Horns Parallel Analysis: Formula, Verified Results, Charts and Interpretation
Horn’s parallel analysis retains a dimension when its observed eigenvalue exceeds the corresponding eigenvalue generated from random data of the same sample size and variable count. A percentile criterion is more conservative than comparing only with the random-data mean. This guide uses the supplied real-data results, native MathML equations, matching charts, and separate Python, R, SPSS or AMOS, and Excel verification.
The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
Horn retained factors = 3 is retained as a distinct supporting quantity for the Horn percentile root comparison; it is not substituted for the primary result.
What Horns Parallel Analysis measures
The exact estimand and the result this method is allowed to support.
Horns Parallel Analysis addresses one defined analytical target: Horn’s parallel analysis retains a dimension when its observed eigenvalue exceeds the corresponding eigenvalue generated from random data of the same sample size and variable count. A percentile criterion is more conservative than comparing only with the random-data mean.
Quantity estimated in this analysis
The horn percentile root comparison 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 Horn percentile root comparison; it is not substituted for the primary result.
For Horns 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
Parallel analysis does not interpret factors, choose a rotation, or prove that retained dimensions are substantively meaningful. PCA-based and common-factor-based simulations answer different retention questions and must be labeled.
For Horns 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 Horns Parallel Analysis
Research scope, neighboring methods, and excluded claims.
Research question answered
The defensible question is whether the Horn percentile root comparison supports the result stated for the declared dataset and analytical specification. It is answered by compare roots one through four explicitly, followed by retain the percentile rather than switching silently to the mean. The evidence is bounded by Horn retained factors = 3 and its named companion quantities.
For Horns 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
Parallel Analysis: This post emphasizes Horn’s original retention logic and the reported percentile roots; the general parallel-analysis post compares implementation choices.
Scree Plot: The scree elbow is visual and subjective; parallel analysis supplies an empirical random-data reference.
These distinctions determine which formula, output table, and chart can legitimately appear in a Horns Parallel Analysis post.
Real data used for Horns Parallel Analysis
Variables, coding, sample or panel size, and the role each input plays.
For Horns 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 Horn percentile root comparison, 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 |
Horns Parallel Analysis assumptions and design requirements
Six conditions checked before the coefficient or decision rule is interpreted.
1. Random data use the same n and p
This condition determines whether the input object matches the formula. In the current Horns Parallel Analysis analysis, the check is to compare roots one through four explicitly while preserving Horn retained factors = 3.
For Horns 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 observed and simulated matrices use the same correlation type
This requirement controls whether the numerical estimate has the interpretation claimed. In the current Horns Parallel Analysis analysis, the check is to retain the percentile rather than switching silently to the mean while preserving Observed eigenvalue 3 = 1.393698.
For Horns 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 PCA or common-factor version is declared
This design condition prevents an attractive coefficient from being attached to the wrong population or model. In the current Horns Parallel Analysis analysis, the check is to verify n = 649 and p = 9 in simulations while preserving Horn 95th percentile root 3 = 1.106209.
For Horns 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 simulations are run
This specification rule keeps the software routes numerically comparable. In the current Horns Parallel Analysis analysis, the check is to record the 500-iteration setting used here while preserving Observed eigenvalue 4 = 0.846560.
For Horns 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. A reproducible random seed is recorded
This diagnostic requirement is checked before a benchmark is applied. In the current Horns Parallel Analysis analysis, the check is to check whether common-factor parallel analysis agrees while preserving Horn 95th percentile root 4 = 1.064106.
For Horns 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. Eigenvalues are compared by ordered root
This final condition governs whether the conclusion can survive replication or sensitivity analysis. In the current Horns Parallel Analysis analysis, the check is to avoid interpreting simulated cutoffs as fit indices while preserving Parallel iterations = 500.
For Horns 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.
Horns Parallel Analysis hypotheses or decision rule
The statistical question is stated at the correct level for this method.
Statistical question
For Horns 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 Horns 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 Horn percentile root comparison; it is not substituted for the primary result.
The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
Horns 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 Horns 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 95th-percentile benchmark is stricter than comparing with the random mean.
Horn’s parallel analysis therefore retains exactly three dimensions.
Symbol and denominator control
Horn’s parallel analysis retains a dimension when its observed eigenvalue exceeds the corresponding eigenvalue generated from random data of the same sample size and variable count. A percentile criterion is more conservative than comparing only with the random-data mean.
For Horns 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
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.
Parallel analysis does not interpret factors, choose a rotation, or prove that retained dimensions are substantively meaningful. PCA-based and common-factor-based simulations answer different retention questions and must be labeled.
Step-by-step Horns 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 Horn percentile root comparison. 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: Compare roots one through four explicitly.
Numerical trace: Horn retained factors = 3; Observed eigenvalue 3 = 1.393698.
Condition: random data use the same n and p. 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: Retain the percentile rather than switching silently to the mean.
Numerical trace: Observed eigenvalue 3 = 1.393698; Horn 95th percentile root 3 = 1.106209.
Condition: the observed and simulated matrices use the same correlation type. 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 n = 649 and p = 9 in simulations.
Numerical trace: Horn 95th percentile root 3 = 1.106209; Observed eigenvalue 4 = 0.846560.
Condition: the PCA or common-factor version is declared. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Apply the decision rule
Action: Record the 500-iteration setting used here.
Numerical trace: Observed eigenvalue 4 = 0.846560; Horn 95th percentile root 4 = 1.064106.
Condition: enough simulations are run. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Inspect local evidence
Action: Check whether common-factor parallel analysis agrees.
Numerical trace: Horn 95th percentile root 4 = 1.064106; Parallel iterations = 500.
Condition: a reproducible random seed is recorded. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Reconcile and report
Action: Avoid interpreting simulated cutoffs as fit indices.
Numerical trace: Parallel iterations = 500; Eigenvalue 1 = 3.195831.
Condition: eigenvalues are compared by ordered root. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Horns Parallel Analysis results and interpretation
Primary and supporting statistics are kept separate and precisely labeled.
Primary result
Horn retained factors
The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
Why the result is internally coherent
Horn retained factors = 3 is retained as a distinct supporting quantity for the Horn percentile root comparison; it is not substituted for the primary result.
Observed eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
For Horns 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 Horn percentile root comparison; 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. |
Horns 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 Horn percentile root comparison 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 compare roots one through four explicitly; the associated design condition is that random data use the same n and p. Parallel analysis does not interpret factors, choose a rotation, or prove that retained dimensions are substantively meaningful. PCA-based and common-factor-based simulations answer different retention questions and must be labeled.
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))
Horns 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 Horn percentile root comparison. 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 retain the percentile rather than switching silently to the mean. 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)Horns 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 Horn percentile root comparison. 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 n = 649 and p = 9 in simulations, while preserving the requirement that the PCA or common-factor version is declared.
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 Horns Parallel Analysis evidence identified in this post.Horns Parallel Analysis in Excel
The workbook exposes source values, intermediate arithmetic, and the final formula.
The Excel workbook is an arithmetic audit for the Horn percentile root comparison. 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 record the 500-iteration setting used here and documents Observed eigenvalue 3 = 1.393698 independently.
Data: 649 rows with documented coding.
Inputs: named cells or ranges required only by Horns 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.Horns 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 Horns 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 Horns-Parallel-Analysis Primary Metrics
This panel reconciles the headline estimate with its principal supporting values for Horns 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 compare roots one through four explicitly. Its interpretation remains valid only when random data use the same n and p. 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 Horns-Parallel-Analysis Horn Common Factor Comparison
This panel places the ordered roots around the retention boundary for Horns 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 retain the percentile rather than switching silently to the mean. Its interpretation remains valid only when the observed and simulated matrices use the same correlation type. 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 Horns-Parallel-Analysis Random Common Eigenvalues
This panel places the ordered roots around the retention boundary for Horns 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 n = 649 and p = 9 in simulations. Its interpretation remains valid only when the PCA or common-factor version is declared. 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 Horns-Parallel-Analysis Reduced Correlation Matrix
This panel shows the cell-level pattern that a single coefficient can conceal for Horns 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 record the 500-iteration setting used here. Its interpretation remains valid only when enough simulations are run. 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 Horns-Parallel-Analysis Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for Horns 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 check whether common-factor parallel analysis agrees. Its interpretation remains valid only when a reproducible random seed is recorded. 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 Horns-Parallel-Analysis Primary Metrics
This panel reconciles the headline estimate with its principal supporting values for Horns 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 avoid interpreting simulated cutoffs as fit indices. Its interpretation remains valid only when eigenvalues are compared by ordered root. 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 Horns-Parallel-Analysis Horn Common Factor Comparison
This panel places the ordered roots around the retention boundary for Horns 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 compare roots one through four explicitly. Its interpretation remains valid only when random data use the same n and p. 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 Horns-Parallel-Analysis Random Common Eigenvalues
This panel places the ordered roots around the retention boundary for Horns 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 retain the percentile rather than switching silently to the mean. Its interpretation remains valid only when the observed and simulated matrices use the same correlation type. 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 Horns-Parallel-Analysis Reduced Correlation Matrix
This panel shows the cell-level pattern that a single coefficient can conceal for Horns 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 n = 649 and p = 9 in simulations. Its interpretation remains valid only when the PCA or common-factor version is declared. 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 Horns-Parallel-Analysis Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for Horns 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 record the 500-iteration setting used here. Its interpretation remains valid only when enough simulations are run. 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.
Horns 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 Horns Parallel Analysis.
1. Compare roots one through four explicitly
Begin by compare roots one through four explicitly. For the Horn percentile root comparison, 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 Horn percentile root comparison; it is not substituted for the primary result.
The governing condition is that random data use the same n and p. 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 Parallel Analysis, because This post emphasizes Horn’s original retention logic and the reported percentile roots; the general parallel-analysis post compares implementation choices.
2. Retain the percentile rather than switching silently to the mean
Next, retain the percentile rather than switching silently to the mean. For the Horn percentile root comparison, 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 observed and simulated matrices use the same correlation type. 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 The scree elbow is visual and subjective; parallel analysis supplies an empirical random-data reference.
3. Verify n = 649 and p = 9 in simulations
The third verification is to verify n = 649 and p = 9 in simulations. For the Horn percentile root comparison, 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 PCA or common-factor version is declared. 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 Kaiser Criterion, because The eigenvalue-greater-than-one rule ignores sampling behavior and is less defensible than parallel analysis.
4. Record the 500-iteration setting used here
After the core arithmetic is stable, record the 500-iteration setting used here. For the Horn percentile root comparison, 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 simulations are run. 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 Parallel Analysis, because This post emphasizes Horn’s original retention logic and the reported percentile roots; the general parallel-analysis post compares implementation choices.
5. Check whether common-factor parallel analysis agrees
A robustness review must check whether common-factor parallel analysis agrees. For the Horn percentile root comparison, 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 a reproducible random seed is recorded. 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 The scree elbow is visual and subjective; parallel analysis supplies an empirical random-data reference.
6. Avoid interpreting simulated cutoffs as fit indices
The final reconciliation should avoid interpreting simulated cutoffs as fit indices. For the Horn percentile root comparison, 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 eigenvalues are compared by ordered root. 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 Kaiser Criterion, because The eigenvalue-greater-than-one rule ignores sampling behavior and is less defensible than parallel analysis.
| # | Verification operation | Condition protected | Saved quantity traced |
|---|---|---|---|
| 1 | compare roots one through four explicitly | random data use the same n and p | Horn retained factors = 3 |
| 2 | retain the percentile rather than switching silently to the mean | the observed and simulated matrices use the same correlation type | Observed eigenvalue 3 = 1.393698 |
| 3 | verify n = 649 and p = 9 in simulations | the PCA or common-factor version is declared | Horn 95th percentile root 3 = 1.106209 |
| 4 | record the 500-iteration setting used here | enough simulations are run | Observed eigenvalue 4 = 0.846560 |
| 5 | check whether common-factor parallel analysis agrees | a reproducible random seed is recorded | Horn 95th percentile root 4 = 1.064106 |
| 6 | avoid interpreting simulated cutoffs as fit indices | eigenvalues are compared by ordered root | Parallel iterations = 500 |
Horns 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 Horns Parallel Analysis formula and output rather than a nearby procedure.
Parallel Analysis
This post emphasizes Horn’s original retention logic and the reported percentile roots; the general parallel-analysis post compares implementation choices.
In the current analysis, Observed eigenvalue 3 = 1.393698 remains evidence for the Horn percentile root comparison; it is not relabeled as a 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
The scree elbow is visual and subjective; parallel analysis supplies an empirical random-data reference.
In the current analysis, Horn 95th percentile root 3 = 1.106209 remains evidence for the Horn percentile root comparison; 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.
Kaiser Criterion
The eigenvalue-greater-than-one rule ignores sampling behavior and is less defensible than parallel analysis.
In the current analysis, Observed eigenvalue 4 = 0.846560 remains evidence for the Horn percentile root comparison; it is not relabeled as a Kaiser Criterion 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 Horns Parallel Analysis
A complete result paragraph includes the value, analytical object, settings, and limitation.
Results paragraph
Horns 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. The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
The report then states the limitation explicitly: Parallel analysis does not interpret factors, choose a rotation, or prove that retained dimensions are substantively meaningful. PCA-based and common-factor-based simulations answer different retention questions and must be labeled.
Settings that must accompany the result
random data use the same n and p; the observed and simulated matrices use the same correlation type; the PCA or common-factor version is declared; enough simulations are run.
For Horns Parallel Analysis, these details identify the exact version of the analysis and make cross-software reconciliation possible.
Verification actions retained in the record
compare roots one through four explicitly; retain the percentile rather than switching silently to the mean; verify n = 649 and p = 9 in simulations; record the 500-iteration setting used here.
The final wording is revised only after those operations reproduce the saved values.
Horns Parallel Analysis decision scenarios
For Horns Parallel Analysis, worked conflicts show how the conclusion changes when an input, assumption, or supporting statistic fails.
Boundary-case interpretation: Compare roots one through four explicitly
Consider a review in which Horn retained factors = 3 is reproduced but Observed eigenvalue 3 = 1.393698 is not. For the Horn percentile root comparison, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to compare roots one through four explicitly and verify that random data use the same n and p.
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: This post emphasizes Horn’s original retention logic and the reported percentile roots; the general parallel-analysis post compares implementation choices. The published conclusion remains The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
Input-definition sensitivity: Retain the percentile rather than switching silently to the mean
Consider a review in which Horn 95th percentile root 3 = 1.106209 is reproduced but Observed eigenvalue 4 = 0.846560 is not. For the Horn percentile root comparison, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to retain the percentile rather than switching silently to the mean and verify that the observed and simulated matrices use the same correlation type.
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: The scree elbow is visual and subjective; parallel analysis supplies an empirical random-data reference. The published conclusion remains The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
Software-definition reconciliation: Verify n = 649 and p = 9 in simulations
Consider a review in which Horn 95th percentile root 4 = 1.064106 is reproduced but Parallel iterations = 500 is not. For the Horn percentile root comparison, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to verify n = 649 and p = 9 in simulations and verify that the PCA or common-factor version is declared.
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 Kaiser Criterion only for method selection: The eigenvalue-greater-than-one rule ignores sampling behavior and is less defensible than parallel analysis. The published conclusion remains The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
Local-chart conflict: Record the 500-iteration setting used here
Consider a review in which Eigenvalue 1 = 3.195831 is reproduced but Eigenvalue 2 = 1.817089 is not. For the Horn percentile root comparison, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to record the 500-iteration setting used here and verify that enough simulations are run.
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: This post emphasizes Horn’s original retention logic and the reported percentile roots; the general parallel-analysis post compares implementation choices. The published conclusion remains The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
Alternative-method challenge: Check whether common-factor parallel analysis agrees
Consider a review in which Eigenvalue 3 = 1.393698 is reproduced but Eigenvalue 4 = 0.846560 is not. For the Horn percentile root comparison, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to check whether common-factor parallel analysis agrees and verify that a reproducible random seed is recorded.
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: The scree elbow is visual and subjective; parallel analysis supplies an empirical random-data reference. The published conclusion remains The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
Replication and reporting decision: Avoid interpreting simulated cutoffs as fit indices
Consider a review in which Three-dimension cumulative variance = 71.1846% is reproduced but Overall KMO = 0.713439 is not. For the Horn percentile root comparison, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to avoid interpreting simulated cutoffs as fit indices and verify that eigenvalues are compared by ordered root.
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 Kaiser Criterion only for method selection: The eigenvalue-greater-than-one rule ignores sampling behavior and is less defensible than parallel analysis. The published conclusion remains The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
Boundary-case interpretation: Compare roots one through four explicitly
Consider a review in which Lowest item MSA = 0.588169 is reproduced but Highest item MSA = 0.867806 is not. For the Horn percentile root comparison, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to compare roots one through four explicitly and verify that random data use the same n and p.
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: This post emphasizes Horn’s original retention logic and the reported percentile roots; the general parallel-analysis post compares implementation choices. The published conclusion remains The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
Input-definition sensitivity: Retain the percentile rather than switching silently to the mean
Consider a review in which Bartlett chi-square = 3018.238 is reproduced but Horn retained factors = 3 is not. For the Horn percentile root comparison, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to retain the percentile rather than switching silently to the mean and verify that the observed and simulated matrices use the same correlation type.
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: The scree elbow is visual and subjective; parallel analysis supplies an empirical random-data reference. The published conclusion remains The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
Software-definition reconciliation: Verify n = 649 and p = 9 in simulations
Consider a review in which Observed eigenvalue 3 = 1.393698 is reproduced but Horn 95th percentile root 3 = 1.106209 is not. For the Horn percentile root comparison, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to verify n = 649 and p = 9 in simulations and verify that the PCA or common-factor version is declared.
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 Kaiser Criterion only for method selection: The eigenvalue-greater-than-one rule ignores sampling behavior and is less defensible than parallel analysis. The published conclusion remains The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
Local-chart conflict: Record the 500-iteration setting used here
Consider a review in which Observed eigenvalue 4 = 0.846560 is reproduced but Horn 95th percentile root 4 = 1.064106 is not. For the Horn percentile root comparison, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to record the 500-iteration setting used here and verify that enough simulations are run.
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: This post emphasizes Horn’s original retention logic and the reported percentile roots; the general parallel-analysis post compares implementation choices. The published conclusion remains The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
Alternative-method challenge: Check whether common-factor parallel analysis agrees
Consider a review in which Parallel iterations = 500 is reproduced but Eigenvalue 1 = 3.195831 is not. For the Horn percentile root comparison, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to check whether common-factor parallel analysis agrees and verify that a reproducible random seed is recorded.
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: The scree elbow is visual and subjective; parallel analysis supplies an empirical random-data reference. The published conclusion remains The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
Horns 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 Horns 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.
Horns Parallel Analysis frequently asked questions
Answers use the worked result and the exact method boundary.
What does Horns Parallel Analysis measure?
Horn’s parallel analysis retains a dimension when its observed eigenvalue exceeds the corresponding eigenvalue generated from random data of the same sample size and variable count. A percentile criterion is more conservative than comparing only with the random-data mean.
What is the main result in this Horns Parallel Analysis analysis?
Horn retained factors = 3. The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.
What does the result not prove?
Parallel analysis does not interpret factors, choose a rotation, or prove that retained dimensions are substantively meaningful. PCA-based and common-factor-based simulations answer different retention questions and must be labeled.
Which supporting value should be reported with the primary result?
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 random data use the same n and p. The result is recomputed if that condition is not satisfied.
What is the most important numerical verification?
The analyst must compare roots one through four explicitly. That operation traces Horn retained factors = 3 to the formula and saved inputs.
Why can software packages disagree on Horns Parallel Analysis?
Disagreement can arise because the observed and simulated matrices use the same correlation type or because the packages implement different estimators, matrices, baselines, rotations, standardizations, bootstrap rules, or coefficient definitions. Matching labels alone is not enough.
How is Horns Parallel Analysis different from Parallel Analysis?
This post emphasizes Horn’s original retention logic and the reported percentile roots; the general parallel-analysis post compares implementation choices.
How should a chart be interpreted?
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 Horns Parallel Analysis be reported?
Report Horn retained factors = 3, the required supporting quantities, sample or panel size, exact method settings, and this qualified conclusion: The first three observed roots exceed the reported 95th-percentile random roots, while the fourth observed root does not. Horn’s criterion therefore retains three dimensions.