Parallel Analysis: Formula, Verified Results, Charts and Interpretation
Parallel analysis is a family of retention procedures that compares observed ordered eigenvalues with eigenvalues from random datasets matched on sample size and variable count. The result depends on whether mean or percentile roots and PCA or common-factor roots are used. This guide uses the supplied real-data results, native MathML equations, matching charts, and separate Python, R, SPSS or AMOS, and Excel verification.
Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Horn retained factors = 3 is retained as a distinct supporting quantity for the simulated-root retention analysis; it is not substituted for the primary result.
What Parallel Analysis measures
The exact estimand and the result this method is allowed to support.
Parallel Analysis addresses one defined analytical target: Parallel analysis is a family of retention procedures that compares observed ordered eigenvalues with eigenvalues from random datasets matched on sample size and variable count. The result depends on whether mean or percentile roots and PCA or common-factor roots are used.
Quantity estimated in this analysis
The simulated-root retention analysis is reconstructed from the exact variables, matrix, model, panel, or resampling design shown below. The primary output is Horn retained factors = 3; Observed eigenvalue 3 = 1.393698 supplies the first supporting check. Horn retained factors = 3 is retained as a distinct supporting quantity for the simulated-root retention analysis; it is not substituted for the primary result.
For Parallel 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
A “parallel analysis” result is incomplete unless the simulation method, correlation type, number of replications, quantile, n, p, and random seed are documented. It does not interpret the retained factors.
For Parallel 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 Parallel Analysis
Research scope, neighboring methods, and excluded claims.
Research question answered
The defensible question is whether the simulated-root retention analysis supports the result stated for the declared dataset and analytical specification. It is answered by state whether the analysis is PCA or common-factor based, followed by report all roots around the retention boundary. The evidence is bounded by Horn retained factors = 3 and its named companion quantities.
For Parallel 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
Horn’s Parallel Analysis: Horn’s post focuses on the named percentile implementation; this post emphasizes general implementation choices and reproducibility.
Scree Plot: Scree inspection is visual; parallel analysis uses simulated reference roots.
These distinctions determine which formula, output table, and chart can legitimately appear in a Parallel Analysis post.
Real data used for Parallel Analysis
Variables, coding, sample or panel size, and the role each input plays.
For Parallel 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 simulated-root retention analysis, 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 |
Parallel Analysis assumptions and design requirements
Six conditions checked before the coefficient or decision rule is interpreted.
1. Simulation dimensions match the observed data
This condition determines whether the input object matches the formula. In the current Parallel Analysis analysis, the check is to state whether the analysis is PCA or common-factor based while preserving Horn retained factors = 3.
For Parallel 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. The same matrix type is used
This requirement controls whether the numerical estimate has the interpretation claimed. In the current Parallel Analysis analysis, the check is to report all roots around the retention boundary while preserving Observed eigenvalue 3 = 1.393698.
For Parallel 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 quantile criterion is prespecified
This design condition prevents an attractive coefficient from being attached to the wrong population or model. In the current Parallel Analysis analysis, the check is to verify the third observed root exceeds its random cutoff while preserving Horn 95th percentile root 3 = 1.106209.
For Parallel 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. Enough replications stabilize the cutoff
This specification rule keeps the software routes numerically comparable. In the current Parallel Analysis analysis, the check is to verify the fourth observed root falls below its cutoff while preserving Observed eigenvalue 4 = 0.846560.
For Parallel 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. Eigenvalues are compared by rank
This diagnostic requirement is checked before a benchmark is applied. In the current Parallel Analysis analysis, the check is to repeat with more simulations as a stability check while preserving Horn 95th percentile root 4 = 1.064106.
For Parallel 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. A seed makes the result reproducible
This final condition governs whether the conclusion can survive replication or sensitivity analysis. In the current Parallel Analysis analysis, the check is to compare with theory and the scree plot while preserving Parallel iterations = 500.
For Parallel 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.
Parallel Analysis hypotheses or decision rule
The statistical question is stated at the correct level for this method.
Statistical question
For Parallel Analysis, the retention decision asks whether each observed ordered root remains larger than its adjacent or simulated reference value; it is not a single omnibus null hypothesis.
For Parallel Analysis, uncertainty is evaluated at the retention boundary, especially the last retained and first rejected dimensions.
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 simulated-root retention analysis; it is not substituted for the primary result.
Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Parallel Analysis formula and worked substitution
Native MathML preserves fractions, roots, summations, matrices, subscripts, and superscripts.
The equation below is the defining mathematical object for Parallel Analysis. Its symbols are connected to the saved inputs and to Horn retained factors = 3, Observed eigenvalue 3 = 1.393698, Horn 95th percentile root 3 = 1.106209, Observed eigenvalue 4 = 0.846560.
The random reference must reproduce the observed sample size, variable count, and correlation construction.
Permutation random-eigenvalue procedure retains three dimensions after 500 iterations.
Symbol and denominator control
Parallel analysis is a family of retention procedures that compares observed ordered eigenvalues with eigenvalues from random datasets matched on sample size and variable count. The result depends on whether mean or percentile roots and PCA or common-factor roots are used.
For Parallel 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
For Parallel Analysis, the spreadsheet and software outputs retain unrounded inputs until the final displayed value. The arithmetic is then reconciled with Horn retained factors = 3 and Observed eigenvalue 3 = 1.393698 .
A “parallel analysis” result is incomplete unless the simulation method, correlation type, number of replications, quantile, n, p, and random seed are documented. It does not interpret the retained factors.
Step-by-step Parallel Analysis calculation
Every stage is tied to a saved value and a method-specific condition.
The worked calculation follows six operations specific to the simulated-root retention analysis. 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: State whether the analysis is PCA or common-factor based.
Numerical trace: Horn retained factors = 3; Observed eigenvalue 3 = 1.393698.
Condition: simulation dimensions match the observed data. 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: Report all roots around the retention boundary.
For Parallel Analysis, numerical trace: Observed eigenvalue 3 = 1.393698; Horn 95th percentile root 3 = 1.106209.
Condition: the same matrix type is used. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Verify the companion quantity
Action: Verify the third observed root exceeds its random cutoff.
For Parallel Analysis, numerical trace: Horn 95th percentile root 3 = 1.106209; Observed eigenvalue 4 = 0.846560.
Condition: the quantile criterion is prespecified. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Apply the decision rule
Action: Verify the fourth observed root falls below its cutoff.
For Parallel Analysis, numerical trace: Observed eigenvalue 4 = 0.846560; Horn 95th percentile root 4 = 1.064106.
Condition: enough replications stabilize the cutoff. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Inspect local evidence
Action: Repeat with more simulations as a stability check.
Numerical trace: Horn 95th percentile root 4 = 1.064106; Parallel iterations = 500.
Condition: eigenvalues are compared by rank. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Reconcile and report
Action: Compare with theory and the scree plot.
Numerical trace: Parallel iterations = 500; Eigenvalue 1 = 3.195831.
Condition: a seed makes the result reproducible. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Parallel Analysis results and interpretation
Primary and supporting statistics are kept separate and precisely labeled.
Primary result
Horn retained factors
Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Why the result is internally coherent
Horn retained factors = 3 is retained as a distinct supporting quantity for the simulated-root retention analysis; it is not substituted for the primary result.
For Parallel Analysis, observed eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
For Parallel 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 simulated-root retention analysis; it is not substituted for the primary result. |
| Observed eigenvalue 3 | 1.393698 | Observed eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Horn 95th percentile root 3 | 1.106209 | Horn 95th percentile root 3 = 1.106209 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Observed eigenvalue 4 | 0.846560 | Observed eigenvalue 4 = 0.846560 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Horn 95th percentile root 4 | 1.064106 | Horn 95th percentile root 4 = 1.064106 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made. |
| Parallel iterations | 500 | Parallel iterations = 500 documents simulation or convergence effort rather than substantive magnitude. |
| 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. |
| 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. |
| Lowest item MSA | 0.588169 | Lowest item MSA = 0.588169 describes the balance between ordinary and partial correlations, so it informs factorability or item adequacy rather than factor retention. |
| Highest item MSA | 0.867806 | Highest item MSA = 0.867806 describes the balance between ordinary and partial correlations, so it informs factorability or item adequacy rather than factor retention. |
Parallel Analysis in Python
The Python route calculates or reconstructs the exact named result.
The Python workflow uses the explicit NumPy/Pandas calculation to calculate or extract the simulated-root retention analysis from the declared data and analytical specification. It must reproduce Horn retained factors = 3 and retain Observed eigenvalue 3 = 1.393698 as a separate supporting quantity.
The code is read as an executable analysis, not as a printed answer. Its critical verification is to state whether the analysis is PCA or common-factor based; the associated design condition is that simulation dimensions match the observed data. A “parallel analysis” result is incomplete unless the simulation method, correlation type, number of replications, quantile, n, p, and random seed are documented. It does not interpret the retained factors.
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()
rng=np.random.default_rng(20260731); p=X.shape[1]; n=X.shape[0]; B=500
obs=np.linalg.eigvalsh(X.corr())[::-1]
null=np.empty((B,p))
for b in range(B):
Z=rng.normal(size=(n,p))
null[b]=np.linalg.eigvalsh(np.corrcoef(Z,rowvar=False))[::-1]
q95=np.quantile(null,.95,axis=0)
print(np.c_[obs,q95,obs>q95]); print("retain",np.sum(obs>q95))
Parallel 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 simulated-root retention analysis. 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 report all roots around the retention boundary. 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)
fa.parallel(X, fa="fa", n.iter=500, quant=.95, plot=TRUE)Parallel 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 simulated-root retention analysis. 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 verify the third observed root exceeds its random cutoff, while preserving the requirement that the quantile criterion is prespecified.
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 Parallel Analysis evidence identified in this post.Parallel Analysis in Excel
The workbook exposes source values, intermediate arithmetic, and the final formula.
The Excel workbook is an arithmetic audit for the simulated-root retention analysis. 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 verify the fourth observed root falls below its cutoff and documents Observed eigenvalue 3 = 1.393698 independently.
Data: 649 rows with documented coding.
Inputs: named cells or ranges required only by Parallel 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.Parallel 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 Parallel 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 Parallel-Analysis Primary Metrics
This panel reconciles the headline estimate with its principal supporting values for Parallel Analysis. Read Horn retained factors = 3 beside Observed eigenvalue 3 = 1.393698; the first quantity is not replaced by the second.
The chart is used to state whether the analysis is PCA or common-factor based. Its interpretation remains valid only when simulation dimensions match the observed data. 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 Parallel-Analysis Pca Parallel Comparison
This panel places the ordered roots around the retention boundary for Parallel Analysis. Read Observed eigenvalue 3 = 1.393698 beside Horn 95th percentile root 3 = 1.106209; the first quantity is not replaced by the second.
The chart is used to report all roots around the retention boundary. Its interpretation remains valid only when the same matrix type is used. 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 Parallel-Analysis Random Pca Eigenvalues
This panel places the ordered roots around the retention boundary for Parallel Analysis. Read Horn 95th percentile root 3 = 1.106209 beside Observed eigenvalue 4 = 0.846560; the first quantity is not replaced by the second.
The chart is used to verify the third observed root exceeds its random cutoff. Its interpretation remains valid only when the quantile criterion is prespecified. 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 Parallel-Analysis Observed Pca Eigenvalues
This panel places the ordered roots around the retention boundary for Parallel Analysis. Read Observed eigenvalue 4 = 0.846560 beside Horn 95th percentile root 4 = 1.064106; the first quantity is not replaced by the second.
The chart is used to verify the fourth observed root falls below its cutoff. Its interpretation remains valid only when enough replications stabilize the cutoff. 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 Parallel-Analysis Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for Parallel Analysis. Read Horn 95th percentile root 4 = 1.064106 beside Parallel iterations = 500; the first quantity is not replaced by the second.
The chart is used to repeat with more simulations as a stability check. Its interpretation remains valid only when eigenvalues are compared by rank. 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 Parallel-Analysis Primary Metrics
This panel reconciles the headline estimate with its principal supporting values for Parallel Analysis. Read Parallel iterations = 500 beside Eigenvalue 1 = 3.195831; the first quantity is not replaced by the second.
The chart is used to compare with theory and the scree plot. Its interpretation remains valid only when a seed makes the result reproducible. 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 Parallel-Analysis Pca Parallel Comparison
This panel places the ordered roots around the retention boundary for Parallel 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 state whether the analysis is PCA or common-factor based. Its interpretation remains valid only when simulation dimensions match the observed data. 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 Parallel-Analysis Random Pca Eigenvalues
This panel places the ordered roots around the retention boundary for Parallel Analysis. Read Eigenvalue 2 = 1.817089 beside Eigenvalue 3 = 1.393698; the first quantity is not replaced by the second.
The chart is used to report all roots around the retention boundary. Its interpretation remains valid only when the same matrix type is used. 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 Parallel-Analysis Observed Pca Eigenvalues
This panel places the ordered roots around the retention boundary for Parallel Analysis. Read Eigenvalue 3 = 1.393698 beside Eigenvalue 4 = 0.846560; the first quantity is not replaced by the second.
The chart is used to verify the third observed root exceeds its random cutoff. Its interpretation remains valid only when the quantile criterion is prespecified. 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 Parallel-Analysis Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for Parallel Analysis. Read Eigenvalue 4 = 0.846560 beside Three-dimension cumulative variance = 71.1846%; the first quantity is not replaced by the second.
The chart is used to verify the fourth observed root falls below its cutoff. Its interpretation remains valid only when enough replications stabilize the cutoff. 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.
Parallel 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 Parallel Analysis.
1. State whether the analysis is PCA or common-factor based
Begin by state whether the analysis is PCA or common-factor based. For the simulated-root retention analysis, this operation directly connects Horn retained factors = 3 with Horn 95th percentile root 3 = 1.106209. Horn retained factors = 3 is retained as a distinct supporting quantity for the simulated-root retention analysis; it is not substituted for the primary result.
The governing condition is that simulation dimensions match the observed data. 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 Horn’s Parallel Analysis, because Horn’s post focuses on the named percentile implementation; this post emphasizes general implementation choices and reproducibility.
2. Report all roots around the retention boundary
Next, report all roots around the retention boundary. For the simulated-root retention analysis, this operation directly connects Observed eigenvalue 3 = 1.393698 with Observed eigenvalue 4 = 0.846560. Observed eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
The governing condition is that the same matrix type is used. 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 Scree Plot, because Scree inspection is visual; parallel analysis uses simulated reference roots.
3. Verify the third observed root exceeds its random cutoff
The third verification is to verify the third observed root exceeds its random cutoff. For the simulated-root retention analysis, this operation directly connects Horn 95th percentile root 3 = 1.106209 with Horn 95th percentile root 4 = 1.064106. Horn 95th percentile root 3 = 1.106209 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
The governing condition is that the quantile criterion is prespecified. 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 Minimum Average Partial, because MAP is another retention criterion based on residual partial correlations and can be used as a sensitivity check.
4. Verify the fourth observed root falls below its cutoff
After the core arithmetic is stable, verify the fourth observed root falls below its cutoff. For the simulated-root retention analysis, this operation directly connects Observed eigenvalue 4 = 0.846560 with Parallel iterations = 500. Observed eigenvalue 4 = 0.846560 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
The governing condition is that enough replications stabilize the cutoff. 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 Horn’s Parallel Analysis, because Horn’s post focuses on the named percentile implementation; this post emphasizes general implementation choices and reproducibility.
5. Repeat with more simulations as a stability check
A robustness review must repeat with more simulations as a stability check. For the simulated-root retention analysis, this operation directly connects Horn 95th percentile root 4 = 1.064106 with Eigenvalue 1 = 3.195831. Horn 95th percentile root 4 = 1.064106 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
The governing condition is that eigenvalues are compared by rank. 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 Scree Plot, because Scree inspection is visual; parallel analysis uses simulated reference roots.
6. Compare with theory and the scree plot
The final reconciliation should compare with theory and the scree plot. For the simulated-root retention analysis, this operation directly connects Parallel iterations = 500 with Eigenvalue 2 = 1.817089. Parallel iterations = 500 documents simulation or convergence effort rather than substantive magnitude.
The governing condition is that a seed makes the result reproducible. 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 Minimum Average Partial, because MAP is another retention criterion based on residual partial correlations and can be used as a sensitivity check.
| # | Verification operation | Condition protected | Saved quantity traced |
|---|---|---|---|
| 1 | state whether the analysis is PCA or common-factor based | simulation dimensions match the observed data | Horn retained factors = 3 |
| 2 | report all roots around the retention boundary | the same matrix type is used | Observed eigenvalue 3 = 1.393698 |
| 3 | verify the third observed root exceeds its random cutoff | the quantile criterion is prespecified | Horn 95th percentile root 3 = 1.106209 |
| 4 | verify the fourth observed root falls below its cutoff | enough replications stabilize the cutoff | Observed eigenvalue 4 = 0.846560 |
| 5 | repeat with more simulations as a stability check | eigenvalues are compared by rank | Horn 95th percentile root 4 = 1.064106 |
| 6 | compare with theory and the scree plot | a seed makes the result reproducible | Parallel iterations = 500 |
Parallel 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 Parallel Analysis formula and output rather than a nearby procedure.
Horn’s Parallel Analysis
Horn’s post focuses on the named percentile implementation; this post emphasizes general implementation choices and reproducibility.
In the current analysis, Observed eigenvalue 3 = 1.393698 remains evidence for the simulated-root retention analysis; it is not relabeled as a Horn’s Parallel Analysis result. Observed eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
Scree Plot
Scree inspection is visual; parallel analysis uses simulated reference roots.
In the current analysis, Horn 95th percentile root 3 = 1.106209 remains evidence for the simulated-root retention analysis; it is not relabeled as a Scree Plot result. Horn 95th percentile root 3 = 1.106209 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
Minimum Average Partial
MAP is another retention criterion based on residual partial correlations and can be used as a sensitivity check.
In the current analysis, Observed eigenvalue 4 = 0.846560 remains evidence for the simulated-root retention analysis; it is not relabeled as a Minimum Average Partial result. Observed eigenvalue 4 = 0.846560 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
How to report Parallel Analysis
A complete result paragraph includes the value, analytical object, settings, and limitation.
Results paragraph
Parallel Analysis was evaluated using the declared data, specification, and software settings. The primary result was Horn retained factors = 3; Observed eigenvalue 3 = 1.393698 and Horn 95th percentile root 3 = 1.106209 supplied supporting context. Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
The report then states the limitation explicitly: A “parallel analysis” result is incomplete unless the simulation method, correlation type, number of replications, quantile, n, p, and random seed are documented. It does not interpret the retained factors.
Settings that must accompany the result
simulation dimensions match the observed data; the same matrix type is used; the quantile criterion is prespecified; enough replications stabilize the cutoff.
For Parallel Analysis, these details identify the exact version of the analysis and make cross-software reconciliation possible.
Verification actions retained in the record
state whether the analysis is PCA or common-factor based; report all roots around the retention boundary; verify the third observed root exceeds its random cutoff; verify the fourth observed root falls below its cutoff.
The final wording is revised only after those operations reproduce the saved values.
Parallel Analysis decision scenarios
For Parallel Analysis, worked conflicts show how the conclusion changes when an input, assumption, or supporting statistic fails.
Boundary-case interpretation: State whether the analysis is PCA or common-factor based
Consider a review in which Horn retained factors = 3 is reproduced but Observed eigenvalue 3 = 1.393698 is not. For the simulated-root retention analysis, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to state whether the analysis is PCA or common-factor based and verify that simulation dimensions match the observed data.
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 Horn’s Parallel Analysis only for method selection: Horn’s post focuses on the named percentile implementation; this post emphasizes general implementation choices and reproducibility. The published conclusion remains Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Input-definition sensitivity: Report all roots around the retention boundary
Consider a review in which Horn 95th percentile root 3 = 1.106209 is reproduced but Observed eigenvalue 4 = 0.846560 is not. For the simulated-root retention analysis, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to report all roots around the retention boundary and verify that the same matrix type is used.
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 Scree Plot only for method selection: Scree inspection is visual; parallel analysis uses simulated reference roots. The published conclusion remains Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Software-definition reconciliation: Verify the third observed root exceeds its random cutoff
Consider a review in which Horn 95th percentile root 4 = 1.064106 is reproduced but Parallel iterations = 500 is not. For the simulated-root retention analysis, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to verify the third observed root exceeds its random cutoff and verify that the quantile criterion is prespecified.
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 Minimum Average Partial only for method selection: MAP is another retention criterion based on residual partial correlations and can be used as a sensitivity check. The published conclusion remains Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Local-chart conflict: Verify the fourth observed root falls below its cutoff
Consider a review in which Eigenvalue 1 = 3.195831 is reproduced but Eigenvalue 2 = 1.817089 is not. For the simulated-root retention analysis, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to verify the fourth observed root falls below its cutoff and verify that enough replications stabilize the cutoff.
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 Horn’s Parallel Analysis only for method selection: Horn’s post focuses on the named percentile implementation; this post emphasizes general implementation choices and reproducibility. The published conclusion remains Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Alternative-method challenge: Repeat with more simulations as a stability check
Consider a review in which Eigenvalue 3 = 1.393698 is reproduced but Eigenvalue 4 = 0.846560 is not. For the simulated-root retention analysis, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to repeat with more simulations as a stability check and verify that eigenvalues are compared by rank.
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 Scree Plot only for method selection: Scree inspection is visual; parallel analysis uses simulated reference roots. The published conclusion remains Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Replication and reporting decision: Compare with theory and the scree plot
Consider a review in which Three-dimension cumulative variance = 71.1846% is reproduced but Overall KMO = 0.713439 is not. For the simulated-root retention analysis, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to compare with theory and the scree plot and verify that a seed makes the result reproducible.
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 Minimum Average Partial only for method selection: MAP is another retention criterion based on residual partial correlations and can be used as a sensitivity check. The published conclusion remains Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Boundary-case interpretation: State whether the analysis is PCA or common-factor based
Consider a review in which Lowest item MSA = 0.588169 is reproduced but Highest item MSA = 0.867806 is not. For the simulated-root retention analysis, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to state whether the analysis is PCA or common-factor based and verify that simulation dimensions match the observed data.
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 Horn’s Parallel Analysis only for method selection: Horn’s post focuses on the named percentile implementation; this post emphasizes general implementation choices and reproducibility. The published conclusion remains Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Input-definition sensitivity: Report all roots around the retention boundary
Consider a review in which Bartlett chi-square = 3018.238 is reproduced but Horn retained factors = 3 is not. For the simulated-root retention analysis, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to report all roots around the retention boundary and verify that the same matrix type is used.
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 Scree Plot only for method selection: Scree inspection is visual; parallel analysis uses simulated reference roots. The published conclusion remains Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Software-definition reconciliation: Verify the third observed root exceeds its random cutoff
Consider a review in which Observed eigenvalue 3 = 1.393698 is reproduced but Horn 95th percentile root 3 = 1.106209 is not. For the simulated-root retention analysis, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to verify the third observed root exceeds its random cutoff and verify that the quantile criterion is prespecified.
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 Minimum Average Partial only for method selection: MAP is another retention criterion based on residual partial correlations and can be used as a sensitivity check. The published conclusion remains Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Local-chart conflict: Verify the fourth observed root falls below its cutoff
Consider a review in which Observed eigenvalue 4 = 0.846560 is reproduced but Horn 95th percentile root 4 = 1.064106 is not. For the simulated-root retention analysis, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to verify the fourth observed root falls below its cutoff and verify that enough replications stabilize the cutoff.
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 Horn’s Parallel Analysis only for method selection: Horn’s post focuses on the named percentile implementation; this post emphasizes general implementation choices and reproducibility. The published conclusion remains Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Alternative-method challenge: Repeat with more simulations as a stability check
Consider a review in which Parallel iterations = 500 is reproduced but Eigenvalue 1 = 3.195831 is not. For the simulated-root retention analysis, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to repeat with more simulations as a stability check and verify that eigenvalues are compared by rank.
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 Scree Plot only for method selection: Scree inspection is visual; parallel analysis uses simulated reference roots. The published conclusion remains Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Replication and reporting decision: Compare with theory and the scree plot
Consider a review in which Eigenvalue 2 = 1.817089 is reproduced but Eigenvalue 3 = 1.393698 is not. For the simulated-root retention analysis, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to compare with theory and the scree plot and verify that a seed makes the result reproducible.
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 Minimum Average Partial only for method selection: MAP is another retention criterion based on residual partial correlations and can be used as a sensitivity check. The published conclusion remains Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
Parallel Analysis downloads and reproducibility files
All linked files belong to the same analysis and remain on onlineinternetcafe.com.
The four files belong to one Parallel 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.
Parallel Analysis frequently asked questions
Answers use the worked result and the exact method boundary.
What does Parallel Analysis measure?
Parallel analysis is a family of retention procedures that compares observed ordered eigenvalues with eigenvalues from random datasets matched on sample size and variable count. The result depends on whether mean or percentile roots and PCA or common-factor roots are used.
What is the main result in this Parallel Analysis analysis?
Horn retained factors = 3. Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.
What does the result not prove?
A “parallel analysis” result is incomplete unless the simulation method, correlation type, number of replications, quantile, n, p, and random seed are documented. It does not interpret the retained factors.
Which supporting value should be reported with the primary result?
For Parallel Analysis, observed eigenvalue 3 = 1.393698 is the first companion quantity. Observed eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
Which assumption is most likely to change the interpretation?
The first requirement is that simulation dimensions match the observed data. The result is recomputed if that condition is not satisfied.
What is the most important numerical verification?
The analyst must state whether the analysis is PCA or common-factor based. That operation traces Horn retained factors = 3 to the formula and saved inputs.
Why can software packages disagree on Parallel Analysis?
Disagreement can arise because the same matrix type is used or because the packages implement different estimators, matrices, baselines, rotations, standardizations, bootstrap rules, or coefficient definitions. Matching labels alone is not enough.
How is Parallel Analysis different from Horn’s Parallel Analysis?
Horn’s post focuses on the named percentile implementation; this post emphasizes general implementation choices and reproducibility.
How should a chart be interpreted?
For Parallel Analysis, each chart is tied to a named output such as Horn 95th percentile root 3 = 1.106209. It supports a local calculation or diagnostic and does not replace the full numerical result.
How should Parallel Analysis be reported?
Report Horn retained factors = 3, the required supporting quantities, sample or panel size, exact method settings, and this qualified conclusion: Under the reported 95th-percentile comparison, three observed eigenvalues exceed their random counterparts and the fourth does not. The retention boundary is therefore three dimensions.