Bartletts Test of Sphericity: Formula, Verified Results, Charts and Interpretation
Bartlett’s test of sphericity is a likelihood-based test of the null hypothesis that the population correlation matrix is an identity matrix. The statistic therefore addresses whether the variables exhibit collective correlation structure, not how many factors should be retained. This guide uses the supplied real-data results, native MathML equations, matching charts, and separate Python, R, SPSS or AMOS, and Excel verification.
The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.
Bartlett chi-square = 3018.238 is read with its degrees of freedom, estimator, sample size, and p-value; it is not a stand-alone effect size.
What Bartletts Test of Sphericity measures
The exact estimand and the result this method is allowed to support.
Bartletts Test of Sphericity addresses one defined analytical target: Bartlett’s test of sphericity is a likelihood-based test of the null hypothesis that the population correlation matrix is an identity matrix. The statistic therefore addresses whether the variables exhibit collective correlation structure, not how many factors should be retained.
Quantity estimated in this analysis
The identity-matrix sphericity test is reconstructed from the exact variables, matrix, model, panel, or resampling design shown below. The primary output is Bartlett chi-square = 3018.238; Bartlett degrees of freedom = 36 supplies the first supporting check. Bartlett chi-square = 3018.238 is read with its degrees of freedom, estimator, sample size, and p-value; it is not a stand-alone effect size.
For Bartletts Test of Sphericity, 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 significant Bartlett result does not prove that a factor model is substantively correct, that every item is adequate, or that a particular number of factors exists. With large samples, even modest departures from identity can be statistically significant.
For Bartletts Test of Sphericity, 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 Bartletts Test of Sphericity
Research scope, neighboring methods, and excluded claims.
Research question answered
The defensible question is whether the identity-matrix sphericity test supports the result stated for the declared dataset and analytical specification. It is answered by recompute the determinant from the same ordered nine-variable correlation matrix, followed by verify df = p(p−1)/2 for p = 9. The evidence is bounded by Bartlett chi-square = 3018.238 and its named companion quantities.
For Bartletts Test of Sphericity, 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
Kaiser–Meyer–Olkin Test: KMO evaluates the balance of zero-order and partial correlations; Bartlett tests identity of the full correlation matrix.
Parallel Analysis: Parallel analysis decides how many dimensions outperform random data; Bartlett does not determine dimensionality.
These distinctions determine which formula, output table, and chart can legitimately appear in a Bartletts Test of Sphericity post.
Real data used for Bartletts Test of Sphericity
Variables, coding, sample or panel size, and the role each input plays.
The worked analysis preserves the exact sample, variables, coding, and model declaration recorded in the supplied reports.
Those inputs are the only basis for Bartlett chi-square = 3018.238 and the accompanying interpretation of the identity-matrix sphericity test.
| 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 |
Bartletts Test of Sphericity assumptions and design requirements
Six conditions checked before the coefficient or decision rule is interpreted.
1. Observations are independent
This condition determines whether the input object matches the formula. In the current Bartletts Test of Sphericity analysis, the check is to recompute the determinant from the same ordered nine-variable correlation matrix while preserving Bartlett chi-square = 3018.238.
For Bartletts Test of Sphericity, 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 correlation matrix is positive definite
This requirement controls whether the numerical estimate has the interpretation claimed. In the current Bartletts Test of Sphericity analysis, the check is to verify df = p(p−1)/2 for p = 9 while preserving Bartlett degrees of freedom = 36.
For Bartletts Test of Sphericity, 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 variables support the selected correlation type
This design condition prevents an attractive coefficient from being attached to the wrong population or model. In the current Bartletts Test of Sphericity analysis, the check is to retain the finite-sample correction in the chi-square formula while preserving Correlation determinant = 0.00922819.
For Bartletts Test of Sphericity, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.
4. The sample size used in the determinant matches the reported n
This specification rule keeps the software routes numerically comparable. In the current Bartletts Test of Sphericity analysis, the check is to distinguish statistical significance from practical factorability while preserving Overall KMO = 0.713439.
For Bartletts Test of Sphericity, 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. Missing data handling is fixed before the test
This diagnostic requirement is checked before a benchmark is applied. In the current Bartletts Test of Sphericity analysis, the check is to inspect item-level MSA even after the global test rejects identity while preserving Eigenvalue 1 = 3.195831.
For Bartletts Test of Sphericity, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.
6. The chi-square approximation is appropriate for the analysis
This final condition governs whether the conclusion can survive replication or sensitivity analysis. In the current Bartletts Test of Sphericity analysis, the check is to avoid using Bartlett significance to choose three factors while preserving Eigenvalue 2 = 1.817089.
For Bartletts Test of Sphericity, 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.
Bartletts Test of Sphericity hypotheses or decision rule
The statistical question is stated at the correct level for this method.
Statistical question
H₀: the population correlation matrix is an identity matrix. H₁: at least one off-diagonal population correlation is nonzero.
The chi-square result tests collective factorability only; it does not select the number of factors or establish a particular loading pattern.
Decision for the worked analysis
The calculation yields Bartlett chi-square = 3018.238. Bartlett chi-square = 3018.238 is read with its degrees of freedom, estimator, sample size, and p-value; it is not a stand-alone effect size.
The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.
Bartletts Test of Sphericity formula and worked substitution
Native MathML preserves fractions, roots, summations, matrices, subscripts, and superscripts.
The equation below is the defining mathematical object for Bartletts Test of Sphericity. Its symbols are connected to the saved inputs and to Bartlett chi-square = 3018.238, Bartlett degrees of freedom = 36, Correlation determinant = 0.00922819, Overall KMO = 0.713439.
The null hypothesis is that the population correlation matrix is an identity matrix.
The correlation matrix is decisively different from an identity matrix, supporting factorability.
Symbol and denominator control
Bartlett’s test of sphericity is a likelihood-based test of the null hypothesis that the population correlation matrix is an identity matrix. The statistic therefore addresses whether the variables exhibit collective correlation structure, not how many factors should be retained.
For Bartletts Test of Sphericity, 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 Bartlett chi-square = 3018.238 and Bartlett degrees of freedom = 36.
A significant Bartlett result does not prove that a factor model is substantively correct, that every item is adequate, or that a particular number of factors exists. With large samples, even modest departures from identity can be statistically significant.
Step-by-step Bartletts Test of Sphericity calculation
Every stage is tied to a saved value and a method-specific condition.
The worked calculation follows six operations specific to the identity-matrix sphericity test. 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: Recompute the determinant from the same ordered nine-variable correlation matrix.
Numerical trace: Bartlett chi-square = 3018.238; Bartlett degrees of freedom = 36.
Condition: observations are independent. 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: Verify df = p(p−1)/2 for p = 9.
Numerical trace: Bartlett degrees of freedom = 36; Correlation determinant = 0.00922819.
Condition: the correlation matrix is positive definite. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Verify the companion quantity
Action: Retain the finite-sample correction in the chi-square formula.
Numerical trace: Correlation determinant = 0.00922819; Overall KMO = 0.713439.
Condition: the variables support the selected correlation type. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Apply the decision rule
Action: Distinguish statistical significance from practical factorability.
Numerical trace: Overall KMO = 0.713439; Eigenvalue 1 = 3.195831.
Condition: the sample size used in the determinant matches the reported n. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Inspect local evidence
Action: Inspect item-level MSA even after the global test rejects identity.
Numerical trace: Eigenvalue 1 = 3.195831; Eigenvalue 2 = 1.817089.
Condition: missing data handling is fixed before the test. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Reconcile and report
Action: Avoid using Bartlett significance to choose three factors.
Numerical trace: Eigenvalue 2 = 1.817089; Eigenvalue 3 = 1.393698.
Condition: the chi-square approximation is appropriate for the analysis. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Bartletts Test of Sphericity results and interpretation
Primary and supporting statistics are kept separate and precisely labeled.
Primary result
Bartlett chi-square
The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.
Why the result is internally coherent
Bartlett chi-square = 3018.238 is read with its degrees of freedom, estimator, sample size, and p-value; it is not a stand-alone effect size.
Bartlett degrees of freedom = 36 is retained as a distinct supporting quantity for the identity-matrix sphericity test; it is not substituted for the primary result.
For Bartletts Test of Sphericity, 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 |
|---|---|---|
| Bartlett chi-square | 3018.238 | Bartlett chi-square = 3018.238 is read with its degrees of freedom, estimator, sample size, and p-value; it is not a stand-alone effect size. |
| Bartlett degrees of freedom | 36 | Bartlett degrees of freedom = 36 is retained as a distinct supporting quantity for the identity-matrix sphericity test; it is not substituted for the primary result. |
| Correlation determinant | 0.00922819 | Correlation determinant = 0.00922819 is an association for the named variables or constructs; it is not a loading, reliability coefficient, or causal effect. |
| 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. |
| 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. |
| Horn retained factors | 3 | Horn retained factors = 3 is retained as a distinct supporting quantity for the identity-matrix sphericity test; it is not substituted for the primary result. |
| Parallel iterations | 500 | Parallel iterations = 500 documents simulation or convergence effort rather than substantive magnitude. |
| 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. |
Bartletts Test of Sphericity in Python
The Python route calculates or reconstructs the exact named result.
The Python workflow uses scipy, chi2 to calculate or extract the identity-matrix sphericity test from the declared data and analytical specification. It must reproduce Bartlett chi-square = 3018.238 and retain Bartlett degrees of freedom = 36 as a separate supporting quantity.
The code is read as an executable analysis, not as a printed answer. Its critical verification is to recompute the determinant from the same ordered nine-variable correlation matrix; the associated design condition is that observations are independent. A significant Bartlett result does not prove that a factor model is substantively correct, that every item is adequate, or that a particular number of factors exists. With large samples, even modest departures from identity can be statistically significant.
import pandas as pd
import numpy as npdf = pd.read_csv("student-por.csv", sep=";")
df["TravelAccess"] = 5 - df["traveltime"]
vars9 = ["G1","G2","G3","Medu","Fedu","TravelAccess","goout","Dalc","Walc"]
X = df[vars9].dropna()
from scipy.stats import chi2
R = X.corr().to_numpy(); n,p = X.shape
detR = np.linalg.det(R)
stat = -(n-1-(2*p+5)/6)*np.log(detR)
df_b = p*(p-1)//2
p_value = chi2.sf(stat, df_b)
print(stat, df_b, p_value, detR)
Bartletts Test of Sphericity 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 identity-matrix sphericity test. Package defaults are made explicit because estimator, matrix type, extraction, rotation, baseline, or bootstrap choices can change the result.
R output is reconciled with Bartlett chi-square = 3018.238 after the analyst verify df = p(p−1)/2 for p = 9. 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)
cortest.bartlett(cor(X), n=nrow(X))Bartletts Test of Sphericity 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 identity-matrix sphericity test. 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 Bartlett chi-square = 3018.238 and the settings needed to reproduce it. The software review specifically retain the finite-sample correction in the chi-square formula, while preserving the requirement that the variables support the selected correlation type.
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 Bartletts Test of Sphericity evidence identified in this post.Bartletts Test of Sphericity in Excel
The workbook exposes source values, intermediate arithmetic, and the final formula.
The Excel workbook is an arithmetic audit for the identity-matrix sphericity test. Named cells retain the inputs, intermediate components, and final formula leading to Bartlett chi-square = 3018.238; 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 distinguish statistical significance from practical factorability and documents Bartlett degrees of freedom = 36 independently.
Data: 649 rows with documented coding.
Inputs: named cells or ranges required only by Bartletts Test of Sphericity.
Calculation: =-(N-1-(2*P+5)/6)*LN(MDETERM(Correlation_Matrix))
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.Bartletts Test of Sphericity 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 Bartletts Test of Sphericity 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 Bartletts-Test-Of-Sphericity Primary Metrics
This panel reconciles the headline estimate with its principal supporting values for Bartletts Test of Sphericity. Read Bartlett chi-square = 3018.238 beside Bartlett degrees of freedom = 36; the first quantity is not replaced by the second.
The chart is used to recompute the determinant from the same ordered nine-variable correlation matrix. Its interpretation remains valid only when observations are independent. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
02 Bartletts-Test-Of-Sphericity Correlation Matrix
This panel shows the cell-level pattern that a single coefficient can conceal for Bartletts Test of Sphericity. Read Bartlett degrees of freedom = 36 beside Correlation determinant = 0.00922819; the first quantity is not replaced by the second.
The chart is used to verify df = p(p−1)/2 for p = 9. Its interpretation remains valid only when the correlation matrix is positive definite. 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 Bartletts-Test-Of-Sphericity Bartlett Components
This panel displays the quantities entering the defining equation for Bartletts Test of Sphericity. Read Correlation determinant = 0.00922819 beside Overall KMO = 0.713439; the first quantity is not replaced by the second.
The chart is used to retain the finite-sample correction in the chi-square formula. Its interpretation remains valid only when the variables support the selected 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 Bartletts-Test-Of-Sphericity Eigenvalues
This panel places the ordered roots around the retention boundary for Bartletts Test of Sphericity. Read Overall KMO = 0.713439 beside Eigenvalue 1 = 3.195831; the first quantity is not replaced by the second.
The chart is used to distinguish statistical significance from practical factorability. Its interpretation remains valid only when the sample size used in the determinant matches the reported n. 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 Bartletts-Test-Of-Sphericity Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for Bartletts Test of Sphericity. Read Eigenvalue 1 = 3.195831 beside Eigenvalue 2 = 1.817089; the first quantity is not replaced by the second.
The chart is used to inspect item-level MSA even after the global test rejects identity. Its interpretation remains valid only when missing data handling is fixed before the test. 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 Bartletts-Test-Of-Sphericity Primary Metrics
This panel reconciles the headline estimate with its principal supporting values for Bartletts Test of Sphericity. Read Eigenvalue 2 = 1.817089 beside Eigenvalue 3 = 1.393698; the first quantity is not replaced by the second.
The chart is used to avoid using Bartlett significance to choose three factors. Its interpretation remains valid only when the chi-square approximation is appropriate for the analysis. 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 Bartletts-Test-Of-Sphericity Correlation Matrix
This panel shows the cell-level pattern that a single coefficient can conceal for Bartletts Test of Sphericity. Read Eigenvalue 3 = 1.393698 beside Eigenvalue 4 = 0.846560; the first quantity is not replaced by the second.
The chart is used to recompute the determinant from the same ordered nine-variable correlation matrix. Its interpretation remains valid only when observations are independent. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.
03 Bartletts-Test-Of-Sphericity Bartlett Components
This panel displays the quantities entering the defining equation for Bartletts Test of Sphericity. 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 df = p(p−1)/2 for p = 9. Its interpretation remains valid only when the correlation matrix is positive definite. 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 Bartletts-Test-Of-Sphericity Eigenvalues
This panel places the ordered roots around the retention boundary for Bartletts Test of Sphericity. Read Three-dimension cumulative variance = 71.1846% beside Horn retained factors = 3; the first quantity is not replaced by the second.
The chart is used to retain the finite-sample correction in the chi-square formula. Its interpretation remains valid only when the variables support the selected 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.
05 Bartletts-Test-Of-Sphericity Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for Bartletts Test of Sphericity. Read Horn retained factors = 3 beside Parallel iterations = 500; the first quantity is not replaced by the second.
The chart is used to distinguish statistical significance from practical factorability. Its interpretation remains valid only when the sample size used in the determinant matches the reported n. 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.
Bartletts Test of Sphericity 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 Bartletts Test of Sphericity.
1. Recompute the determinant from the same ordered nine-variable correlation matrix
Begin by recompute the determinant from the same ordered nine-variable correlation matrix. For the identity-matrix sphericity test, this operation directly connects Bartlett chi-square = 3018.238 with Correlation determinant = 0.00922819. Bartlett chi-square = 3018.238 is read with its degrees of freedom, estimator, sample size, and p-value; it is not a stand-alone effect size.
The governing condition is that observations are independent. 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 Kaiser–Meyer–Olkin Test, because KMO evaluates the balance of zero-order and partial correlations; Bartlett tests identity of the full correlation matrix.
2. Verify df = p(p−1)/2 for p = 9
Next, verify df = p(p−1)/2 for p = 9. For the identity-matrix sphericity test, this operation directly connects Bartlett degrees of freedom = 36 with Overall KMO = 0.713439. Bartlett degrees of freedom = 36 is retained as a distinct supporting quantity for the identity-matrix sphericity test; it is not substituted for the primary result.
The governing condition is that the correlation matrix is positive definite. 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 Parallel Analysis, because Parallel analysis decides how many dimensions outperform random data; Bartlett does not determine dimensionality.
3. Retain the finite-sample correction in the chi-square formula
The third verification is to retain the finite-sample correction in the chi-square formula. For the identity-matrix sphericity test, this operation directly connects Correlation determinant = 0.00922819 with Eigenvalue 1 = 3.195831. Correlation determinant = 0.00922819 is an association for the named variables or constructs; it is not a loading, reliability coefficient, or causal effect.
The governing condition is that the variables support the selected correlation type. 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 Bartlett Test of Homogeneity of Variances, because The homogeneity test concerns group variances and is a different procedure despite sharing Bartlett’s name.
4. Distinguish statistical significance from practical factorability
After the core arithmetic is stable, distinguish statistical significance from practical factorability. For the identity-matrix sphericity test, this operation directly connects Overall KMO = 0.713439 with Eigenvalue 2 = 1.817089. Overall KMO = 0.713439 describes the balance between ordinary and partial correlations, so it informs factorability or item adequacy rather than factor retention.
The governing condition is that the sample size used in the determinant matches the reported n. 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 Kaiser–Meyer–Olkin Test, because KMO evaluates the balance of zero-order and partial correlations; Bartlett tests identity of the full correlation matrix.
5. Inspect item-level MSA even after the global test rejects identity
A robustness review must inspect item-level MSA even after the global test rejects identity. For the identity-matrix sphericity test, this operation directly connects Eigenvalue 1 = 3.195831 with Eigenvalue 3 = 1.393698. Eigenvalue 1 = 3.195831 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
The governing condition is that missing data handling is fixed before the test. 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 Parallel Analysis, because Parallel analysis decides how many dimensions outperform random data; Bartlett does not determine dimensionality.
6. Avoid using Bartlett significance to choose three factors
The final reconciliation should avoid using Bartlett significance to choose three factors. For the identity-matrix sphericity test, this operation directly connects Eigenvalue 2 = 1.817089 with Eigenvalue 4 = 0.846560. Eigenvalue 2 = 1.817089 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
The governing condition is that the chi-square approximation is appropriate for the analysis. 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 Bartlett Test of Homogeneity of Variances, because The homogeneity test concerns group variances and is a different procedure despite sharing Bartlett’s name.
| # | Verification operation | Condition protected | Saved quantity traced |
|---|---|---|---|
| 1 | recompute the determinant from the same ordered nine-variable correlation matrix | observations are independent | Bartlett chi-square = 3018.238 |
| 2 | verify df = p(p−1)/2 for p = 9 | the correlation matrix is positive definite | Bartlett degrees of freedom = 36 |
| 3 | retain the finite-sample correction in the chi-square formula | the variables support the selected correlation type | Correlation determinant = 0.00922819 |
| 4 | distinguish statistical significance from practical factorability | the sample size used in the determinant matches the reported n | Overall KMO = 0.713439 |
| 5 | inspect item-level MSA even after the global test rejects identity | missing data handling is fixed before the test | Eigenvalue 1 = 3.195831 |
| 6 | avoid using Bartlett significance to choose three factors | the chi-square approximation is appropriate for the analysis | Eigenvalue 2 = 1.817089 |
Bartletts Test of Sphericity 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 Bartletts Test of Sphericity formula and output rather than a nearby procedure.
Kaiser–Meyer–Olkin Test
KMO evaluates the balance of zero-order and partial correlations; Bartlett tests identity of the full correlation matrix.
In the current analysis, Bartlett degrees of freedom = 36 remains evidence for the identity-matrix sphericity test; it is not relabeled as a Kaiser–Meyer–Olkin Test result. Bartlett degrees of freedom = 36 is retained as a distinct supporting quantity for the identity-matrix sphericity test; it is not substituted for the primary result.
Parallel Analysis
Parallel analysis decides how many dimensions outperform random data; Bartlett does not determine dimensionality.
In the current analysis, Correlation determinant = 0.00922819 remains evidence for the identity-matrix sphericity test; it is not relabeled as a Parallel Analysis result. Correlation determinant = 0.00922819 is an association for the named variables or constructs; it is not a loading, reliability coefficient, or causal effect.
Bartlett Test of Homogeneity of Variances
The homogeneity test concerns group variances and is a different procedure despite sharing Bartlett’s name.
In the current analysis, Overall KMO = 0.713439 remains evidence for the identity-matrix sphericity test; it is not relabeled as a Bartlett Test of Homogeneity of Variances result. Overall KMO = 0.713439 describes the balance between ordinary and partial correlations, so it informs factorability or item adequacy rather than factor retention.
How to report Bartletts Test of Sphericity
A complete result paragraph includes the value, analytical object, settings, and limitation.
Results paragraph
Bartletts Test of Sphericity was evaluated using the declared data, specification, and software settings. The primary result was Bartlett chi-square = 3018.238; Bartlett degrees of freedom = 36 and Correlation determinant = 0.00922819 supplied supporting context. The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.
The report then states the limitation explicitly: A significant Bartlett result does not prove that a factor model is substantively correct, that every item is adequate, or that a particular number of factors exists. With large samples, even modest departures from identity can be statistically significant.
Settings that must accompany the result
observations are independent; the correlation matrix is positive definite; the variables support the selected correlation type; the sample size used in the determinant matches the reported n.
For Bartletts Test of Sphericity, these details identify the exact version of the analysis and make cross-software reconciliation possible.
Verification actions retained in the record
recompute the determinant from the same ordered nine-variable correlation matrix; verify df = p(p−1)/2 for p = 9; retain the finite-sample correction in the chi-square formula; distinguish statistical significance from practical factorability.
The final wording is revised only after those operations reproduce the saved values.
Bartletts Test of Sphericity decision scenarios
For Bartletts Test of Sphericity, worked conflicts show how the conclusion changes when an input, assumption, or supporting statistic fails.
Boundary-case interpretation: Recompute the determinant from the same ordered nine-variable correlation matrix
Consider a review in which Bartlett chi-square = 3018.238 is reproduced but Bartlett degrees of freedom = 36 is not. For the identity-matrix sphericity test, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to recompute the determinant from the same ordered nine-variable correlation matrix and verify that observations are independent.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Kaiser–Meyer–Olkin Test only for method selection: KMO evaluates the balance of zero-order and partial correlations; Bartlett tests identity of the full correlation matrix. The published conclusion remains The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.
Input-definition sensitivity: Verify df = p(p−1)/2 for p = 9
Consider a review in which Correlation determinant = 0.00922819 is reproduced but Overall KMO = 0.713439 is not. For the identity-matrix sphericity test, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to verify df = p(p−1)/2 for p = 9 and verify that the correlation matrix is positive definite.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis decides how many dimensions outperform random data; Bartlett does not determine dimensionality. The published conclusion remains The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.
Software-definition reconciliation: Retain the finite-sample correction in the chi-square formula
Consider a review in which Eigenvalue 1 = 3.195831 is reproduced but Eigenvalue 2 = 1.817089 is not. For the identity-matrix sphericity test, 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 finite-sample correction in the chi-square formula and verify that the variables support the selected 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 Bartlett Test of Homogeneity of Variances only for method selection: The homogeneity test concerns group variances and is a different procedure despite sharing Bartlett’s name. The published conclusion remains The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.
Local-chart conflict: Distinguish statistical significance from practical factorability
Consider a review in which Eigenvalue 3 = 1.393698 is reproduced but Eigenvalue 4 = 0.846560 is not. For the identity-matrix sphericity test, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to distinguish statistical significance from practical factorability and verify that the sample size used in the determinant matches the reported n.
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–Meyer–Olkin Test only for method selection: KMO evaluates the balance of zero-order and partial correlations; Bartlett tests identity of the full correlation matrix. The published conclusion remains The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.
Alternative-method challenge: Inspect item-level MSA even after the global test rejects identity
Consider a review in which Three-dimension cumulative variance = 71.1846% is reproduced but Horn retained factors = 3 is not. For the identity-matrix sphericity test, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to inspect item-level MSA even after the global test rejects identity and verify that missing data handling is fixed before the test.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis decides how many dimensions outperform random data; Bartlett does not determine dimensionality. The published conclusion remains The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.
Replication and reporting decision: Avoid using Bartlett significance to choose three factors
Consider a review in which Parallel iterations = 500 is reproduced but Observed eigenvalue 3 = 1.393698 is not. For the identity-matrix sphericity test, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to avoid using Bartlett significance to choose three factors and verify that the chi-square approximation is appropriate for the analysis.
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 Bartlett Test of Homogeneity of Variances only for method selection: The homogeneity test concerns group variances and is a different procedure despite sharing Bartlett’s name. The published conclusion remains The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.
Boundary-case interpretation: Recompute the determinant from the same ordered nine-variable correlation matrix
Consider a review in which Horn 95th percentile root 3 = 1.106209 is reproduced but Observed eigenvalue 4 = 0.846560 is not. For the identity-matrix sphericity test, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to recompute the determinant from the same ordered nine-variable correlation matrix and verify that observations are independent.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Kaiser–Meyer–Olkin Test only for method selection: KMO evaluates the balance of zero-order and partial correlations; Bartlett tests identity of the full correlation matrix. The published conclusion remains The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.
Input-definition sensitivity: Verify df = p(p−1)/2 for p = 9
Consider a review in which Horn 95th percentile root 4 = 1.064106 is reproduced but Bartlett chi-square = 3018.238 is not. For the identity-matrix sphericity test, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to verify df = p(p−1)/2 for p = 9 and verify that the correlation matrix is positive definite.
If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis decides how many dimensions outperform random data; Bartlett does not determine dimensionality. The published conclusion remains The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.
Software-definition reconciliation: Retain the finite-sample correction in the chi-square formula
Consider a review in which Bartlett degrees of freedom = 36 is reproduced but Correlation determinant = 0.00922819 is not. For the identity-matrix sphericity test, 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 finite-sample correction in the chi-square formula and verify that the variables support the selected 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 Bartlett Test of Homogeneity of Variances only for method selection: The homogeneity test concerns group variances and is a different procedure despite sharing Bartlett’s name. The published conclusion remains The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.
Bartletts Test of Sphericity downloads and reproducibility files
All linked files belong to the same analysis and remain on onlineinternetcafe.com.
The four files belong to one Bartletts Test of Sphericity 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.
Bartletts Test of Sphericity frequently asked questions
Answers use the worked result and the exact method boundary.
What does Bartletts Test of Sphericity measure?
Bartlett’s test of sphericity is a likelihood-based test of the null hypothesis that the population correlation matrix is an identity matrix. The statistic therefore addresses whether the variables exhibit collective correlation structure, not how many factors should be retained.
What is the main result in this Bartletts Test of Sphericity analysis?
Bartlett chi-square = 3018.238. The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.
What does the result not prove?
A significant Bartlett result does not prove that a factor model is substantively correct, that every item is adequate, or that a particular number of factors exists. With large samples, even modest departures from identity can be statistically significant.
Which supporting value should be reported with the primary result?
Bartlett degrees of freedom = 36 is the first companion quantity. Bartlett degrees of freedom = 36 is retained as a distinct supporting quantity for the identity-matrix sphericity test; it is not substituted for the primary result.
Which assumption is most likely to change the interpretation?
The first requirement is that observations are independent. The result is recomputed if that condition is not satisfied.
What is the most important numerical verification?
The analyst must recompute the determinant from the same ordered nine-variable correlation matrix. That operation traces Bartlett chi-square = 3018.238 to the formula and saved inputs.
Why can software packages disagree on Bartletts Test of Sphericity?
Disagreement can arise because the correlation matrix is positive definite or because the packages implement different estimators, matrices, baselines, rotations, standardizations, bootstrap rules, or coefficient definitions. Matching labels alone is not enough.
How is Bartletts Test of Sphericity different from Kaiser–Meyer–Olkin Test?
KMO evaluates the balance of zero-order and partial correlations; Bartlett tests identity of the full correlation matrix.
How should a chart be interpreted?
Each chart is tied to a named output such as Correlation determinant = 0.00922819. It supports a local calculation or diagnostic and does not replace the full numerical result.
How should Bartletts Test of Sphericity be reported?
Report Bartlett chi-square = 3018.238, the required supporting quantities, sample or panel size, exact method settings, and this qualified conclusion: The very large chi-square relative to 36 degrees of freedom rejects the identity-matrix null. Factor analysis is statistically defensible, but retention and interpretation must come from KMO/MSA, communalities, loadings, residuals, and parallel analysis.