Chapter map

1. 1. Observed counts
2. 2. Expected counts
3. 3. Observed proportions
4. 4. Expected proportions
5. 5. Residual vector
6. 6. Distance matrix
7. 7. Closeness kernel
8. 8. Diagonal convention
9. 9. Matrix product
10. 10. Pair contribution
11. 11. Scale parameter
12. 12. Profile p-value
13. 13. Monte Carlo null
14. 14. Influence analysis
15. 15. Alternative weights
16. 16. External validation

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1. Observed counts

The central mathematical idea is concrete. Record Oi across common regions and periods.

A concrete situation

Counts are 5,4,1.

What the mathematics does not license

Different ascertainment creates false residuals. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Audit definitions. 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. Expected counts

Here the geometry becomes operational. Build Ei from population or standardization.

A concrete situation

Equal E gives p=(1/3,1/3,1/3).

What the mathematics does not license

Wrong baseline creates apparent clusters. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Version the model. 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. Observed proportions

This step deserves a careful pause. Divide each Oi by O+.

A concrete situation

r=(.5,.4,.1).

What the mathematics does not license

Proportions share a fixed total. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use the correct covariance. 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. Expected proportions

The useful question is not only how to calculate, but what the calculation assumes. Divide Ei by E+.

A concrete situation

p is one third each.

What the mathematics does not license

Estimated p carries uncertainty. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Refit in simulation if needed. 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 vector

A small example makes the mechanism visible. Subtract expected from observed shares.

A concrete situation

d≈(.1667,.0667,−.2333).

What the mathematics does not license

Residuals sum to zero. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Keep signed maps. 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 matrix

This is where modelling choices begin to affect evidence. Calculate centroid or network distances.

A concrete situation

D12 is smaller than D13.

What the mathematics does not license

Centroids simplify irregular regions. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Compare defensible metrics. 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. Closeness kernel

The central mathematical idea is concrete. Set Aij=exp(−Dij/φ) or another declared rule.

A concrete situation

A12=.8, A13=.1, A23=.2.

What the mathematics does not license

Kernel choice sets the alternative. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Publish A. 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. Diagonal convention

Here the geometry becomes operational. Set self-weights consistently with the method.

A concrete situation

Aii=1.

What the mathematics does not license

Changing diagonals changes dispersion contribution. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Match theory and software. 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. Matrix product

This step deserves a careful pause. Calculate d′Ad.

A concrete situation

The toy value is about .0904.

What the mathematics does not license

Rounding can hide contributions. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Retain precision. 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. Pair contribution

The useful question is not only how to calculate, but what the calculation assumes. Inspect 2Aij di dj for nearby pairs.

A concrete situation

Two positive residuals add positive evidence.

What the mathematics does not license

Opposite signs can reduce the statistic. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Map signed contributions. 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. Scale parameter

A small example makes the mechanism visible. Repeat predeclared φ values.

A concrete situation

Small φ is local; large φ is broad.

What the mathematics does not license

One chosen φ may miss the process. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use a profile plan. 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. Profile p-value

This is where modelling choices begin to affect evidence. Find the most unusual calibrated scale.

A concrete situation

The minimum p occurs at 8 km.

What the mathematics does not license

A minimum unadjusted p is biased. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Simulate the full search. 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. Monte Carlo null

The central mathematical idea is concrete. Allocate cases using p and rebuild the profile.

A concrete situation

9999 maps create a global reference.

What the mathematics does not license

Too few replicates limit precision. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report Monte Carlo error. 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. Influence analysis

Here the geometry becomes operational. Measure how each region affects C.

A concrete situation

Region one drives half the statistic.

What the mathematics does not license

Influence is not proof of error. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Inspect data quality. 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. Alternative weights

This step deserves a careful pause. Compare adjacency and distance decay.

A concrete situation

Queen adjacency changes the result.

What the mathematics does not license

Weight shopping inflates evidence. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Predeclare a small family. 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. External validation

The useful question is not only how to calculate, but what the calculation assumes. Repeat fixed choices in held-out data.

A concrete situation

The same scale recurs later.

What the mathematics does not license

Stable clustering remains descriptive. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Investigate causes separately. 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

Tango’s C statistic compares the observed share of cases in every region with the share expected from its population or standardized expected count. The residual vector is r−p, where ri=Oi/O+ and pi=Ei/E+. A closeness matrix A tells the statistic which pairs of regions should reinforce one another. The quadratic form C=(r−p)′A(r−p) becomes large when deviations align at the spatial scales encoded by A.

A frequently used closeness is Aij=exp(−Dij/φ), where Dij is the distance between region centroids and φ controls the range of influence. Small φ emphasizes very local agreement; larger φ allows distant residuals to interact. This scale is part of the hypothesis. Tango’s later profile approach searches across φ values and adjusts the minimum profile p-value by Monte Carlo simulation instead of reporting the most favourable unadjusted scale.

For three equal-expected regions, let observed proportions be 0.5,0.4,0.1, so d=r−p is approximately 0.1667,0.0667,−0.2333. With diagonal weights one and off-diagonal closeness values A12=.8, A13=.1 and A23=.2, d′Ad is about 0.0904. The value is meaningful only relative to the null distribution under the same expected proportions and matrix.

The method can detect several clusters rather than only one most-likely window, but the weight matrix, centroids, expected counts and φ search shape the answer. A significant global statistic does not identify a unique cause or boundary. Good practice publishes the residual map, matrix construction, scale profile, Monte Carlo algorithm, influential regions and confirmation in reserved data.

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Frequently asked questions

What is Tango’s C?

It is a quadratic form in observed-minus-expected regional proportions weighted by spatial closeness.

What does the matrix A do?

A assigns greater interaction to nearby regions, commonly through exponential distance decay.

What does φ control?

It controls the spatial scale over which residuals reinforce one another.

Why profile across φ?

The relevant cluster size may be unknown, but the scale search requires multiplicity adjustment.

How is significance assessed?

Simulate or permute under the expected spatial distribution and repeat the full statistic or profile search.

Does a large C locate one cluster?

No. It signals general spatial clustering; residual and local diagnostics are needed for interpretation.

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Primary source and fact-check note

Toshiro Tango, “A class of tests for detecting ‘general’ and ‘focused’ clustering of rare diseases,” Statistics in Medicine 14(21–22), 2323–2334 (1995), DOI 10.1002/sim.4780142105. 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

• Potthoff–Whittinghill overdispersion tests
• Moran’s I spatial autocorrelation
• How Mathematics Works

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: topic-specific validation and responsible use

Data audit

Verify case definitions, denominators, coordinates and observation periods.

Geometry audit

Publish foci, centroids, distances, windows and weight matrices.

Calculation audit

Reproduce one complete toy statistic by hand.

Null audit

State conditioning, expected probabilities and simulation design.

Multiplicity audit

Control centres, scales, sources and local diagnostics.

Sensitivity audit

Repeat only scientifically plausible geometries and baselines.

Transfer audit

Reserve later periods or separate regions.

Final claim

Report spatial evidence without automatic causal attribution.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Observed counts

Begin with this mechanism: Record Oi across common regions and periods. Recreate the concrete case: Counts are 5,4,1. Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Different ascertainment creates false residuals. Complete the transfer step: Audit definitions. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 2: Expected counts

Begin with this mechanism: Build Ei from population or standardization. Recreate the concrete case: Equal E gives p=(1/3,1/3,1/3). Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Wrong baseline creates apparent clusters. Complete the transfer step: Version the model. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 3: Observed proportions

Begin with this mechanism: Divide each Oi by O+. Recreate the concrete case: r=(.5,.4,.1). Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Proportions share a fixed total. Complete the transfer step: Use the correct covariance. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 4: Expected proportions

Begin with this mechanism: Divide Ei by E+. Recreate the concrete case: p is one third each. Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Estimated p carries uncertainty. Complete the transfer step: Refit in simulation if needed. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 5: Residual vector

Begin with this mechanism: Subtract expected from observed shares. Recreate the concrete case: d≈(.1667,.0667,−.2333). Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Residuals sum to zero. Complete the transfer step: Keep signed maps. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 6: Distance matrix

Begin with this mechanism: Calculate centroid or network distances. Recreate the concrete case: D12 is smaller than D13. Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Centroids simplify irregular regions. Complete the transfer step: Compare defensible metrics. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 7: Closeness kernel

Begin with this mechanism: Set Aij=exp(−Dij/φ) or another declared rule. Recreate the concrete case: A12=.8, A13=.1, A23=.2. Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Kernel choice sets the alternative. Complete the transfer step: Publish A. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 8: Diagonal convention

Begin with this mechanism: Set self-weights consistently with the method. Recreate the concrete case: Aii=1. Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Changing diagonals changes dispersion contribution. Complete the transfer step: Match theory and software. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 9: Matrix product

Begin with this mechanism: Calculate d′Ad. Recreate the concrete case: The toy value is about .0904. Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Rounding can hide contributions. Complete the transfer step: Retain precision. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 10: Pair contribution

Begin with this mechanism: Inspect 2Aij di dj for nearby pairs. Recreate the concrete case: Two positive residuals add positive evidence. Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Opposite signs can reduce the statistic. Complete the transfer step: Map signed contributions. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 11: Scale parameter

Begin with this mechanism: Repeat predeclared φ values. Recreate the concrete case: Small φ is local; large φ is broad. Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: One chosen φ may miss the process. Complete the transfer step: Use a profile plan. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 12: Profile p-value

Begin with this mechanism: Find the most unusual calibrated scale. Recreate the concrete case: The minimum p occurs at 8 km. Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: A minimum unadjusted p is biased. Complete the transfer step: Simulate the full search. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 13: Monte Carlo null

Begin with this mechanism: Allocate cases using p and rebuild the profile. Recreate the concrete case: 9999 maps create a global reference. Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Too few replicates limit precision. Complete the transfer step: Report Monte Carlo error. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 14: Influence analysis

Begin with this mechanism: Measure how each region affects C. Recreate the concrete case: Region one drives half the statistic. Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Influence is not proof of error. Complete the transfer step: Inspect data quality. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 15: Alternative weights

Begin with this mechanism: Compare adjacency and distance decay. Recreate the concrete case: Queen adjacency changes the result. Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Weight shopping inflates evidence. Complete the transfer step: Predeclare a small family. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 16: External validation

Begin with this mechanism: Repeat fixed choices in held-out data. Recreate the concrete case: The same scale recurs later. Write observed counts, expected counts, observed proportions, expected proportions, residuals, distances, closeness weights, matrix products, pair contributions, scale values, profile p-values and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Stable clustering remains descriptive. Complete the transfer step: Investigate causes separately. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.