 93.127
#> 6 0.412 6.873 100.000
#> -------------------------------------------------------------------------
#> Sums = sums of squared loadings.
#> Extraction Method: Principal Component Analysis.
#>
#> Component Matrix (unrotated)
#> ----------------------------------------
#> PC1 PC2 PC3
#> political_orientation Political orientation (1=lef... -0.885
#> environmental_concern Environmental concern (1=low... 0.885
#> trust_science Trust in science (1=none, 5=... 0.672
#> trust_government Trust in government (1=none,... 0.547
#> trust_media Trust in media (1=none, 5=co... 0.524 0.448
#> life_satisfaction Life satisfaction (1=dissati... 0.809
#> Extraction Method: Principal Component Analysis.
#>
#> Rotated Component Matrix
#> ----------------------------------------
#> PC1 PC2 PC3
#> political_orientation Political orientation (1=lef... -0.887
#> environmental_concern Environmental concern (1=low... 0.884
#> trust_science Trust in science (1=none, 5=... 0.762
#> trust_government Trust in government (1=none,... 0.566
#> life_satisfaction Life satisfaction (1=dissati... 0.789
#> trust_media Trust in media (1=none, 5=co... 0.620
#> Extraction Method: Principal Component Analysis.
#> Rotation Method: Varimax with Kaiser Normalization.
#> Rotation converged in 4 iterations.