Fornell Larcker Criterion: Formula, Verified Results, Charts and Interpretation
The Fornell–Larcker criterion compares each construct’s square root of AVE with the absolute latent correlations involving that construct. The square-root AVE values form the diagonal of the assessment matrix, while latent correlations occupy off-diagonal cells. This guide uses the supplied real-data results, native MathML equations, matching charts, and separate Python, R, SPSS or AMOS, and Excel verification.
Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
Academic sqrt AVE = 0.934138 is a diagonal construct value that must exceed the absolute interconstruct correlations in its row and column.
What Fornell Larcker Criterion measures
The exact estimand and the result this method is allowed to support.
Fornell Larcker Criterion addresses one defined analytical target: The Fornell–Larcker criterion compares each construct’s square root of AVE with the absolute latent correlations involving that construct. The square-root AVE values form the diagonal of the assessment matrix, while latent correlations occupy off-diagonal cells.
Quantity estimated in this analysis
The square-root-of-ave comparison is reconstructed from the exact variables, matrix, model, panel, or resampling design shown below. The primary output is Academic sqrt AVE = 0.934138; Educational sqrt AVE = 0.683380 supplies the first supporting check. Academic sqrt AVE = 0.934138 is a diagonal construct value that must exceed the absolute interconstruct correlations in its row and column.
For Fornell Larcker Criterion, 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
The rule is not applied to raw observed correlations, and it is not a sensitive stand-alone test of discriminant validity. Passing the criterion cannot overrule high HTMT, problematic cross-loadings, or conceptual overlap.
For Fornell Larcker Criterion, 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 Fornell Larcker Criterion
Research scope, neighboring methods, and excluded claims.
Research question answered
The defensible question is whether the square-root-of-AVE comparison supports the result stated for the declared dataset and analytical specification. It is answered by verify square roots of all three AVE values, followed by compare each diagonal value with every off-diagonal value in its row and column. The evidence is bounded by Academic sqrt AVE = 0.934138 and its named companion quantities.
For Fornell Larcker Criterion, 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
HTMT Ratio: HTMT is more sensitive to discriminant-validity problems and should generally be prioritized.
Average Variance Extracted: AVE is calculated first; Fornell–Larcker then uses its square root in a correlation comparison.
These distinctions determine which formula, output table, and chart can legitimately appear in a Fornell Larcker Criterion post.
Real data used for Fornell Larcker Criterion
Variables, coding, sample or panel size, and the role each input plays.
For Fornell Larcker Criterion, the worked measurement evidence uses 649 complete records unless the method is based on expert ratings. The reflective blocks are Academic Achievement, Educational Advantage, and Social-Alcohol Exposure, with TravelAccess reverse-coded so that higher values indicate easier travel.
The square-root-of-ave comparison is evaluated from the loadings, residual variances, construct correlations, external criterion, or expert judgments appropriate to this method. The post does not transfer a coefficient from another evidence source simply because the same scale names appear.
| 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 |
Fornell Larcker Criterion assumptions and design requirements
Six conditions checked before the coefficient or decision rule is interpreted.
1. AVE comes from the same admissible reflective model
This condition determines whether the input object matches the formula. In the current Fornell Larcker Criterion analysis, the check is to verify square roots of all three AVE values while preserving Academic sqrt AVE = 0.934138.
For Fornell Larcker Criterion, 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. Latent correlations use the same estimator and sample
This requirement controls whether the numerical estimate has the interpretation claimed. In the current Fornell Larcker Criterion analysis, the check is to compare each diagonal value with every off-diagonal value in its row and column while preserving Educational sqrt AVE = 0.683380.
For Fornell Larcker Criterion, 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 square root—not raw AVE—is placed on the diagonal
This design condition prevents an attractive coefficient from being attached to the wrong population or model. In the current Fornell Larcker Criterion analysis, the check is to avoid using observed indicator correlations while preserving Social-Alcohol sqrt AVE = 0.700640.
For Fornell Larcker Criterion, 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. Absolute correlations are used for comparison
This specification rule keeps the software routes numerically comparable. In the current Fornell Larcker Criterion analysis, the check is to check rounding near the diagonal/off-diagonal boundary while preserving Academic–Education factor correlation = 0.313995.
For Fornell Larcker Criterion, 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. Construct ordering is consistent across matrices
This diagnostic requirement is checked before a benchmark is applied. In the current Fornell Larcker Criterion analysis, the check is to label Fornell–Larcker as supplementary when HTMT is available while preserving Academic–Social factor correlation = 0.200278.
For Fornell Larcker Criterion, 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. HTMT and cross-loadings are also reviewed
This final condition governs whether the conclusion can survive replication or sensitivity analysis. In the current Fornell Larcker Criterion analysis, the check is to investigate conceptual overlap even when numeric comparisons pass while preserving Education–Social factor correlation = 0.006832.
For Fornell Larcker Criterion, 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.
Fornell Larcker Criterion hypotheses or decision rule
The statistical question is stated at the correct level for this method.
Statistical question
For Fornell Larcker Criterion, the decision is defined by the named coefficient or evidence criterion. When a bootstrap interval or parameter test is available, its null concerns that exact coefficient or construct pair.
For Fornell Larcker Criterion, a threshold result is one component of a validity argument and cannot by itself establish the intended score interpretation.
Decision for the worked analysis
The calculation yields Academic sqrt AVE = 0.934138. Academic sqrt AVE = 0.934138 is a diagonal construct value that must exceed the absolute interconstruct correlations in its row and column.
Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
Fornell Larcker Criterion formula and worked substitution
Native MathML preserves fractions, roots, summations, matrices, subscripts, and superscripts.
The equation below is the defining mathematical object for Fornell Larcker Criterion. Its symbols are connected to the saved inputs and to Academic sqrt AVE = 0.934138, Educational sqrt AVE = 0.683380, Social-Alcohol sqrt AVE = 0.700640, Academic–Education factor correlation = 0.313995.
The diagonal square-root AVE entries are compared with off-diagonal latent correlations, not with arbitrary observed correlations.
Each diagonal value exceeds the corresponding absolute inter-construct correlations in this model.
Symbol and denominator control
The Fornell–Larcker criterion compares each construct’s square root of AVE with the absolute latent correlations involving that construct. The square-root AVE values form the diagonal of the assessment matrix, while latent correlations occupy off-diagonal cells.
For Fornell Larcker Criterion, 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 Academic sqrt AVE = 0.934138 and Educational sqrt AVE = 0.683380.
The rule is not applied to raw observed correlations, and it is not a sensitive stand-alone test of discriminant validity. Passing the criterion cannot overrule high HTMT, problematic cross-loadings, or conceptual overlap.
Step-by-step Fornell Larcker Criterion calculation
Every stage is tied to a saved value and a method-specific condition.
The worked calculation follows six operations specific to the square-root-of-AVE 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: Verify square roots of all three AVE values.
Numerical trace: Academic sqrt AVE = 0.934138; Educational sqrt AVE = 0.683380.
Condition: AVE comes from the same admissible reflective model. 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: Compare each diagonal value with every off-diagonal value in its row and column.
Numerical trace: Educational sqrt AVE = 0.683380; Social-Alcohol sqrt AVE = 0.700640.
Condition: latent correlations use the same estimator and sample. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Verify the companion quantity
Action: Avoid using observed indicator correlations.
Numerical trace: Social-Alcohol sqrt AVE = 0.700640; Academic–Education factor correlation = 0.313995.
Condition: the square root—not raw AVE—is placed on the diagonal. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Apply the decision rule
Action: Check rounding near the diagonal/off-diagonal boundary.
Numerical trace: Academic–Education factor correlation = 0.313995; Academic–Social factor correlation = 0.200278.
Condition: absolute correlations are used for comparison. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Inspect local evidence
Action: Label Fornell–Larcker as supplementary when HTMT is available.
Numerical trace: Academic–Social factor correlation = 0.200278; Education–Social factor correlation = 0.006832.
Condition: construct ordering is consistent across matrices. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Reconcile and report
Action: Investigate conceptual overlap even when numeric comparisons pass.
Numerical trace: Education–Social factor correlation = 0.006832; Academic Achievement AVE = 0.872614.
Condition: HTMT and cross-loadings are also reviewed. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.
Fornell Larcker Criterion results and interpretation
Primary and supporting statistics are kept separate and precisely labeled.
Primary result
Academic sqrt AVE
Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
Why the result is internally coherent
Academic sqrt AVE = 0.934138 is a diagonal construct value that must exceed the absolute interconstruct correlations in its row and column.
Educational sqrt AVE = 0.683380 is a diagonal construct value that must exceed the absolute interconstruct correlations in its row and column.
For Fornell Larcker Criterion, 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 |
|---|---|---|
| Academic sqrt AVE | 0.934138 | Academic sqrt AVE = 0.934138 is a diagonal construct value that must exceed the absolute interconstruct correlations in its row and column. |
| Educational sqrt AVE | 0.683380 | Educational sqrt AVE = 0.683380 is a diagonal construct value that must exceed the absolute interconstruct correlations in its row and column. |
| Social-Alcohol sqrt AVE | 0.700640 | Social-Alcohol sqrt AVE = 0.700640 is a diagonal construct value that must exceed the absolute interconstruct correlations in its row and column. |
| Academic–Education factor correlation | 0.313995 | Academic–Education factor correlation = 0.313995 is an association for the named variables or constructs; it is not a loading, reliability coefficient, or causal effect. |
| Academic–Social factor correlation | 0.200278 | Academic–Social factor correlation = 0.200278 is an association for the named variables or constructs; it is not a loading, reliability coefficient, or causal effect. |
| Education–Social factor correlation | 0.006832 | Education–Social factor correlation = 0.006832 is an association for the named variables or constructs; it is not a loading, reliability coefficient, or causal effect. |
| Academic Achievement AVE | 0.872614 | Academic Achievement AVE = 0.872614 is above the .50 captured-variance reference; the judgment applies to the named construct rather than the whole instrument. |
| Educational Advantage AVE | 0.467009 | Educational Advantage AVE = 0.467009 is below the .50 captured-variance reference; the judgment applies to the named construct rather than the whole instrument. |
| Social-Alcohol Exposure AVE | 0.490896 | Social-Alcohol Exposure AVE = 0.490896 is below the .50 captured-variance reference; the judgment applies to the named construct rather than the whole instrument. |
| Academic Achievement CR | 0.953517 | Academic Achievement CR = 0.953517 summarizes loading-weighted consistency; it is interpreted with the loadings and residual variances used in the same fitted measurement model. |
| Educational Advantage CR | 0.696353 | Educational Advantage CR = 0.696353 summarizes loading-weighted consistency; it is interpreted with the loadings and residual variances used in the same fitted measurement model. |
| Social-Alcohol Exposure CR | 0.724808 | Social-Alcohol Exposure CR = 0.724808 summarizes loading-weighted consistency; it is interpreted with the loadings and residual variances used in the same fitted measurement model. |
| HTMT Academic Achievement vs Educational Advantage | 0.361500 | HTMT Academic Achievement vs Educational Advantage = 0.361500 is a pairwise construct-separation ratio; a bootstrap interval is preferable when the result lies near the selected boundary. |
| HTMT Academic Achievement vs Social-Alcohol Exposure | 0.244980 | HTMT Academic Achievement vs Social-Alcohol Exposure = 0.244980 is a pairwise construct-separation ratio; a bootstrap interval is preferable when the result lies near the selected boundary. |
Fornell Larcker Criterion 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 square-root-of-AVE comparison from the declared data and analytical specification. It must reproduce Academic sqrt AVE = 0.934138 and retain Educational sqrt AVE = 0.683380 as a separate supporting quantity.
The code is read as an executable analysis, not as a printed answer. Its critical verification is to verify square roots of all three AVE values; the associated design condition is that AVE comes from the same admissible reflective model. The rule is not applied to raw observed correlations, and it is not a sensitive stand-alone test of discriminant validity. Passing the criterion cannot overrule high HTMT, problematic cross-loadings, or conceptual overlap.
import numpy as np
import pandas as pdconstructs = ["Academic Achievement", "Educational Advantage", "Social-Alcohol Exposure"]
ave = np.array([0.872614, 0.467009, 0.490896])
latent_r = np.array([
[1.000000, 0.313995, 0.200278],
[0.313995, 1.000000, 0.006832],
[0.200278, 0.006832, 1.000000],
])
fl = latent_r.copy()
np.fill_diagonal(fl, np.sqrt(ave))
result = pd.DataFrame(fl, index=constructs, columns=constructs)
print(result.round(6))
for j in range(3):
assert np.sqrt(ave[j]) > np.max(np.abs(np.delete(latent_r[j], j)))
Fornell Larcker Criterion in R
The R route declares package, estimator, extraction, rotation, or resampling settings.
The R route uses semTools and the displayed arguments to estimate the square-root-of-AVE 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 Academic sqrt AVE = 0.934138 after the analyst compare each diagonal value with every off-diagonal value in its row and column. 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(semTools)
# Fit the CFA model, then request HTMT and AVE from the fitted lavaan object.
htmt(model, data=d); reliability(fit); AVE(fit)Fornell Larcker Criterion 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 square-root-of-AVE 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 Academic sqrt AVE = 0.934138 and the settings needed to reproduce it. The software review specifically avoid using observed indicator correlations, while preserving the requirement that the square root—not raw AVE—is placed on the diagonal.
* Prepare the standardized loading, residual-variance, construct-correlation, or expert-rating table required for Fornell Larcker Criterion.
* Use MATRIX, AMOS, or validated R/Python integration when base SPSS does not expose the coefficient.
* Do not rename a different SPSS statistic as Fornell Larcker Criterion.Fornell Larcker Criterion in Excel
The workbook exposes source values, intermediate arithmetic, and the final formula.
The Excel workbook is an arithmetic audit for the square-root-of-AVE comparison. Named cells retain the inputs, intermediate components, and final formula leading to Academic sqrt AVE = 0.934138; 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 check rounding near the diagonal/off-diagonal boundary and documents Educational sqrt AVE = 0.683380 independently.
Data: 649 rows with documented coding.
Inputs: named cells or ranges required only by Fornell Larcker Criterion.
Calculation: =SQRT(AVE_Cell) then compare with ABS(Latent_Correlation_Cell)
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.Fornell Larcker Criterion 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 Fornell Larcker Criterion 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 Fornell-Larcker-Criterion Fornell Larcker Matrix
This panel shows the cell-level pattern that a single coefficient can conceal for Fornell Larcker Criterion. Read Academic sqrt AVE = 0.934138 beside Educational sqrt AVE = 0.683380; the first quantity is not replaced by the second.
The chart is used to verify square roots of all three AVE values. Its interpretation remains valid only when AVE comes from the same admissible reflective model. 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 Fornell-Larcker-Criterion Construct Margins
This panel provides a visual diagnostic tied to the method’s exact decision rule for Fornell Larcker Criterion. Read Educational sqrt AVE = 0.683380 beside Social-Alcohol sqrt AVE = 0.700640; the first quantity is not replaced by the second.
The chart is used to compare each diagonal value with every off-diagonal value in its row and column. Its interpretation remains valid only when latent correlations use the same estimator and sample. 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 Fornell-Larcker-Criterion Latent Correlation Matrix
This panel shows the cell-level pattern that a single coefficient can conceal for Fornell Larcker Criterion. Read Social-Alcohol sqrt AVE = 0.700640 beside Academic–Education factor correlation = 0.313995; the first quantity is not replaced by the second.
The chart is used to avoid using observed indicator correlations. Its interpretation remains valid only when the square root—not raw AVE—is placed on the diagonal. 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 Fornell-Larcker-Criterion Source G1 Distribution
This panel shows sampling or resampling uncertainty around the reported estimate for Fornell Larcker Criterion. Read Academic–Education factor correlation = 0.313995 beside Academic–Social factor correlation = 0.200278; the first quantity is not replaced by the second.
The chart is used to check rounding near the diagonal/off-diagonal boundary. Its interpretation remains valid only when absolute correlations are used for comparison. 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 Fornell-Larcker-Criterion Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for Fornell Larcker Criterion. Read Academic–Social factor correlation = 0.200278 beside Education–Social factor correlation = 0.006832; the first quantity is not replaced by the second.
The chart is used to label Fornell–Larcker as supplementary when HTMT is available. Its interpretation remains valid only when construct ordering is consistent across matrices. 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 Fornell-Larcker-Criterion Primary Metrics
This panel reconciles the headline estimate with its principal supporting values for Fornell Larcker Criterion. Read Education–Social factor correlation = 0.006832 beside Academic Achievement AVE = 0.872614; the first quantity is not replaced by the second.
The chart is used to investigate conceptual overlap even when numeric comparisons pass. Its interpretation remains valid only when HTMT and cross-loadings are also reviewed. 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 Fornell-Larcker-Criterion Fornell Larcker Matrix
This panel shows the cell-level pattern that a single coefficient can conceal for Fornell Larcker Criterion. Read Academic Achievement AVE = 0.872614 beside Educational Advantage AVE = 0.467009; the first quantity is not replaced by the second.
The chart is used to verify square roots of all three AVE values. Its interpretation remains valid only when AVE comes from the same admissible reflective model. 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 Fornell-Larcker-Criterion Construct Margins
This panel provides a visual diagnostic tied to the method’s exact decision rule for Fornell Larcker Criterion. Read Educational Advantage AVE = 0.467009 beside Social-Alcohol Exposure AVE = 0.490896; the first quantity is not replaced by the second.
The chart is used to compare each diagonal value with every off-diagonal value in its row and column. Its interpretation remains valid only when latent correlations use the same estimator and sample. 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 Fornell-Larcker-Criterion Latent Correlation Matrix
This panel shows the cell-level pattern that a single coefficient can conceal for Fornell Larcker Criterion. Read Social-Alcohol Exposure AVE = 0.490896 beside Academic Achievement CR = 0.953517; the first quantity is not replaced by the second.
The chart is used to avoid using observed indicator correlations. Its interpretation remains valid only when the square root—not raw AVE—is placed on the diagonal. 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 Fornell-Larcker-Criterion Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for Fornell Larcker Criterion. Read Academic Achievement CR = 0.953517 beside Educational Advantage CR = 0.696353; the first quantity is not replaced by the second.
The chart is used to check rounding near the diagonal/off-diagonal boundary. Its interpretation remains valid only when absolute correlations are used for comparison. 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.
Fornell Larcker Criterion 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 Fornell Larcker Criterion.
1. Verify square roots of all three AVE values
Begin by verify square roots of all three AVE values. For the square-root-of-AVE comparison, this operation directly connects Academic sqrt AVE = 0.934138 with Social-Alcohol sqrt AVE = 0.700640. Academic sqrt AVE = 0.934138 is a diagonal construct value that must exceed the absolute interconstruct correlations in its row and column.
The governing condition is that AVE comes from the same admissible reflective model. 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 HTMT Ratio, because HTMT is more sensitive to discriminant-validity problems and should generally be prioritized.
2. Compare each diagonal value with every off-diagonal value in its row and column
Next, compare each diagonal value with every off-diagonal value in its row and column. For the square-root-of-AVE comparison, this operation directly connects Educational sqrt AVE = 0.683380 with Academic–Education factor correlation = 0.313995. Educational sqrt AVE = 0.683380 is a diagonal construct value that must exceed the absolute interconstruct correlations in its row and column.
The governing condition is that latent correlations use the same estimator and sample. 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 Average Variance Extracted, because AVE is calculated first; Fornell–Larcker then uses its square root in a correlation comparison.
3. Avoid using observed indicator correlations
The third verification is to avoid using observed indicator correlations. For the square-root-of-AVE comparison, this operation directly connects Social-Alcohol sqrt AVE = 0.700640 with Academic–Social factor correlation = 0.200278. Social-Alcohol sqrt AVE = 0.700640 is a diagonal construct value that must exceed the absolute interconstruct correlations in its row and column.
The governing condition is that the square root—not raw AVE—is placed on the diagonal. 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 Cross-Loadings, because Cross-loading inspection localizes indicator overlap but is also less reliable as a sole discriminant-validity test.
4. Check rounding near the diagonal/off-diagonal boundary
After the core arithmetic is stable, check rounding near the diagonal/off-diagonal boundary. For the square-root-of-AVE comparison, this operation directly connects Academic–Education factor correlation = 0.313995 with Education–Social factor correlation = 0.006832. Academic–Education factor correlation = 0.313995 is an association for the named variables or constructs; it is not a loading, reliability coefficient, or causal effect.
The governing condition is that absolute correlations are used for comparison. 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 HTMT Ratio, because HTMT is more sensitive to discriminant-validity problems and should generally be prioritized.
5. Label Fornell–Larcker as supplementary when HTMT is available
A robustness review must label Fornell–Larcker as supplementary when HTMT is available. For the square-root-of-AVE comparison, this operation directly connects Academic–Social factor correlation = 0.200278 with Academic Achievement AVE = 0.872614. Academic–Social factor correlation = 0.200278 is an association for the named variables or constructs; it is not a loading, reliability coefficient, or causal effect.
The governing condition is that construct ordering is consistent across matrices. 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 Average Variance Extracted, because AVE is calculated first; Fornell–Larcker then uses its square root in a correlation comparison.
6. Investigate conceptual overlap even when numeric comparisons pass
The final reconciliation should investigate conceptual overlap even when numeric comparisons pass. For the square-root-of-AVE comparison, this operation directly connects Education–Social factor correlation = 0.006832 with Educational Advantage AVE = 0.467009. Education–Social factor correlation = 0.006832 is an association for the named variables or constructs; it is not a loading, reliability coefficient, or causal effect.
The governing condition is that HTMT and cross-loadings are also reviewed. 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 Cross-Loadings, because Cross-loading inspection localizes indicator overlap but is also less reliable as a sole discriminant-validity test.
| # | Verification operation | Condition protected | Saved quantity traced |
|---|---|---|---|
| 1 | verify square roots of all three AVE values | AVE comes from the same admissible reflective model | Academic sqrt AVE = 0.934138 |
| 2 | compare each diagonal value with every off-diagonal value in its row and column | latent correlations use the same estimator and sample | Educational sqrt AVE = 0.683380 |
| 3 | avoid using observed indicator correlations | the square root—not raw AVE—is placed on the diagonal | Social-Alcohol sqrt AVE = 0.700640 |
| 4 | check rounding near the diagonal/off-diagonal boundary | absolute correlations are used for comparison | Academic–Education factor correlation = 0.313995 |
| 5 | label Fornell–Larcker as supplementary when HTMT is available | construct ordering is consistent across matrices | Academic–Social factor correlation = 0.200278 |
| 6 | investigate conceptual overlap even when numeric comparisons pass | HTMT and cross-loadings are also reviewed | Education–Social factor correlation = 0.006832 |
Fornell Larcker Criterion 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 Fornell Larcker Criterion formula and output rather than a nearby procedure.
HTMT Ratio
HTMT is more sensitive to discriminant-validity problems and should generally be prioritized.
In the current analysis, Educational sqrt AVE = 0.683380 remains evidence for the square-root-of-AVE comparison; it is not relabeled as a HTMT Ratio result. Educational sqrt AVE = 0.683380 is a diagonal construct value that must exceed the absolute interconstruct correlations in its row and column.
Average Variance Extracted
AVE is calculated first; Fornell–Larcker then uses its square root in a correlation comparison.
In the current analysis, Social-Alcohol sqrt AVE = 0.700640 remains evidence for the square-root-of-AVE comparison; it is not relabeled as a Average Variance Extracted result. Social-Alcohol sqrt AVE = 0.700640 is a diagonal construct value that must exceed the absolute interconstruct correlations in its row and column.
Cross-Loadings
Cross-loading inspection localizes indicator overlap but is also less reliable as a sole discriminant-validity test.
In the current analysis, Academic–Education factor correlation = 0.313995 remains evidence for the square-root-of-AVE comparison; it is not relabeled as a Cross-Loadings result. Academic–Education factor correlation = 0.313995 is an association for the named variables or constructs; it is not a loading, reliability coefficient, or causal effect.
How to report Fornell Larcker Criterion
A complete result paragraph includes the value, analytical object, settings, and limitation.
Results paragraph
Fornell Larcker Criterion was evaluated using the declared data, specification, and software settings. The primary result was Academic sqrt AVE = 0.934138; Educational sqrt AVE = 0.683380 and Social-Alcohol sqrt AVE = 0.700640 supplied supporting context. Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
The report then states the limitation explicitly: The rule is not applied to raw observed correlations, and it is not a sensitive stand-alone test of discriminant validity. Passing the criterion cannot overrule high HTMT, problematic cross-loadings, or conceptual overlap.
Settings that must accompany the result
AVE comes from the same admissible reflective model; latent correlations use the same estimator and sample; the square root—not raw AVE—is placed on the diagonal; absolute correlations are used for comparison.
For Fornell Larcker Criterion, these details identify the exact version of the analysis and make cross-software reconciliation possible.
Verification actions retained in the record
verify square roots of all three AVE values; compare each diagonal value with every off-diagonal value in its row and column; avoid using observed indicator correlations; check rounding near the diagonal/off-diagonal boundary.
The final wording is revised only after those operations reproduce the saved values.
Fornell Larcker Criterion decision scenarios
For Fornell Larcker Criterion, worked conflicts show how the conclusion changes when an input, assumption, or supporting statistic fails.
Boundary-case interpretation: Verify square roots of all three AVE values
Consider a review in which Academic sqrt AVE = 0.934138 is reproduced but Educational sqrt AVE = 0.683380 is not. For the square-root-of-AVE 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 square roots of all three AVE values and verify that AVE comes from the same admissible reflective model.
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 HTMT Ratio only for method selection: HTMT is more sensitive to discriminant-validity problems and should generally be prioritized. The published conclusion remains Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
Input-definition sensitivity: Compare each diagonal value with every off-diagonal value in its row and column
Consider a review in which Social-Alcohol sqrt AVE = 0.700640 is reproduced but Academic–Education factor correlation = 0.313995 is not. For the square-root-of-AVE 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 each diagonal value with every off-diagonal value in its row and column and verify that latent correlations use the same estimator and sample.
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 Average Variance Extracted only for method selection: AVE is calculated first; Fornell–Larcker then uses its square root in a correlation comparison. The published conclusion remains Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
Software-definition reconciliation: Avoid using observed indicator correlations
Consider a review in which Academic–Social factor correlation = 0.200278 is reproduced but Education–Social factor correlation = 0.006832 is not. For the square-root-of-AVE 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 using observed indicator correlations and verify that the square root—not raw AVE—is placed on the diagonal.
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 Cross-Loadings only for method selection: Cross-loading inspection localizes indicator overlap but is also less reliable as a sole discriminant-validity test. The published conclusion remains Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
Local-chart conflict: Check rounding near the diagonal/off-diagonal boundary
Consider a review in which Academic Achievement AVE = 0.872614 is reproduced but Educational Advantage AVE = 0.467009 is not. For the square-root-of-AVE 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 rounding near the diagonal/off-diagonal boundary and verify that absolute correlations are used for comparison.
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 HTMT Ratio only for method selection: HTMT is more sensitive to discriminant-validity problems and should generally be prioritized. The published conclusion remains Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
Alternative-method challenge: Label Fornell–Larcker as supplementary when HTMT is available
Consider a review in which Social-Alcohol Exposure AVE = 0.490896 is reproduced but Academic Achievement CR = 0.953517 is not. For the square-root-of-AVE comparison, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to label Fornell–Larcker as supplementary when HTMT is available and verify that construct ordering is consistent across matrices.
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 Average Variance Extracted only for method selection: AVE is calculated first; Fornell–Larcker then uses its square root in a correlation comparison. The published conclusion remains Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
Replication and reporting decision: Investigate conceptual overlap even when numeric comparisons pass
Consider a review in which Educational Advantage CR = 0.696353 is reproduced but Social-Alcohol Exposure CR = 0.724808 is not. For the square-root-of-AVE comparison, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to investigate conceptual overlap even when numeric comparisons pass and verify that HTMT and cross-loadings are also reviewed.
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 Cross-Loadings only for method selection: Cross-loading inspection localizes indicator overlap but is also less reliable as a sole discriminant-validity test. The published conclusion remains Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
Boundary-case interpretation: Verify square roots of all three AVE values
Consider a review in which HTMT Academic Achievement vs Educational Advantage = 0.361500 is reproduced but HTMT Academic Achievement vs Social-Alcohol Exposure = 0.244980 is not. For the square-root-of-AVE 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 square roots of all three AVE values and verify that AVE comes from the same admissible reflective model.
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 HTMT Ratio only for method selection: HTMT is more sensitive to discriminant-validity problems and should generally be prioritized. The published conclusion remains Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
Input-definition sensitivity: Compare each diagonal value with every off-diagonal value in its row and column
Consider a review in which HTMT Educational Advantage vs Social-Alcohol Exposure = 0.087231 is reproduced but Academic sqrt AVE = 0.934138 is not. For the square-root-of-AVE 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 each diagonal value with every off-diagonal value in its row and column and verify that latent correlations use the same estimator and sample.
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 Average Variance Extracted only for method selection: AVE is calculated first; Fornell–Larcker then uses its square root in a correlation comparison. The published conclusion remains Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
Software-definition reconciliation: Avoid using observed indicator correlations
Consider a review in which Educational sqrt AVE = 0.683380 is reproduced but Social-Alcohol sqrt AVE = 0.700640 is not. For the square-root-of-AVE 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 using observed indicator correlations and verify that the square root—not raw AVE—is placed on the diagonal.
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 Cross-Loadings only for method selection: Cross-loading inspection localizes indicator overlap but is also less reliable as a sole discriminant-validity test. The published conclusion remains Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
Local-chart conflict: Check rounding near the diagonal/off-diagonal boundary
Consider a review in which Academic–Education factor correlation = 0.313995 is reproduced but Academic–Social factor correlation = 0.200278 is not. For the square-root-of-AVE 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 rounding near the diagonal/off-diagonal boundary and verify that absolute correlations are used for comparison.
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 HTMT Ratio only for method selection: HTMT is more sensitive to discriminant-validity problems and should generally be prioritized. The published conclusion remains Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
Fornell Larcker Criterion downloads and reproducibility files
All linked files belong to the same analysis and remain on onlineinternetcafe.com.
The four files belong to one Fornell Larcker Criterion 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.
Fornell Larcker Criterion frequently asked questions
Answers use the worked result and the exact method boundary.
What does Fornell Larcker Criterion measure?
The Fornell–Larcker criterion compares each construct’s square root of AVE with the absolute latent correlations involving that construct. The square-root AVE values form the diagonal of the assessment matrix, while latent correlations occupy off-diagonal cells.
What is the main result in this Fornell Larcker Criterion analysis?
Academic sqrt AVE = 0.934138. Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.
What does the result not prove?
The rule is not applied to raw observed correlations, and it is not a sensitive stand-alone test of discriminant validity. Passing the criterion cannot overrule high HTMT, problematic cross-loadings, or conceptual overlap.
Which supporting value should be reported with the primary result?
Educational sqrt AVE = 0.683380 is the first companion quantity. Educational sqrt AVE = 0.683380 is a diagonal construct value that must exceed the absolute interconstruct correlations in its row and column.
Which assumption is most likely to change the interpretation?
The first requirement is that AVE comes from the same admissible reflective model. The result is recomputed if that condition is not satisfied.
What is the most important numerical verification?
The analyst must verify square roots of all three AVE values. That operation traces Academic sqrt AVE = 0.934138 to the formula and saved inputs.
Why can software packages disagree on Fornell Larcker Criterion?
Disagreement can arise because latent correlations use the same estimator and sample or because the packages implement different estimators, matrices, baselines, rotations, standardizations, bootstrap rules, or coefficient definitions. Matching labels alone is not enough.
How is Fornell Larcker Criterion different from HTMT Ratio?
HTMT is more sensitive to discriminant-validity problems and should generally be prioritized.
How should a chart be interpreted?
Each chart is tied to a named output such as Social-Alcohol sqrt AVE = 0.700640. It supports a local calculation or diagnostic and does not replace the full numerical result.
How should Fornell Larcker Criterion be reported?
Report Academic sqrt AVE = 0.934138, the required supporting quantities, sample or panel size, exact method settings, and this qualified conclusion: Each square-root AVE value exceeds the corresponding absolute factor correlations in the worked model. The criterion therefore passes, but HTMT and its bootstrap interval provide the stronger separation check.