This calculator performs comprehensive Exploratory Factor Analysis (EFA), a statistical method used to identify underlying latent factors that explain patterns of correlations among observed variables. EFA is widely used in psychology, social sciences, and market research to understand construct validity and reduce data complexity.
What You'll Get:
- KMO Test: Kaiser-Meyer-Olkin measure of sampling adequacy to assess data suitability
- Bartlett's Test: Test of sphericity to check if variables are correlated
- Factor Loadings: Correlations between variables and extracted factors with rotation
- Communalities: Proportion of variance explained for each variable
- Variance Explained: How much variance each factor accounts for
- Scree Plot: Visual aid for determining optimal number of factors
- Factor Correlations: Relationships between factors (for oblique rotations)
- APA-Formatted Report: Professional statistical reporting ready for publication
💡 Pro Tip: Use KMO (> 0.5) and Bartlett's test (p < .05) to verify your data is suitable for factor analysis. Choose varimax rotation for independent factors or promax for correlated factors. For dimensionality reduction, consider Principal Component Analysis.
Software Implementation Differences
While factor analysis follows established statistical theory, different software packages may produce slightly different results due to implementation details:
- R (
psych::fa()) and Python (factor_analyzer) typically differ by less than 0.01 in factor loadings - Differences arise from varimax rotation algorithms, convergence criteria, and numerical precision
- SPSS, SAS, Stata, Mplus: May show similar small variations
- Important: These differences are statistically negligible and do not affect interpretation or conclusions
- Communalities, eigenvalues, and overall variance explained should match very closely across software
Ready to explore latent factors in your data? Load our sample dataset (psychological test scores) to see EFA in action, or upload your own data to discover the underlying structure in your variables.
Calculator
1. Load Your Data
2. Select Variables & Options
Selected: 0 of 0 variables
Leave empty to use eigenvalue > 1 criterion
Varimax for independent factors, Promax for correlated
ML is recommended for most analyses
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Definition
Exploratory Factor Analysis (EFA) is a multivariate statistical technique used to identify underlying latent factors that explain the pattern of correlations among a set of observed variables. Unlike confirmatory factor analysis, EFA does not require a priori hypotheses about the factor structure and is used for theory development and scale construction.
When to Use Factor Analysis
Use Factor Analysis when you want to:
- Identify latent constructs: Discover underlying dimensions in questionnaire or test data
- Scale development: Determine which items measure the same underlying construct
- Data reduction: Reduce many correlated variables to fewer factors while preserving information
- Validate theoretical structures: Test whether observed variables reflect hypothesized constructs
- Remove redundancy: Identify and eliminate variables that measure the same thing
EFA vs CFA vs PCA: Key Differences
| Aspect | EFA | CFA | PCA |
|---|---|---|---|
| Purpose | Seeks to explain correlations among variables using underlying latent factors. Separates shared variance from unique variance. | Tests a pre-specified factor structure based on theory. Confirms hypotheses about relationships between observed variables and latent factors. | Focuses on explaining total variance and creating orthogonal components. Does not distinguish shared from unique variance. |
| Approach | Exploratory - discovers underlying structure without prior hypotheses | Confirmatory - tests specific hypothesized factor structures | Descriptive - reduces dimensionality for data simplification |
| When to Use | Theory building, scale development, understanding construct validity | Theory testing, validating measurement models, assessing model fit to data | Data reduction, feature extraction, eliminating multicollinearity |
Interpreting Factor Loadings
Factor loadings represent the correlation between each variable and the factor. General guidelines for interpretation:
- >= 0.7: Excellent - Variable is a strong indicator of the factor
- 0.5 - 0.7: Good - Variable moderately relates to the factor
- 0.3 - 0.5: Fair - Variable weakly relates to the factor
- < 0.3: Poor - Consider removing the variable
- Cross-loadings: Variables loading highly on multiple factors may need revision
Rotation Methods
- Varimax: Most common; maximizes variance of squared loadings, produces clearer factor structure
- Quartimax: Simplifies variables rather than factors
- Promax: Faster than other oblique methods; good when factors are expected to correlate
Choose oblique rotation when you expect factors to be related (common in social sciences). Choose orthogonal when theoretical independence is important.
Example Code
R Code
library(psych)
library(tidyverse)
# Psychological test scores data
data <- tibble(
verbal = c(65, 72, 58, 68, 75, 62, 70, 66, 71, 69, 63, 74, 60, 67, 73),
numerical = c(62, 68, 55, 65, 71, 60, 68, 63, 70, 66, 61, 72, 58, 64, 70),
logical = c(68, 74, 60, 70, 78, 65, 73, 69, 75, 71, 66, 76, 62, 69, 77),
spatial = c(58, 65, 52, 62, 68, 56, 64, 60, 66, 63, 58, 67, 54, 61, 69),
memory = c(70, 76, 63, 72, 80, 68, 75, 71, 77, 73, 69, 79, 65, 72, 81)
)
# Check sampling adequacy
KMO(data)
# Bartlett's test
cortest.bartlett(cor(data), n = nrow(data))
# Perform EFA with varimax rotation
fa_result <- fa(data, nfactors = 2, rotate = "varimax", fm = "ml")
# View results
print(fa_result, digits = 3)
# Loadings
print(fa_result$loadings, cutoff = 0.3)
# Communalities
fa_result$communality
# Scree plot
scree(data, main = "Scree Plot")
# Factor scores
head(fa_result$scores)Python Code
import numpy as np
import pandas as pd
from factor_analyzer import FactorAnalyzer, calculate_kmo, calculate_bartlett_sphericity
import matplotlib.pyplot as plt
import seaborn as sns
# Psychological test scores data
data = pd.DataFrame({
'verbal': [65, 72, 58, 68, 75, 62, 70, 66, 71, 69, 63, 74, 60, 67, 73],
'numerical': [62, 68, 55, 65, 71, 60, 68, 63, 70, 66, 61, 72, 58, 64, 70],
'logical': [68, 74, 60, 70, 78, 65, 73, 69, 75, 71, 66, 76, 62, 69, 77],
'spatial': [58, 65, 52, 62, 68, 56, 64, 60, 66, 63, 58, 67, 54, 61, 69],
'memory': [70, 76, 63, 72, 80, 68, 75, 71, 77, 73, 69, 79, 65, 72, 81]
})
# KMO test
kmo_all, kmo_model = calculate_kmo(data)
print(f"KMO: {kmo_model:.3f}")
# Bartlett's test
chi_square_value, p_value = calculate_bartlett_sphericity(data)
print(f"Bartlett's test: χ² = {chi_square_value:.2f}, p = {p_value:.4f}")
# Determine number of factors (scree plot)
fa = FactorAnalyzer(n_factors=len(data.columns), rotation=None)
fa.fit(data)
eigenvalues, _ = fa.get_eigenvalues()
plt.figure(figsize=(8, 5))
plt.plot(range(1, len(eigenvalues) + 1), eigenvalues, 'bo-')
plt.axhline(y=1, color='r', linestyle='--', label='Kaiser criterion')
plt.xlabel('Factor')
plt.ylabel('Eigenvalue')
plt.title('Scree Plot')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()
# Perform factor analysis with varimax rotation
fa = FactorAnalyzer(n_factors=2, rotation='varimax', method='ml')
fa.fit(data)
# Loadings
loadings = pd.DataFrame(
fa.loadings_,
index=data.columns,
columns=['Factor1', 'Factor2']
)
print("\nFactor Loadings:")
print(loadings)
# Communalities
communalities = fa.get_communalities()
print("\nCommunalities:")
print(pd.Series(communalities, index=data.columns))
# Variance explained
variance = fa.get_factor_variance()
print("\nVariance Explained:")
print(pd.DataFrame(variance,
index=['SS Loadings', 'Proportion Var', 'Cumulative Var'],
columns=['Factor1', 'Factor2']))