Exploratory Factor Analysis (EFA)

• You already know which items should load on which factor Confirmatory Factor Analysis (CFA) →
• You want to check that a translated scale has the original's factors Confirmatory Factor Analysis (CFA) →
• You want the reliability of a scale you've already settled McDonald's Omega →
• You want to sort people into groups, not items into factors Cluster Analysis →
• You want to know how items relate to a trait level and difficulty Item Response Theory (IRT) →
• You have only two or three variables to relate Correlation and Association →

• Don'tUse principal components analysis and call it factor analysis.

  InsteadPCA summarizes all of each item's variance, unique part included; factor analysis models only the shared part. Use maximum likelihood or principal axis factoring, and name it.

• Don'tIgnore a Heywood case, a communality of 1 or more, or a failure to converge.

  InsteadIt usually means too many factors, or a factor with fewer than three items. Try fewer factors.

• Don'tName a factor by its strongest item and treat the name as proven.

  InsteadName it cautiously from all its items, and test the structure in new data with a confirmatory factor analysis.

• Don'tDelete every item with a cross-loading.

  InsteadLook at its content first. An item that belongs to two related traits may still be a good item. Score it on the factor it loads on most, or leave it out of the subscales, and say which.

• Don'tRun the EFA and a confirmatory factor analysis on the same sample and call the CFA a confirmation.

  InsteadSplit the sample, or confirm the structure in a new one.

• Don'tKeep every factor whose eigenvalue is above 1 because that's the software's default.

  InsteadDecide with parallel analysis and the scree, and check that each factor makes sense. Random data alone give leading eigenvalues above 1.

• Don'tUse varimax out of habit when the traits you're measuring are likely to be related.

  InsteadUse an oblique rotation such as oblimin or promax, and report the factor correlations.

Equation Form

Table 1. Pattern Matrix for the Twelve Wellbeing Items

Table 2. Eigenvalues and Parallel Analysis

Full Text (APA Style)

The equation form above, piece by piece.

What Your Data Should Look Like

The whole test, one idea at a time. Under each heading, a line says why the step matters, and the green note reads the equation in plain English with the wellbeing study's numbers. Collapse all to read just the headings: they tell the whole story on their own.

The code below reads the running example's data from data.csv; save it in the same folder you run the code from.

FACTOR
  /VARIABLES cheerful joyful content laugh absorbed lose_time focused
    challenge supported confide loved together
  /PRINT INITIAL EXTRACTION ROTATION KMO
  /FORMAT SORT BLANK(.30)
  /PLOT EIGEN
  /CRITERIA FACTORS(3) ITERATE(100) NOKAISER
  /EXTRACTION ML
  /ROTATION OBLIMIN(0).

library(psych)
wellbeing <- read.csv("data.csv")
items <- wellbeing[, -1]   # drop the person column

# 1. Enough shared variance to factor?
KMO(items)$MSA
cortest.bartlett(cor(items), n = nrow(items))

# 2. How many factors? Parallel analysis on principal components (Horn, 1965):
#    eigenvalues of 500 random-normal data sets the same size as ours
set.seed(2026)
random <- replicate(500, eigen(cor(matrix(rnorm(nrow(items) * ncol(items)), ncol = ncol(items))))$values)
round(data.frame(observed = eigen(cor(items))$values,
                 random_mean = rowMeans(random),
                 random_95 = apply(random, 1, quantile, 0.95))[1:6, ], 2)

# 3. Three factors by maximum likelihood, oblimin rotation
fit <- fa(items, nfactors = 3, fm = "ml", rotate = "oblimin")
print(fit$loadings, digits = 2, cutoff = 0.30)
round(fit$communality, 2)
round(fit$Phi, 2)
c(chisq = fit$STATISTIC, df = fit$dof, p = fit$PVAL)
max(abs(fit$residual[lower.tri(fit$residual)]))   # largest residual correlation

[1] 0.8156448
$chisq
[1] 1022.864

$p.value
[1] 1.50276e-171

$df
[1] 66

  observed random_mean random_95
1     3.71        1.34      1.42
2     1.97        1.25      1.31
3     1.54        1.18      1.23
4     0.68        1.12      1.16
5     0.66        1.07      1.10
6     0.60        1.01      1.05

Loadings:
          ML3   ML2   ML1
cheerful   0.71
joyful     0.78
content    0.66
laugh      0.59
absorbed         0.78
lose_time        0.71
focused          0.61
challenge        0.54
supported              0.77
confide                0.63
loved                  0.74
together   0.36        0.45

                ML3  ML2  ML1
SS loadings    2.05 1.79 1.76
Proportion Var 0.17 0.15 0.15
Cumulative Var 0.17 0.32 0.47
 cheerful    joyful   content     laugh  absorbed lose_time   focused challenge
     0.51      0.61      0.47      0.34      0.61      0.51      0.39      0.27
supported   confide     loved  together
     0.60      0.40      0.52      0.47
     ML3  ML2  ML1
ML3 1.00 0.41 0.33
ML2 0.41 1.00 0.16
ML1 0.33 0.16 1.00
     chisq         df          p
23.9253695 33.0000000  0.8760736
[1] 0.04272299

import numpy as np
import pandas as pd
from factor_analyzer import FactorAnalyzer
from factor_analyzer.factor_analyzer import calculate_kmo, calculate_bartlett_sphericity

wellbeing = pd.read_csv("data.csv")
items = wellbeing.drop(columns="person")
calculate_kmo(items)[1]                # 0.8156
calculate_bartlett_sphericity(items)   # (1022.86, 1.50e-171)

fa = FactorAnalyzer(n_factors=3, method="ml", rotation="oblimin").fit(items)
fa.get_eigenvalues()[0][:4].round(2)   # [3.71 1.97 1.54 0.68]
print(pd.DataFrame(fa.loadings_, index=items.columns).round(2))

# Communalities from the unrotated fit (see below)
print(FactorAnalyzer(n_factors=3, method="ml", rotation=None).fit(items).get_communalities().round(2))

              0     1     2
cheerful   0.71 -0.05  0.05
joyful     0.78  0.02 -0.03
content    0.66  0.06 -0.03
laugh      0.59 -0.01 -0.03
absorbed   0.01  0.78 -0.02
lose_time -0.02  0.71  0.04
focused    0.03  0.61  0.00
challenge -0.02  0.54 -0.05
supported  0.03  0.00  0.77
confide   -0.05  0.06  0.63
loved     -0.05 -0.04  0.74
together   0.36  0.06  0.45
[0.51 0.61 0.47 0.34 0.61 0.51 0.39 0.27 0.6  0.4  0.52 0.47]

None. This page estimates rather than tests, so there are no follow-up comparisons to run.

• Related DesignConfirmatory Factor Analysis (CFA): The same common-factor model with the structure fixed in advance: each item allowed to load only on the factors you specify, and the fit of that structure tested. EFA suggests a structure; CFA tests one, ideally in new data.
• Related DesignMcDonald's Omega: Omega is built on a one-factor model. Once EFA has shown which items form each factor, omega gives the reliability of each subscale's total score.
• Related DesignCronbach's Alpha: Alpha estimates a total score's reliability but says nothing about how many traits its items measure. Use EFA first to check that a subscale is one factor, then report alpha or omega for it.
• Related DesignCluster Analysis: Cluster analysis asks the same kind of question turned sideways: instead of grouping variables that people answer alike, it groups people who answer alike.
• AlternativeMultilevel SEM: An alternative if “Independent Observations” doesn't hold. Model the factors within and between clusters.

• Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30(2), 179–185. https://doi.org/10.1007/BF02289447

  “A procedure based on the generation of random variables is given for estimating the component which needs to be subtracted.” (Abstract)

• Fabrigar, L. R., Wegener, D. T., MacCallum, R. C., & Strahan, E. J. (1999). Evaluating the use of exploratory factor analysis in psychological research. Psychological Methods, 4(3), 272–299. https://doi.org/10.1037/1082-989X.4.3.272

  “Despite the widespread use of exploratory factor analysis in psychological research, researchers often make questionable decisions when conducting these analyses.” (Abstract)

• Costello, A. B., & Osborne, J. W. (2005). Best practices in exploratory factor analysis: Four recommendations for getting the most from your analysis. Practical Assessment, Research, and Evaluation, 10, Article 7. https://doi.org/10.7275/jyj1-4868

  “In particular, this paper provides practical information on making decisions regarding (a) extraction, (b) rotation, (c) the number of factors to interpret, and (d) sample size.” (Abstract)

• Stefana, A., Damiani, S., Granziol, U., Provenzani, U., Solmi, M., Youngstrom, E. A., & Fusar-Poli, P. (2025). Psychological, psychiatric, and behavioral sciences measurement scales: Best practice guidelines for their development and validation. Frontiers in Psychology, 15, Article 1494261. https://doi.org/10.3389/fpsyg.2024.1494261

  “A combination of theoretical reasoned reflection, parallel analysis (Horn, 1965) with replications of the simulated comparison data, and visual scree test (Cattell, 1966; Horn and Engstrom, 1979) should be used to determine the exact number of factors to retain” (Step 12: Extract the factors)

  “Items with factor loadings or slope coefficients less than 0.30 are deemed insufficient because they contribute less than 10% variance to the latent construct measured” (Step 13: Identify the best items)

  “Communalities are generally considered high if above 0.80, though 0.40–0.70 is more frequent in social sciences” (Step 11: Administer to an appropriately large and representative sample)

  “In the absence of such a theory, oblique rotations generally offer more accurate data representations, as psychological/psychiatric constructs are often interrelated.” (Rotation method)

  “As a first step, the suitability of the data for factor analysis must be evaluated using Bartlett’s sphericity test (p ≤ 0.05) and Kaiser-Meyer-Olkin measure of sampling adequacy (KMO ≥ 0.60)” (Step 12: Extract the factors)

• Osborne, J. W. (2015). What is rotating in exploratory factor analysis? Practical Assessment, Research, and Evaluation, 20, Article 2. https://doi.org/10.7275/hb2g-m060

  “orthogonal rotations are probably not a best practice, as oblique rotations can accurately model uncorrelated and correlated factors, whereas orthogonal rotations cannot handle correlated factors as effectively.” (p. 6)

  “Oblique rotations do not force factors to be correlated, and so in that instance, the factors would be allowed to assume a correlation of zero, and the solution would be the same as that of an orthogonal rotation.” (p. 5)

• Bernaards, C. A., & Jennrich, R. I. (2005). Gradient projection algorithms and software for arbitrary rotation criteria in factor analysis. Educational and Psychological Measurement, 65(5), 676–696. https://doi.org/10.1177/0013164404272507
• Revelle, W. (2026). psych: Procedures for psychological, psychometric, and personality research (R package version 2.6.5). Northwestern University. https://CRAN.R-project.org/package=psych

  “The eigen value of 1 rule, although the default for many programs, seems to be a rough way of dividing the number of variables by 3.” (nfactors help page, Details)

• Kaiser, H. F. (1970). A second generation little jiffy. Psychometrika, 35(4), 401–415. https://doi.org/10.1007/BF02291817
• Kaiser, H. F., & Rice, J. (1974). Little Jiffy, Mark IV. Educational and Psychological Measurement, 34(1), 111–117. https://doi.org/10.1177/001316447403400115