Exploratory Factor Analysis: Formula, Verified Results, Charts and Interpretation
Exploratory factor analysis models observed variables as functions of a smaller set of common factors plus unique variance while allowing the loading pattern to be discovered. Extraction, retention, and rotation are separate decisions and must all be documented. This guide uses the supplied real-data results, native MathML equations, matching charts, and separate Python, R, SPSS or AMOS, and Excel verification.
Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Horn retained factors = 3 is retained as a distinct supporting quantity for the common-factor exploration; it is not substituted for the primary result.
What Exploratory Factor Analysis measures
The exact estimand and the result this method is allowed to support.
Exploratory Factor Analysis addresses one defined analytical target: Exploratory factor analysis models observed variables as functions of a smaller set of common factors plus unique variance while allowing the loading pattern to be discovered. Extraction, retention, and rotation are separate decisions and must all be documented.
Quantity estimated in this analysis
The common-factor exploration is reconstructed from the exact variables, matrix, model, panel, or resampling design shown below. The primary output is Horn retained factors = 3; PAF iterations = 374 supplies the first supporting check. Horn retained factors = 3 is retained as a distinct supporting quantity for the common-factor exploration; it is not substituted for the primary result.
For Exploratory Factor Analysis, the calculation retains full precision until the final display. That matters because the software reports, spreadsheet formulas, chart labels, and narrative must refer to one identical result rather than separately rounded approximations.
Interpretation that is not permitted
EFA is not PCA, and an exploratory solution is not automatically confirmatory because the same sample suggested and evaluated the structure. Rotation does not improve model fit; it changes the coordinate representation of the retained factor space.
For Exploratory Factor Analysis, this boundary is substantive. A nearby coefficient may share the same data or model, yet it answers a different question. The article therefore names every supporting statistic instead of using broad labels such as “valid,” “good,” or “significant” without the object being evaluated.
When to use Exploratory Factor Analysis
Research scope, neighboring methods, and excluded claims.
Research question answered
The defensible question is whether the common-factor exploration supports the result stated for the declared dataset and analytical specification. It is answered by confirm KMO and Bartlett before extraction, followed by use common-factor extraction rather than the PCA default. The evidence is bounded by Horn retained factors = 3 and its named companion quantities.
For Exploratory Factor Analysis, changing the case set, expert panel, item block, estimator, factor count, rotation, baseline model, bootstrap design, or criterion definition changes the question. Such a change requires a new result rather than a revision of the wording around the old value.
Nearest methods that answer different questions
Principal Component Analysis: PCA decomposes total variance; EFA separates common and unique variance.
Confirmatory Factor Analysis: CFA tests a prespecified loading structure rather than exploring broad cross-loadings.
These distinctions determine which formula, output table, and chart can legitimately appear in a Exploratory Factor Analysis post.
Real data used for Exploratory Factor Analysis
Variables, coding, sample or panel size, and the role each input plays.
For Exploratory Factor Analysis, the factor-oriented analysis uses 649 complete records and nine ordered variables. TravelAccess is defined as 5 − traveltime so larger values represent easier travel. The correlation matrix, eigenvalues, communalities, loading matrices, rotation output, and simulation cutoffs all preserve the same variable order.
For the common-factor exploration, the data are not merely background. A change in correlation type, standardization, missing-case rule, variable order, or retained dimension count changes the matrix on which Horn retained factors = 3 was obtained.
| Variable | Meaning | Mean | SD | Range | Construct |
|---|---|---|---|---|---|
| G1 | first-period grade | 11.3991 | 2.7453 | 0–19 | Academic Achievement |
| G2 | second-period grade | 11.5701 | 2.9136 | 0–19 | Academic Achievement |
| G3 | final grade | 11.9060 | 3.2307 | 0–19 | Academic Achievement |
| Medu | mother’s education | 2.5146 | 1.1346 | 0–4 | Educational Advantage |
| Fedu | father’s education | 2.3066 | 1.0999 | 0–4 | Educational Advantage |
| TravelAccess | reverse-coded travel accessibility | 3.4314 | 0.7487 | 1–4 | Educational Advantage |
| goout | frequency of going out | 3.1849 | 1.1758 | 1–5 | Social-Alcohol Exposure |
| Dalc | workday alcohol use | 1.5023 | 0.9248 | 1–5 | Social-Alcohol Exposure |
| Walc | weekend alcohol use | 2.2804 | 1.2844 | 1–5 | Social-Alcohol Exposure |
Exploratory Factor Analysis assumptions and design requirements
Six conditions checked before the coefficient or decision rule is interpreted.
1. The selected variables support meaningful correlations
This condition determines whether the input object matches the formula. In the current Exploratory Factor Analysis analysis, the check is to confirm KMO and Bartlett before extraction while preserving Horn retained factors = 3.
For Exploratory Factor Analysis, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.
2. Observations are independent
This requirement controls whether the numerical estimate has the interpretation claimed. In the current Exploratory Factor Analysis analysis, the check is to use common-factor extraction rather than the PCA default while preserving PAF iterations = 374.
For Exploratory Factor Analysis, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.
3. The common-factor model is plausible
This design condition prevents an attractive coefficient from being attached to the wrong population or model. In the current Exploratory Factor Analysis analysis, the check is to retain three factors from parallel analysis while preserving Overall KMO = 0.713439.
For Exploratory Factor Analysis, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.
4. The extraction method converges
This specification rule keeps the software routes numerically comparable. In the current Exploratory Factor Analysis analysis, the check is to inspect pattern and structure matrices under oblique rotation while preserving Bartlett chi-square = 3018.238.
For Exploratory Factor Analysis, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.
5. Factor count is justified independently
This diagnostic requirement is checked before a benchmark is applied. In the current Exploratory Factor Analysis analysis, the check is to review communalities and Heywood cases while preserving TravelAccess communality = 0.096087.
For Exploratory Factor Analysis, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.
6. The rotation matches expected factor correlations
This final condition governs whether the conclusion can survive replication or sensitivity analysis. In the current Exploratory Factor Analysis analysis, the check is to cross-validate the exploratory structure in a separate CFA sample while preserving G2 communality = 0.959989.
For Exploratory Factor Analysis, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.
Exploratory Factor Analysis hypotheses or decision rule
The statistical question is stated at the correct level for this method.
Statistical question
The primary question concerns factorability, reproduced variance, loading structure, or component retention as defined by Exploratory Factor Analysis; no universal significance test covers all of those quantities.
For Exploratory Factor Analysis, where inferential tests exist, they are reported separately from descriptive coefficients and retention rules.
Decision for the worked analysis
The calculation yields Horn retained factors = 3. Horn retained factors = 3 is retained as a distinct supporting quantity for the common-factor exploration; it is not substituted for the primary result.
Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Exploratory Factor Analysis formula and worked substitution
Native MathML preserves fractions, roots, summations, matrices, subscripts, and superscripts.
The equation below is the defining mathematical object for Exploratory Factor Analysis. Its symbols are connected to the saved inputs and to Horn retained factors = 3, PAF iterations = 374, Overall KMO = 0.713439, Bartlett chi-square = 3018.238.
Communalities are sums of squared loadings across retained common factors.
The three-factor solution is supported by parallel analysis but contains one clearly weak indicator.
Symbol and denominator control
Exploratory factor analysis models observed variables as functions of a smaller set of common factors plus unique variance while allowing the loading pattern to be discovered. Extraction, retention, and rotation are separate decisions and must all be documented.
For Exploratory Factor Analysis, the numerator, denominator, matrix order, degrees of freedom, factor count, or panel size shown in the MathML card is retained exactly. A formula from a neighboring method is not substituted even when both produce values on a similar scale.
Full-precision substitution
The spreadsheet and software outputs retain unrounded inputs until the final displayed value. The arithmetic is then reconciled with Horn retained factors = 3 and PAF iterations = 374.
EFA is not PCA, and an exploratory solution is not automatically confirmatory because the same sample suggested and evaluated the structure. Rotation does not improve model fit; it changes the coordinate representation of the retained factor space.
Step-by-step Exploratory Factor Analysis calculation
Every stage is tied to a saved value and a method-specific condition.
The worked calculation follows six operations specific to the common-factor exploration. Each operation produces a quantity used by the next step, so a discrepancy is resolved where it originates rather than hidden by rounding.
Establish the analytical object
Action: Confirm KMO and Bartlett before extraction.
Numerical trace: Horn retained factors = 3; PAF iterations = 374.
Condition: the selected variables support meaningful correlations. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Reconstruct the first required quantity
Action: Use common-factor extraction rather than the PCA default.
Numerical trace: PAF iterations = 374; Overall KMO = 0.713439.
For Exploratory Factor Analysis, condition: observations are independent. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Verify the companion quantity
Action: Retain three factors from parallel analysis.
Numerical trace: Overall KMO = 0.713439; Bartlett chi-square = 3018.238.
Condition: the common-factor model is plausible. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Apply the decision rule
Action: Inspect pattern and structure matrices under oblique rotation.
Numerical trace: Bartlett chi-square = 3018.238; TravelAccess communality = 0.096087.
Condition: the extraction method converges. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Inspect local evidence
Action: Review communalities and Heywood cases.
Numerical trace: TravelAccess communality = 0.096087; G2 communality = 0.959989.
Condition: factor count is justified independently. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Reconcile and report
Action: Cross-validate the exploratory structure in a separate CFA sample.
Numerical trace: G2 communality = 0.959989; Oblimin G2 dominant loading = -0.983154.
Condition: the rotation matches expected factor correlations. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Exploratory Factor Analysis results and interpretation
Primary and supporting statistics are kept separate and precisely labeled.
Primary result
Horn retained factors
Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Why the result is internally coherent
Horn retained factors = 3 is retained as a distinct supporting quantity for the common-factor exploration; it is not substituted for the primary result.
PAF iterations = 374 documents simulation or convergence effort rather than substantive magnitude.
For Exploratory Factor Analysis, the two quantities are reported together because one is primary and the other supplies context; neither is renamed as the other.
| Result item | Exact value | Interpretation restricted to this method |
|---|---|---|
| Horn retained factors | 3 | Horn retained factors = 3 is retained as a distinct supporting quantity for the common-factor exploration; it is not substituted for the primary result. |
| PAF iterations | 374 | PAF iterations = 374 documents simulation or convergence effort rather than substantive magnitude. |
| Overall KMO | 0.713439 | Overall KMO = 0.713439 describes the balance between ordinary and partial correlations, so it informs factorability or item adequacy rather than factor retention. |
| Bartlett chi-square | 3018.238 | Bartlett chi-square = 3018.238 is read with its degrees of freedom, estimator, sample size, and p-value; it is not a stand-alone effect size. |
| TravelAccess communality | 0.096087 | TravelAccess communality = 0.096087 is below the .50 captured-variance reference; the judgment applies to the named construct rather than the whole instrument. |
| G2 communality | 0.959989 | G2 communality = 0.959989 is the proportion of that variable’s variance reproduced by the retained common factors, not an instrument reliability coefficient. |
| Oblimin G2 dominant loading | -0.983154 | Oblimin G2 dominant loading = -0.983154 is tied to a named indicator and matrix; its sign, standardization, primary dimension, and cross-coefficients must remain explicit. |
| Oblimin Walc dominant loading | -0.976024 | Oblimin Walc dominant loading = -0.976024 is tied to a named indicator and matrix; its sign, standardization, primary dimension, and cross-coefficients must remain explicit. |
| Oblimin Medu dominant loading | 0.910157 | Oblimin Medu dominant loading = 0.910157 is tied to a named indicator and matrix; its sign, standardization, primary dimension, and cross-coefficients must remain explicit. |
| Eigenvalue 1 | 3.195831 | Eigenvalue 1 = 3.195831 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Eigenvalue 2 | 1.817089 | Eigenvalue 2 = 1.817089 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Eigenvalue 3 | 1.393698 | Eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Eigenvalue 4 | 0.846560 | Eigenvalue 4 = 0.846560 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Three-dimension cumulative variance | 71.1846% | Three-dimension cumulative variance = 71.1846% belongs to the declared matrix and retained dimensions and must not be relabeled as model fit. |
Exploratory Factor Analysis in Python
The Python route calculates or reconstructs the exact named result.
The Python workflow uses factor_analyzer, FactorAnalyzer to calculate or extract the common-factor exploration from the declared data and analytical specification. It must reproduce Horn retained factors = 3 and retain PAF iterations = 374 as a separate supporting quantity.
The code is read as an executable analysis, not as a printed answer. Its critical verification is to confirm KMO and Bartlett before extraction; the associated design condition is that the selected variables support meaningful correlations. EFA is not PCA, and an exploratory solution is not automatically confirmatory because the same sample suggested and evaluated the structure. Rotation does not improve model fit; it changes the coordinate representation of the retained factor space.
import pandas as pd
import numpy as npdf = pd.read_csv("student-por.csv", sep=";")
df["TravelAccess"] = 5 - df["traveltime"]
vars9 = ["G1","G2","G3","Medu","Fedu","TravelAccess","goout","Dalc","Walc"]
X = df[vars9].dropna()
from factor_analyzer import FactorAnalyzer
fa=FactorAnalyzer(n_factors=3,method="principal",rotation="oblimin")
fa.fit(X)
print("loadings",fa.loadings_)
print("communalities",fa.get_communalities())
print("uniquenesses",fa.get_uniquenesses())
Exploratory Factor Analysis in R
The R route declares package, estimator, extraction, rotation, or resampling settings.
The R route uses psych and the displayed arguments to estimate the common-factor exploration. Package defaults are made explicit because estimator, matrix type, extraction, rotation, baseline, or bootstrap choices can change the result.
R output is reconciled with Horn retained factors = 3 after the analyst use common-factor extraction rather than the PCA default. Agreement is expected only when the case set, variable order, and method settings match the Python and workbook calculations.
d <- read.csv2("student-por.csv")
d$TravelAccess <- 5 - d$traveltime
vars9 <- c("G1","G2","G3","Medu","Fedu","TravelAccess","goout","Dalc","Walc")
X <- d[vars9]
library(psych)
fit <- fa(X,nfactors=3,fm="pa",rotate="oblimin")
print(fit$loadings,cutoff=0); print(fit$communality); print(fit$Phi)Exploratory Factor Analysis in SPSS or AMOS
The procedure is labeled honestly when base SPSS does not expose the coefficient.
The SPSS or AMOS section shows the procedure that is actually available for the common-factor exploration. When base SPSS does not expose the coefficient, the syntax prepares the correct matrix or model and the coefficient is obtained through AMOS, MATRIX operations, or a validated integration rather than by renaming a different test.
The output must identify Horn retained factors = 3 and the settings needed to reproduce it. The software review specifically retain three factors from parallel analysis, while preserving the requirement that the common-factor model is plausible.
COMPUTE TravelAccess = 5 - traveltime.
EXECUTE.
FACTOR
/VARIABLES G1 G2 G3 Medu Fedu TravelAccess goout Dalc Walc
/MISSING LISTWISE
/PRINT INITIAL KMO EXTRACTION ROTATION
/PLOT EIGEN
/CRITERIA FACTORS(3) ITERATE(500)
/EXTRACTION PAF
/ROTATION OBLIMIN
/METHOD=CORRELATION.
* Read only the Exploratory Factor Analysis evidence identified in this post.Exploratory Factor Analysis in Excel
The workbook exposes source values, intermediate arithmetic, and the final formula.
The Excel workbook is an arithmetic audit for the common-factor exploration. Named cells retain the inputs, intermediate components, and final formula leading to Horn retained factors = 3; no rounded constant is pasted over a formula cell.
Excel can verify visible calculations and cross-software agreement, but it does not replace estimation, optimization, rotation, or resampling that must occur in statistical software. The workbook therefore focuses on the check to inspect pattern and structure matrices under oblique rotation and documents PAF iterations = 374 independently.
Data: 649 rows with documented coding.
Inputs: named cells or ranges required only by Exploratory Factor Analysis.
Calculation: Use the native MathML formula shown above with named ranges for every input
Audit: compare full-precision Excel output with the Python, R, and SPSS/AMOS values.
Decision: reference the exact result and diagnostics; never paste a rounded value over the formula cell.Exploratory Factor Analysis charts and visual diagnostics
Each supplied image is interpreted through its own values and analytical purpose.
Every image below is interpreted as part of the same Exploratory Factor Analysis analysis. The captions identify what the panel contributes, the exact values visible in the result set, and the condition that would invalidate the reading.
01 Exploratory-Factor-Analysis Primary Metrics
This panel reconciles the headline estimate with its principal supporting values for Exploratory Factor Analysis. Read Horn retained factors = 3 beside PAF iterations = 374; the first quantity is not replaced by the second.
The chart is used to confirm KMO and Bartlett before extraction. Its interpretation remains valid only when the selected variables support meaningful correlations. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
02 Exploratory-Factor-Analysis Principal Axis Unrotated
This panel provides a visual diagnostic tied to the method’s exact decision rule for Exploratory Factor Analysis. Read PAF iterations = 374 beside Overall KMO = 0.713439; the first quantity is not replaced by the second.
The chart is used to use common-factor extraction rather than the PCA default. Its interpretation remains valid only when observations are independent. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
03 Exploratory-Factor-Analysis Oblimin Pattern
This panel provides a visual diagnostic tied to the method’s exact decision rule for Exploratory Factor Analysis. Read Overall KMO = 0.713439 beside Bartlett chi-square = 3018.238; the first quantity is not replaced by the second.
The chart is used to retain three factors from parallel analysis. Its interpretation remains valid only when the common-factor model is plausible. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
04 Exploratory-Factor-Analysis Efa Factorability
This panel provides a visual diagnostic tied to the method’s exact decision rule for Exploratory Factor Analysis. Read Bartlett chi-square = 3018.238 beside TravelAccess communality = 0.096087; the first quantity is not replaced by the second.
The chart is used to inspect pattern and structure matrices under oblique rotation. Its interpretation remains valid only when the extraction method converges. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
05 Exploratory-Factor-Analysis Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for Exploratory Factor Analysis. Read TravelAccess communality = 0.096087 beside G2 communality = 0.959989; the first quantity is not replaced by the second.
The chart is used to review communalities and Heywood cases. Its interpretation remains valid only when factor count is justified independently. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
01 Exploratory-Factor-Analysis Primary Metrics
This panel reconciles the headline estimate with its principal supporting values for Exploratory Factor Analysis. Read G2 communality = 0.959989 beside Oblimin G2 dominant loading = -0.983154; the first quantity is not replaced by the second.
The chart is used to cross-validate the exploratory structure in a separate CFA sample. Its interpretation remains valid only when the rotation matches expected factor correlations. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
02 Exploratory-Factor-Analysis Principal Axis Unrotated
This panel provides a visual diagnostic tied to the method’s exact decision rule for Exploratory Factor Analysis. Read Oblimin G2 dominant loading = -0.983154 beside Oblimin Walc dominant loading = -0.976024; the first quantity is not replaced by the second.
The chart is used to confirm KMO and Bartlett before extraction. Its interpretation remains valid only when the selected variables support meaningful correlations. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
03 Exploratory-Factor-Analysis Oblimin Pattern
This panel provides a visual diagnostic tied to the method’s exact decision rule for Exploratory Factor Analysis. Read Oblimin Walc dominant loading = -0.976024 beside Oblimin Medu dominant loading = 0.910157; the first quantity is not replaced by the second.
The chart is used to use common-factor extraction rather than the PCA default. Its interpretation remains valid only when observations are independent. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
04 Exploratory-Factor-Analysis Efa Factorability
This panel provides a visual diagnostic tied to the method’s exact decision rule for Exploratory Factor Analysis. Read Oblimin Medu dominant loading = 0.910157 beside Eigenvalue 1 = 3.195831; the first quantity is not replaced by the second.
The chart is used to retain three factors from parallel analysis. Its interpretation remains valid only when the common-factor model is plausible. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
05 Exploratory-Factor-Analysis Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for Exploratory Factor Analysis. Read Eigenvalue 1 = 3.195831 beside Eigenvalue 2 = 1.817089; the first quantity is not replaced by the second.
The chart is used to inspect pattern and structure matrices under oblique rotation. Its interpretation remains valid only when the extraction method converges. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
Exploratory Factor Analysis verification and sensitivity analysis
Six failure modes are checked against the formula, data, output, and charts.
The following diagnostics are not a general checklist. Each one targets a failure mode that can change the calculation or interpretation of Exploratory Factor Analysis.
1. Confirm KMO and Bartlett before extraction
Begin by confirm KMO and Bartlett before extraction. For the common-factor exploration, this operation directly connects Horn retained factors = 3 with Overall KMO = 0.713439. Horn retained factors = 3 is retained as a distinct supporting quantity for the common-factor exploration; it is not substituted for the primary result.
The governing condition is that the selected variables support meaningful correlations. If it fails, the primary coefficient may be attached to the wrong input object. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Principal Component Analysis, because PCA decomposes total variance; EFA separates common and unique variance.
2. Use common-factor extraction rather than the PCA default
Next, use common-factor extraction rather than the PCA default. For the common-factor exploration, this operation directly connects PAF iterations = 374 with Bartlett chi-square = 3018.238. PAF iterations = 374 documents simulation or convergence effort rather than substantive magnitude.
The governing condition is that observations are independent. If it fails, the companion statistic may no longer describe the same model or sample. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Confirmatory Factor Analysis, because CFA tests a prespecified loading structure rather than exploring broad cross-loadings.
3. Retain three factors from parallel analysis
The third verification is to retain three factors from parallel analysis. For the common-factor exploration, this operation directly connects Overall KMO = 0.713439 with TravelAccess communality = 0.096087. Overall KMO = 0.713439 describes the balance between ordinary and partial correlations, so it informs factorability or item adequacy rather than factor retention.
The governing condition is that the common-factor model is plausible. If it fails, the decision boundary can move because the required quantity has changed. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Parallel Analysis, because Parallel analysis is a retention tool used within an EFA workflow, not a substitute for extraction and interpretation.
4. Inspect pattern and structure matrices under oblique rotation
After the core arithmetic is stable, inspect pattern and structure matrices under oblique rotation. For the common-factor exploration, this operation directly connects Bartlett chi-square = 3018.238 with G2 communality = 0.959989. Bartlett chi-square = 3018.238 is read with its degrees of freedom, estimator, sample size, and p-value; it is not a stand-alone effect size.
The governing condition is that the extraction method converges. If it fails, software agreement can be artificial if unlike definitions are compared. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Principal Component Analysis, because PCA decomposes total variance; EFA separates common and unique variance.
5. Review communalities and Heywood cases
A robustness review must review communalities and Heywood cases. For the common-factor exploration, this operation directly connects TravelAccess communality = 0.096087 with Oblimin G2 dominant loading = -0.983154. TravelAccess communality = 0.096087 is below the .50 captured-variance reference; the judgment applies to the named construct rather than the whole instrument.
The governing condition is that factor count is justified independently. If it fails, a favorable average can conceal a local failure. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Confirmatory Factor Analysis, because CFA tests a prespecified loading structure rather than exploring broad cross-loadings.
6. Cross-validate the exploratory structure in a separate CFA sample
The final reconciliation should cross-validate the exploratory structure in a separate CFA sample. For the common-factor exploration, this operation directly connects G2 communality = 0.959989 with Oblimin Walc dominant loading = -0.976024. G2 communality = 0.959989 is the proportion of that variable’s variance reproduced by the retained common factors, not an instrument reliability coefficient.
The governing condition is that the rotation matches expected factor correlations. If it fails, the published conclusion can exceed the evidence actually reproduced. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Parallel Analysis, because Parallel analysis is a retention tool used within an EFA workflow, not a substitute for extraction and interpretation.
| # | Verification operation | Condition protected | Saved quantity traced |
|---|---|---|---|
| 1 | confirm KMO and Bartlett before extraction | the selected variables support meaningful correlations | Horn retained factors = 3 |
| 2 | use common-factor extraction rather than the PCA default | observations are independent | PAF iterations = 374 |
| 3 | retain three factors from parallel analysis | the common-factor model is plausible | Overall KMO = 0.713439 |
| 4 | inspect pattern and structure matrices under oblique rotation | the extraction method converges | Bartlett chi-square = 3018.238 |
| 5 | review communalities and Heywood cases | factor count is justified independently | TravelAccess communality = 0.096087 |
| 6 | cross-validate the exploratory structure in a separate CFA sample | the rotation matches expected factor correlations | G2 communality = 0.959989 |
Exploratory Factor Analysis compared with related methods
Differences in estimand, formula, and conclusion determine the correct choice.
Method choice depends on the estimand, model, and data structure. These three comparisons explain why the post uses the Exploratory Factor Analysis formula and output rather than a nearby procedure.
Principal Component Analysis
PCA decomposes total variance; EFA separates common and unique variance.
In the current analysis, PAF iterations = 374 remains evidence for the common-factor exploration; it is not relabeled as a Principal Component Analysis result. PAF iterations = 374 documents simulation or convergence effort rather than substantive magnitude.
Confirmatory Factor Analysis
CFA tests a prespecified loading structure rather than exploring broad cross-loadings.
In the current analysis, Overall KMO = 0.713439 remains evidence for the common-factor exploration; it is not relabeled as a Confirmatory Factor Analysis result. Overall KMO = 0.713439 describes the balance between ordinary and partial correlations, so it informs factorability or item adequacy rather than factor retention.
Parallel Analysis
Parallel analysis is a retention tool used within an EFA workflow, not a substitute for extraction and interpretation.
In the current analysis, Bartlett chi-square = 3018.238 remains evidence for the common-factor exploration; it is not relabeled as a Parallel Analysis result. Bartlett chi-square = 3018.238 is read with its degrees of freedom, estimator, sample size, and p-value; it is not a stand-alone effect size.
How to report Exploratory Factor Analysis
A complete result paragraph includes the value, analytical object, settings, and limitation.
Results paragraph
Exploratory Factor Analysis was evaluated using the declared data, specification, and software settings. The primary result was Horn retained factors = 3; PAF iterations = 374 and Overall KMO = 0.713439 supplied supporting context. Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
The report then states the limitation explicitly: EFA is not PCA, and an exploratory solution is not automatically confirmatory because the same sample suggested and evaluated the structure. Rotation does not improve model fit; it changes the coordinate representation of the retained factor space.
Settings that must accompany the result
the selected variables support meaningful correlations; observations are independent; the common-factor model is plausible; the extraction method converges.
For Exploratory Factor Analysis, these details identify the exact version of the analysis and make cross-software reconciliation possible.
Verification actions retained in the record
confirm KMO and Bartlett before extraction; use common-factor extraction rather than the PCA default; retain three factors from parallel analysis; inspect pattern and structure matrices under oblique rotation.
The final wording is revised only after those operations reproduce the saved values.
Exploratory Factor Analysis decision scenarios
For Exploratory Factor Analysis, worked conflicts show how the conclusion changes when an input, assumption, or supporting statistic fails.
Boundary-case interpretation: Confirm KMO and Bartlett before extraction
Consider a review in which Horn retained factors = 3 is reproduced but PAF iterations = 374 is not. For the common-factor exploration, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to confirm KMO and Bartlett before extraction and verify that the selected variables support meaningful correlations.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Principal Component Analysis only for method selection: PCA decomposes total variance; EFA separates common and unique variance. The published conclusion remains Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Input-definition sensitivity: Use common-factor extraction rather than the PCA default
Consider a review in which Overall KMO = 0.713439 is reproduced but Bartlett chi-square = 3018.238 is not. For the common-factor exploration, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to use common-factor extraction rather than the PCA default and verify that observations are independent.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Confirmatory Factor Analysis only for method selection: CFA tests a prespecified loading structure rather than exploring broad cross-loadings. The published conclusion remains Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Software-definition reconciliation: Retain three factors from parallel analysis
Consider a review in which TravelAccess communality = 0.096087 is reproduced but G2 communality = 0.959989 is not. For the common-factor exploration, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to retain three factors from parallel analysis and verify that the common-factor model is plausible.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis is a retention tool used within an EFA workflow, not a substitute for extraction and interpretation. The published conclusion remains Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Local-chart conflict: Inspect pattern and structure matrices under oblique rotation
Consider a review in which Oblimin G2 dominant loading = -0.983154 is reproduced but Oblimin Walc dominant loading = -0.976024 is not. For the common-factor exploration, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to inspect pattern and structure matrices under oblique rotation and verify that the extraction method converges.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Principal Component Analysis only for method selection: PCA decomposes total variance; EFA separates common and unique variance. The published conclusion remains Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Alternative-method challenge: Review communalities and Heywood cases
Consider a review in which Oblimin Medu dominant loading = 0.910157 is reproduced but Eigenvalue 1 = 3.195831 is not. For the common-factor exploration, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to review communalities and Heywood cases and verify that factor count is justified independently.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Confirmatory Factor Analysis only for method selection: CFA tests a prespecified loading structure rather than exploring broad cross-loadings. The published conclusion remains Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Replication and reporting decision: Cross-validate the exploratory structure in a separate CFA sample
Consider a review in which Eigenvalue 2 = 1.817089 is reproduced but Eigenvalue 3 = 1.393698 is not. For the common-factor exploration, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to cross-validate the exploratory structure in a separate CFA sample and verify that the rotation matches expected factor correlations.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis is a retention tool used within an EFA workflow, not a substitute for extraction and interpretation. The published conclusion remains Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Boundary-case interpretation: Confirm KMO and Bartlett before extraction
Consider a review in which Eigenvalue 4 = 0.846560 is reproduced but Three-dimension cumulative variance = 71.1846% is not. For the common-factor exploration, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to confirm KMO and Bartlett before extraction and verify that the selected variables support meaningful correlations.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Principal Component Analysis only for method selection: PCA decomposes total variance; EFA separates common and unique variance. The published conclusion remains Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Input-definition sensitivity: Use common-factor extraction rather than the PCA default
Consider a review in which Parallel iterations = 500 is reproduced but Horn retained factors = 3 is not. For the common-factor exploration, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to use common-factor extraction rather than the PCA default and verify that observations are independent.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Confirmatory Factor Analysis only for method selection: CFA tests a prespecified loading structure rather than exploring broad cross-loadings. The published conclusion remains Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Software-definition reconciliation: Retain three factors from parallel analysis
Consider a review in which PAF iterations = 374 is reproduced but Overall KMO = 0.713439 is not. For the common-factor exploration, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to retain three factors from parallel analysis and verify that the common-factor model is plausible.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis is a retention tool used within an EFA workflow, not a substitute for extraction and interpretation. The published conclusion remains Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Local-chart conflict: Inspect pattern and structure matrices under oblique rotation
Consider a review in which Bartlett chi-square = 3018.238 is reproduced but TravelAccess communality = 0.096087 is not. For the common-factor exploration, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to inspect pattern and structure matrices under oblique rotation and verify that the extraction method converges.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Principal Component Analysis only for method selection: PCA decomposes total variance; EFA separates common and unique variance. The published conclusion remains Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Alternative-method challenge: Review communalities and Heywood cases
Consider a review in which G2 communality = 0.959989 is reproduced but Oblimin G2 dominant loading = -0.983154 is not. For the common-factor exploration, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to review communalities and Heywood cases and verify that factor count is justified independently.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Confirmatory Factor Analysis only for method selection: CFA tests a prespecified loading structure rather than exploring broad cross-loadings. The published conclusion remains Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Replication and reporting decision: Cross-validate the exploratory structure in a separate CFA sample
Consider a review in which Oblimin Walc dominant loading = -0.976024 is reproduced but Oblimin Medu dominant loading = 0.910157 is not. For the common-factor exploration, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to cross-validate the exploratory structure in a separate CFA sample and verify that the rotation matches expected factor correlations.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis is a retention tool used within an EFA workflow, not a substitute for extraction and interpretation. The published conclusion remains Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Boundary-case interpretation: Confirm KMO and Bartlett before extraction
Consider a review in which Eigenvalue 1 = 3.195831 is reproduced but Eigenvalue 2 = 1.817089 is not. For the common-factor exploration, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to confirm KMO and Bartlett before extraction and verify that the selected variables support meaningful correlations.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Principal Component Analysis only for method selection: PCA decomposes total variance; EFA separates common and unique variance. The published conclusion remains Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
Exploratory Factor Analysis downloads and reproducibility files
All linked files belong to the same analysis and remain on onlineinternetcafe.com.
The four files belong to one Exploratory Factor Analysis analysis. Their primary values, variable order, method settings, and chart labels must agree; a mismatch is resolved in the source calculation before the WordPress draft is published.
Exploratory Factor Analysis frequently asked questions
Answers use the worked result and the exact method boundary.
What does Exploratory Factor Analysis measure?
Exploratory factor analysis models observed variables as functions of a smaller set of common factors plus unique variance while allowing the loading pattern to be discovered. Extraction, retention, and rotation are separate decisions and must all be documented.
What is the main result in this Exploratory Factor Analysis analysis?
Horn retained factors = 3. Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.
What does the result not prove?
EFA is not PCA, and an exploratory solution is not automatically confirmatory because the same sample suggested and evaluated the structure. Rotation does not improve model fit; it changes the coordinate representation of the retained factor space.
Which supporting value should be reported with the primary result?
PAF iterations = 374 is the first companion quantity. PAF iterations = 374 documents simulation or convergence effort rather than substantive magnitude.
Which assumption is most likely to change the interpretation?
The first requirement is that the selected variables support meaningful correlations. The result is recomputed if that condition is not satisfied.
What is the most important numerical verification?
The analyst must confirm KMO and Bartlett before extraction. That operation traces Horn retained factors = 3 to the formula and saved inputs.
Why can software packages disagree on Exploratory Factor Analysis?
Disagreement can arise because observations are independent or because the packages implement different estimators, matrices, baselines, rotations, standardizations, bootstrap rules, or coefficient definitions. Matching labels alone is not enough.
How is Exploratory Factor Analysis different from Principal Component Analysis?
PCA decomposes total variance; EFA separates common and unique variance.
How should a chart be interpreted?
Each chart is tied to a named output such as Overall KMO = 0.713439. It supports a local calculation or diagnostic and does not replace the full numerical result.
How should Exploratory Factor Analysis be reported?
Report Horn retained factors = 3, the required supporting quantities, sample or panel size, exact method settings, and this qualified conclusion: Factorability is adequate and parallel analysis supports three factors. The solution is interpretable overall, but TravelAccess and goout have weak communalities that limit the corresponding factor definitions.