Chapter map

1. 1. Dissimilarity order
2. 2. Configuration distances
3. 3. Fitted disparities
4. 4. Residuals
5. 5. Residual squares
6. 6. Distance squares
7. 7. Ratio
8. 8. Square root
9. 9. Competing disparities
10. 10. Competing residuals
11. 11. Competing stress
12. 12. Ties
13. 13. Missing proximities
14. 14. Dimension
15. 15. Starts
16. 16. Claim

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1. Dissimilarity order

This step deserves a careful pause. Use three ordered dissimilarities 1<2<3.

A concrete situation

Nonmetric MDS preserves order through disparities.

What the mathematics does not license

Numeric gaps need not be literal distances. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Record order. 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. Configuration distances

The useful question is not only how to calculate, but what the calculation assumes. Use [1,2,3].

A concrete situation

These are geometric distances in the candidate configuration.

What the mathematics does not license

Keep pairs aligned. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record list. 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. Fitted disparities

A small example makes the mechanism visible. Use [1.2,1.9,2.9].

A concrete situation

The sequence is monotone.

What the mathematics does not license

These are treated as already fitted for the example. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

State assumption. 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. Residuals

This is where modelling choices begin to affect evidence. Compute distance minus disparity.

A concrete situation

Residuals are [-.2,.1,.1].

What the mathematics does not license

Declare sign convention. A trustworthy workflow records the choice before seeing the final test result.

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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5. Residual squares

The central mathematical idea is concrete. Square and sum.

A concrete situation

.04+.01+.01=.06.

What the mathematics does not license

Stress uses all weighted observed pairs. Evidence becomes stronger when the limitation is tested instead of hidden.

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. Distance squares

Here the geometry becomes operational. Compute 1^2+2^2+3^2.

A concrete situation

The denominator is 14.

What the mathematics does not license

This is Stress-1, not Stress-2. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Name convention. 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. Ratio

This step deserves a careful pause. Divide .06 by 14.

A concrete situation

The normalized squared residual is .0042857.

What the mathematics does not license

Keep precision. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Record ratio. 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. Square root

The useful question is not only how to calculate, but what the calculation assumes. Take the square root.

A concrete situation

Stress-1=.065465.

What the mathematics does not license

Lower is better within a comparable setup. Evidence becomes stronger when the limitation is tested instead of hidden.

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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9. Competing disparities

A small example makes the mechanism visible. Use [1.4,1.7,2.7].

A concrete situation

They are also monotone.

What the mathematics does not license

A real fit refits disparities for its configuration. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

State scope. 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. Competing residuals

This is where modelling choices begin to affect evidence. Compute [-.4,.3,.3].

A concrete situation

Squared sum=.34.

What the mathematics does not license

Use the same pair weights. A trustworthy workflow records the choice before seeing the final test result.

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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11. Competing stress

The central mathematical idea is concrete. Compute sqrt(.34/14).

A concrete situation

Stress-1=.155838.

What the mathematics does not license

The first example fits better. Evidence becomes stronger when the limitation is tested instead of hidden.

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. Ties

Here the geometry becomes operational. Give two dissimilarities the same value.

A concrete situation

Primary or secondary tie treatment can change disparities.

What the mathematics does not license

Document software convention. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Stress-test. 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. Missing proximities

This step deserves a careful pause. Remove one pair.

A concrete situation

Weights and denominators change.

What the mathematics does not license

Do not silently impute. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Record mask. 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. Dimension

The useful question is not only how to calculate, but what the calculation assumes. Add another coordinate dimension.

A concrete situation

Stress cannot be compared without noting flexibility.

What the mathematics does not license

Use a prespecified dimension rule. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Plot curve. 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. Starts

A small example makes the mechanism visible. Repeat the optimiser.

A concrete situation

Local minima produce a stress distribution.

What the mathematics does not license

Keep worst credible runs visible. Keeping this boundary clear makes the result easier to audit and transfer.

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

This is where modelling choices begin to affect evidence. State formula, weights, ties, dimension and starts.

A concrete situation

First configuration has lower Stress-1 for the fitted disparities.

What the mathematics does not license

Do not imply causal structure. A trustworthy workflow records the choice before seeing the final test result.

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

Kruskal Stress-1 evaluates how closely configuration distances match fitted disparities in nonmetric multidimensional scaling. The fitted disparities obey the order of observed dissimilarities, usually through monotone regression. Stress-1 is the square root of the residual sum of squares divided by the sum of squared configuration distances.

Suppose three configuration distances are 1, 2 and 3 and the already-fitted monotone disparities are 1.2, 1.9 and 2.9. Residuals are -.2, .1 and .1. Their squared sum is .04+.01+.01=.06, while the distance-square denominator is 1+4+9=14. Stress-1 is sqrt(.06/14)=.065465.

A competing configuration with fitted disparities 1.4, 1.7 and 2.7 has squared residuals .16+.09+.09=.34. Its Stress-1 is sqrt(.34/14)=.155838. The first fit better follows the fitted monotone relation under this convention. In a full analysis the disparities must be refitted for each configuration, not borrowed from a rival.

Stress conventions differ. Stress-1, Stress-2, raw stress, normalized stress and software-specific loss values need not match. Stress decreases when dimensions or flexibility are added, and a low value can still hide local patterns, influential pairs or a dissimilarity measure that does not represent the practical question. Report the exact formula, weights, missing pairs, tie handling, dimension and optimisation starts.

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Frequently asked questions

What is Kruskal Stress-1?

The square root of squared distance-versus-disparity residuals divided by squared configuration distances under the stated weighting convention.

What is a disparity?

A fitted value that relates observed dissimilarity order to configuration distance.

Which direction is better?

Lower stress is better within a comparable model, dimension, weights and convention.

Why must the convention be named?

Raw stress, Stress-1, Stress-2 and software losses use different normalizations.

Does low stress prove the map is true?

No. It measures fit to supplied dissimilarities and model assumptions.

What must be reported?

Dissimilarity direction, transformation, weights, missing pairs, ties, dimension, formula, starts, convergence and residual diagnostics.

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Primary source and fact-check note

Joseph B. Kruskal, “Multidimensional Scaling by Optimizing Goodness of Fit to a Nonmetric Hypothesis,” Psychometrika 29(1), 1–27 (1964), DOI 10.1007/BF02289565. 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

• Shepard diagrams and fitted disparities
• Co-ranking matrices and rank errors
• Johnson–Lindenstrauss projections
• Vector spaces in AI
• How Mathematics Works: Statistics

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: distance-fit evidence and responsible use

Disparity ledger

Keep observed dissimilarities, configuration distances, fitted disparities, residuals, weights and tie blocks on separate columns.

Convention audit

Write the exact Stress-1 formula beside any software output and do not compare it directly with raw stress, Stress-2 or another package convention.

Dimension audit

Plot stress across prespecified dimensions, include flexibility and stability, and avoid treating a visually convenient elbow as proof of intrinsic dimension.

Optimisation audit

Use multiple starts, record local minima and inspect pair-level residuals for the selected configuration.

Mechanism workshop: calculate, challenge and transfer

Manual reconstruction

Rebuild the complete worked example from identifiers. Record dissimilarities, configuration distances, fitted disparities, residuals, weights, denominator and Stress-1. Explain every denominator, transformation and pair convention before using software.

One-pair experiment

Change exactly one pair distance or dissimilarity while holding every other input fixed. Recalculate the affected residual, weight, fitted block, correlation or score and explain the direction.

Scaling experiment

Multiply all coordinates by one positive constant, then stretch only one feature. Separate invariance of a formula from sensitivity of the chosen geometry.

Tie experiment

Create equal distances or dissimilarities at a boundary. Apply the documented tie convention and report whether the comparison survives another defensible rule.

Dimension experiment

Repeat in one, two and three embedding dimensions. Record how fit, flexibility and interpretability change without assuming that the lowest score proves the true dimension.

Stability experiment

Repeat initialisations, samples and plausible preprocessing choices. Report a distribution and preserve one difficult run for inspection.

Parent explanation

Explain why a polished two-dimensional picture can still distort pair relationships and why diagnostics improve decisions without making every projection useless.

Student transfer

Apply the mechanism to a school map, image retrieval set or music-similarity activity. Define the objects, pair evidence and claim boundary.

Limit statement

Write one supported sentence, one overclaim and one repaired claim naming the distance, dimension, preprocessing, formula and uncertainty.

Next measurement

Choose a complementary local, global or task-specific criterion and explain the different question that it answers.

Evidence notebook: reconstruct every mathematical step

Evidence notebook 1: Dissimilarity order

Notebook step 1 isolates dissimilarity order. Use three ordered dissimilarities 1<2<3. The reproducible result is: Nonmetric MDS preserves order through disparities. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: Numeric gaps need not be literal distances. Record order. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 2: Configuration distances

Notebook step 2 isolates configuration distances. Use [1,2,3]. The reproducible result is: These are geometric distances in the candidate configuration. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: Keep pairs aligned. Record list. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 3: Fitted disparities

Notebook step 3 isolates fitted disparities. Use [1.2,1.9,2.9]. The reproducible result is: The sequence is monotone. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: These are treated as already fitted for the example. State assumption. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 4: Residuals

Notebook step 4 isolates residuals. Compute distance minus disparity. The reproducible result is: Residuals are [-.2,.1,.1]. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: Declare sign convention. Record values. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 5: Residual squares

Notebook step 5 isolates residual squares. Square and sum. The reproducible result is: .04+.01+.01=.06. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: Stress uses all weighted observed pairs. Record numerator. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 6: Distance squares

Notebook step 6 isolates distance squares. Compute 1^2+2^2+3^2. The reproducible result is: The denominator is 14. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: This is Stress-1, not Stress-2. Name convention. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 7: Ratio

Notebook step 7 isolates ratio. Divide .06 by 14. The reproducible result is: The normalized squared residual is .0042857. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: Keep precision. Record ratio. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 8: Square root

Notebook step 8 isolates square root. Take the square root. The reproducible result is: Stress-1=.065465. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: Lower is better within a comparable setup. Record score. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 9: Competing disparities

Notebook step 9 isolates competing disparities. Use [1.4,1.7,2.7]. The reproducible result is: They are also monotone. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: A real fit refits disparities for its configuration. State scope. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 10: Competing residuals

Notebook step 10 isolates competing residuals. Compute [-.4,.3,.3]. The reproducible result is: Squared sum=.34. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: Use the same pair weights. Record numerator. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 11: Competing stress

Notebook step 11 isolates competing stress. Compute sqrt(.34/14). The reproducible result is: Stress-1=.155838. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: The first example fits better. Compare. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 12: Ties

Notebook step 12 isolates ties. Give two dissimilarities the same value. The reproducible result is: Primary or secondary tie treatment can change disparities. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: Document software convention. Stress-test. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 13: Missing proximities

Notebook step 13 isolates missing proximities. Remove one pair. The reproducible result is: Weights and denominators change. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: Do not silently impute. Record mask. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 14: Dimension

Notebook step 14 isolates dimension. Add another coordinate dimension. The reproducible result is: Stress cannot be compared without noting flexibility. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: Use a prespecified dimension rule. Plot curve. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 15: Starts

Notebook step 15 isolates starts. Repeat the optimiser. The reproducible result is: Local minima produce a stress distribution. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: Keep worst credible runs visible. Summarise. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Evidence notebook 16: Claim

Notebook step 16 isolates claim. State formula, weights, ties, dimension and starts. The reproducible result is: First configuration has lower Stress-1 for the fitted disparities. The evidence concerns fit between configuration distances and monotone disparities under the exact Stress-1 normalization. Write object or pair identifiers beside every number so a reader can reconstruct the same order. The safeguard is: Do not imply causal structure. Bound conclusion. Change one input while holding the comparison rules fixed, then recalculate the affected distance, disparity, residual, weight, fitted block, centred sum or score. Explain whether the change came from measurement, preprocessing, graph construction, model dimension or optimisation. If the value cannot be reproduced from the notebook, it is not ready for a chart. Finish by stating the narrow conclusion supported here and the separate local, global or task evidence still required.

Failure cases and responsible recovery

Wrong stress convention

Two packages print different values from the same configuration. Compare formulas, weights and denominators before treating the difference as a software error. Work through a small counterexample, repeat it under a second defensible setting and record what changes. Explain the result once to a student and once to a parent. End with the extra evidence that would resolve the remaining uncertainty.

Unfitted disparities

Using raw dissimilarities where the nonmetric model requires fitted monotone disparities changes the question. Preserve both columns and document the regression step. Work through a small counterexample, repeat it under a second defensible setting and record what changes. Explain the result once to a student and once to a parent. End with the extra evidence that would resolve the remaining uncertainty.

Tie ambiguity

Equal dissimilarities can be handled with primary or secondary approaches. Test the declared convention and avoid strong claims from a fragile tie block. Work through a small counterexample, repeat it under a second defensible setting and record what changes. Explain the result once to a student and once to a parent. End with the extra evidence that would resolve the remaining uncertainty.

Dimension overfit

Adding coordinates reduces stress but also increases flexibility. Compare prespecified dimensions with stability and usefulness, not score alone. Work through a small counterexample, repeat it under a second defensible setting and record what changes. Explain the result once to a student and once to a parent. End with the extra evidence that would resolve the remaining uncertainty.

Missing pairs

An incomplete proximity design changes weights and normalization. Preserve the observation mask and avoid silent imputation. Work through a small counterexample, repeat it under a second defensible setting and record what changes. Explain the result once to a student and once to a parent. End with the extra evidence that would resolve the remaining uncertainty.

Local minimum

A poor start can settle at higher stress. Run multiple starts and inspect configurations, not only final decimals. Work through a small counterexample, repeat it under a second defensible setting and record what changes. Explain the result once to a student and once to a parent. End with the extra evidence that would resolve the remaining uncertainty.

Influential pairs

A few large residuals may drive stress. Use a Shepard diagram and pair ledger to localize them before changing the model. Work through a small counterexample, repeat it under a second defensible setting and record what changes. Explain the result once to a student and once to a parent. End with the extra evidence that would resolve the remaining uncertainty.

Responsible conclusion

State goodness of fit to the supplied dissimilarity order and formula. Do not claim a uniquely true psychological or scientific space. Work through a small counterexample, repeat it under a second defensible setting and record what changes. Explain the result once to a student and once to a parent. End with the extra evidence that would resolve the remaining uncertainty.