Chapter map

1. 1. Variable alignment
2. 2. Reference loadings
3. 3. Candidate loadings
4. 4. Cross-products
5. 5. Dot product
6. 6. Reference length
7. 7. Candidate length
8. 8. Denominator
9. 9. Congruence
10. 10. Competitor
11. 11. Competitor score
12. 12. Sign reversal
13. 13. Scale invariance
14. 14. Permutation
15. 15. Sampling
16. 16. Claim

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

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1. Variable alignment

Here the geometry becomes operational. Use the same four variables in the same order.

A concrete situation

Correspondence is fixed.

What the mathematics does not license

Mismatched rows invalidate the coefficient. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record names. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. Reference loadings

This step deserves a careful pause. Set a=[.8,.6,.2,-.1].

A concrete situation

Four signed loadings define one factor vector.

What the mathematics does not license

Pattern and structure loadings differ. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Name type. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. Candidate loadings

The useful question is not only how to calculate, but what the calculation assumes. Set b=[.75,.55,.25,-.05].

A concrete situation

Signs and relative magnitudes are similar.

What the mathematics does not license

Rotation must be documented. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record b. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. Cross-products

A small example makes the mechanism visible. Multiply corresponding entries.

A concrete situation

Products are .60,.33,.05,.005.

What the mathematics does not license

Keep the negative-negative product positive. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record terms. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Dot product

This is where modelling choices begin to affect evidence. Add cross-products.

A concrete situation

a·b=.985.

What the mathematics does not license

Do not centre loadings. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record numerator. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. Reference length

The central mathematical idea is concrete. Sum a².

A concrete situation

||a||²=1.05.

What the mathematics does not license

Use all variables. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record value. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Candidate length

Here the geometry becomes operational. Sum b².

A concrete situation

||b||²=.93.

What the mathematics does not license

Missing loadings change the vector. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record value. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Denominator

This step deserves a careful pause. Compute sqrt(1.05×.93).

A concrete situation

Denominator=.988180.

What the mathematics does not license

Keep precision. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record value. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. Congruence

The useful question is not only how to calculate, but what the calculation assumes. Divide .985 by .988180.

A concrete situation

φ=.996782.

What the mathematics does not license

It is an uncentred cosine. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record score. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. Competitor

A small example makes the mechanism visible. Use b₂=[.7,.2,.3,-.2].

A concrete situation

Dot=.76 and ||b₂||²=.66.

What the mathematics does not license

Same variables remain aligned. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record values. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Competitor score

This is where modelling choices begin to affect evidence. Compute .76/sqrt(1.05×.66).

A concrete situation

φ=.912950.

What the mathematics does not license

First vector is more proportional. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Compare. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. Sign reversal

The central mathematical idea is concrete. Multiply every b loading by -1.

A concrete situation

φ becomes -.996782.

What the mathematics does not license

Factor signs may be indeterminate. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Align deliberately. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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13. Scale invariance

Here the geometry becomes operational. Multiply b by three.

A concrete situation

φ is unchanged.

What the mathematics does not license

Proportionality ignores length. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Verify. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. Permutation

This step deserves a careful pause. Swap candidate factors.

A concrete situation

Best matches can change.

What the mathematics does not license

Avoid post-hoc matching without disclosure. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record algorithm. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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15. Sampling

The useful question is not only how to calculate, but what the calculation assumes. Refit factors in bootstrap samples.

A concrete situation

Congruence has uncertainty.

What the mathematics does not license

Cross-loadings and rotation can destabilise. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Summarise. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. Claim

A small example makes the mechanism visible. State loadings, rotation, matching and signs.

A concrete situation

First vector is more congruent with a.

What the mathematics does not license

Do not claim measurement invariance. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Bound conclusion. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

Tucker’s coefficient of congruence compares two factor-loading vectors as an uncentred cosine. Multiply corresponding loadings, sum the products and divide by the product of the two vector lengths. Values near one indicate proportional vectors in the same direction, values near minus one indicate opposite directions and values near zero indicate near-orthogonality.

Let factor loading vector a=[.8,.6,.2,-.1] and b=[.75,.55,.25,-.05] for the same four variables in the same order. Their dot product is .985. Squared lengths are 1.05 and .93, so the denominator is sqrt(1.05×.93)=.988180. Congruence is .985/.988180=.996782.

A competitor b₂=[.7,.2,.3,-.2] has dot product .76 and squared length .66. Its denominator is sqrt(1.05×.66)=.832466, giving congruence .912950. The first vector is more proportional to a. If every sign in b is reversed, congruence becomes -.996782; factor-analysis sign indeterminacy must be handled explicitly.

Congruence is not a centred Pearson correlation and has no context-free pass mark. Variable order, loading type, rotation, factor matching, sign alignment, scaling, sampling error and cross-loadings matter. High similarity does not establish measurement invariance, equal scores, causal equivalence or predictive validity. Report both vectors, matching procedure, uncertainty and substantive interpretation.

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Frequently asked questions

What is Tucker’s congruence coefficient?

The uncentred cosine between two factor-loading vectors on the same ordered variables.

How does it differ from correlation?

It uses raw cross-products rather than deviations from each vector mean.

What does a negative value mean?

The loading vectors point in opposite directions; factor sign indeterminacy may make a global sign flip equivalent.

Is there a universal cutoff?

No. Published rules of thumb require context, matching, uncertainty and substantive judgment.

Does high congruence prove measurement invariance?

No. Stronger model and parameter tests are required.

What must be reported?

Variables, loading type, rotation, factor matching, sign policy, both vectors, coefficient, uncertainty, cross-loadings and substantive interpretation.

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Primary source and fact-check note

Ledyard R. Tucker, “A Method for Synthesis of Factor Analysis Studies,” Personnel Research Section Report No. 984, Department of the Army (1951); interpretation checked against Lorenzo-Seva and ten Berge (2006), DOI 10.1027/1614-2241.2.2.57. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• Procrustes analysis for configurations
• Representational similarity analysis
• Vector spaces in AI
• How Mathematics Works: Statistics
• eduKateSingapore learning routes

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: configuration evidence and responsible use

Loading audit

Use the same variables, order, loading type, rotation and missing-value rule in both vectors.

Matching audit

Record factor permutation and sign alignment without searching every option and reporting only the maximum.

Arithmetic audit

Show dot product, squared lengths, denominator and signed coefficient at full precision.

Replicability audit

Use resampling or independent samples and inspect cross-loadings instead of relying on one threshold.

Mechanism workshop: calculate, challenge and transfer

Manual reconstruction

Rebuild the example with identifiers and record loading vectors, cross-products, squared lengths, denominator, signed congruence, factor match and uncertainty. Explain every transformation before software.

One-entry experiment

Change one distance, coordinate, loading or row while holding the comparison rule fixed. Recalculate every affected matrix cell and score.

Scaling experiment

Apply uniform and unequal scaling. State which changes are removed by the method and which alter the evidence.

Sign or reflection experiment

Reverse an axis or mirror a configuration. Apply the declared invariance policy and explain the scientific consequence.

Correspondence experiment

Swap two object, landmark or variable labels. Show why arithmetic can remain tidy while the comparison becomes meaningless.

Stability experiment

Repeat samples, starts, rotations, factor matches or preprocessing choices and report a distribution.

Parent explanation

Explain why two maps can look different yet match after alignment, or look similar while their distance or loading structure differs.

Student transfer

Apply the mechanism to school maps, sensor tables, questionnaire factors or image embeddings using real identifiers.

Limit statement

Write one supported claim, one overclaim and one repaired claim naming preprocessing, invariances and uncertainty.

Next measurement

Choose a complementary local, global, predictive or substantive check and explain its separate question.

Evidence notebook: reconstruct every mathematical step

Evidence notebook 1: Variable alignment

Notebook step 1 isolates variable alignment. Use the same four variables in the same order. The reproducible result is: Correspondence is fixed. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: Mismatched rows invalidate the coefficient. Record names. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 2: Reference loadings

Notebook step 2 isolates reference loadings. Set a=[.8,.6,.2,-.1]. The reproducible result is: Four signed loadings define one factor vector. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: Pattern and structure loadings differ. Name type. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 3: Candidate loadings

Notebook step 3 isolates candidate loadings. Set b=[.75,.55,.25,-.05]. The reproducible result is: Signs and relative magnitudes are similar. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: Rotation must be documented. Record b. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 4: Cross-products

Notebook step 4 isolates cross-products. Multiply corresponding entries. The reproducible result is: Products are .60,.33,.05,.005. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: Keep the negative-negative product positive. Record terms. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 5: Dot product

Notebook step 5 isolates dot product. Add cross-products. The reproducible result is: a·b=.985. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: Do not centre loadings. Record numerator. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 6: Reference length

Notebook step 6 isolates reference length. Sum a². The reproducible result is: ||a||²=1.05. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: Use all variables. Record value. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 7: Candidate length

Notebook step 7 isolates candidate length. Sum b². The reproducible result is: ||b||²=.93. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: Missing loadings change the vector. Record value. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 8: Denominator

Notebook step 8 isolates denominator. Compute sqrt(1.05×.93). The reproducible result is: Denominator=.988180. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: Keep precision. Record value. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 9: Congruence

Notebook step 9 isolates congruence. Divide .985 by .988180. The reproducible result is: φ=.996782. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: It is an uncentred cosine. Record score. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 10: Competitor

Notebook step 10 isolates competitor. Use b₂=[.7,.2,.3,-.2]. The reproducible result is: Dot=.76 and ||b₂||²=.66. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: Same variables remain aligned. Record values. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 11: Competitor score

Notebook step 11 isolates competitor score. Compute .76/sqrt(1.05×.66). The reproducible result is: φ=.912950. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: First vector is more proportional. Compare. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 12: Sign reversal

Notebook step 12 isolates sign reversal. Multiply every b loading by -1. The reproducible result is: φ becomes -.996782. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: Factor signs may be indeterminate. Align deliberately. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 13: Scale invariance

Notebook step 13 isolates scale invariance. Multiply b by three. The reproducible result is: φ is unchanged. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: Proportionality ignores length. Verify. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 14: Permutation

Notebook step 14 isolates permutation. Swap candidate factors. The reproducible result is: Best matches can change. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: Avoid post-hoc matching without disclosure. Record algorithm. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 15: Sampling

Notebook step 15 isolates sampling. Refit factors in bootstrap samples. The reproducible result is: Congruence has uncertainty. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: Cross-loadings and rotation can destabilise. Summarise. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 16: Claim

Notebook step 16 isolates claim. State loadings, rotation, matching and signs. The reproducible result is: First vector is more congruent with a. The evidence concerns proportionality of matched loading vectors. Keep object, landmark, variable or row identities beside every value. The safeguard is: Do not claim measurement invariance. Bound conclusion. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Failure cases and responsible recovery

Factor permutation

The same factors appear in another column order. Use a declared matching rule and report the full congruence matrix. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.

Global sign flip

A factor reverses sign but represents the same axis in factor analysis. Align signs transparently before interpretation. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.

Uncentred versus centred

Pearson correlation and congruence answer different questions. Compute the stated uncentred cosine. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.

Cross-loadings

A high target-factor coefficient can coexist with changed cross-loadings. Inspect the entire loading pattern. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.

Rotation dependence

Different rotations produce different loading vectors. Match like with like and preserve the rotation record. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.

Threshold worship

A rule-of-thumb cutoff cannot replace sampling uncertainty, content coverage and substantive judgment. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.

Missing variables

Dropping difficult variables can inflate similarity. Report the common set and exclusions. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.

Responsible conclusion

Claim proportional loading patterns under stated choices, not measurement invariance or score equivalence. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.