Chapter map

1. 1. Prespecified focus
2. 2. Area geometry
3. 3. Distance measure
4. 4. Observed counts
5. 5. Expected counts
6. 6. Distance ordering
7. 7. Cumulative observed
8. 8. Cumulative expected
9. 9. Cumulative ratio
10. 10. Monotone risks
11. 11. Likelihood contrast
12. 12. Bootstrap null
13. 13. Multiple sources
14. 14. Directional sensitivity
15. 15. Influence audit
16. 16. External evidence

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. Prespecified focus

The central mathematical idea is concrete. Name the source without using the outcome map.

A concrete situation

A registered industrial stack defines the origin.

What the mathematics does not license

Choosing the highest-rate area as focus is circular. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Document provenance. 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. Area geometry

Here the geometry becomes operational. Store boundaries and centroids in one coordinate system.

A concrete situation

Four census areas surround the focus.

What the mathematics does not license

Centroids may misrepresent large regions. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare boundary distances. 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. Distance measure

This step deserves a careful pause. Calculate separation or another exposure proxy.

A concrete situation

Areas are ordered 1,2,3,4 km away.

What the mathematics does not license

Wind may make direction more important. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Test a transport-informed metric. 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. Observed counts

The useful question is not only how to calculate, but what the calculation assumes. Count eligible events in a fixed period.

A concrete situation

O=(4,1,0,1).

What the mathematics does not license

Reporting completeness may vary. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Audit ascertainment. 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. Expected counts

A small example makes the mechanism visible. Model baseline burden from population and covariates.

A concrete situation

E=(1,1,1,1).

What the mathematics does not license

Crude expectations ignore age structure. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Standardize appropriately. 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 ordering

This is where modelling choices begin to affect evidence. Sort area pairs by exposure distance.

A concrete situation

Area A is nearest, D farthest.

What the mathematics does not license

Ties need a rule. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Publish ordering and ties. 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. Cumulative observed

The central mathematical idea is concrete. Add counts from nearest outward.

A concrete situation

Cumulative O is 4,5,5,6.

What the mathematics does not license

A prefix is selected after inspection. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use a maximum reference. 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. Cumulative expected

Here the geometry becomes operational. Add expected counts in the same prefixes.

A concrete situation

Cumulative E is 1,2,3,4.

What the mathematics does not license

Expected values may be estimated. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Refit them in simulation. 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. Cumulative ratio

This step deserves a careful pause. Divide cumulative O by cumulative E.

A concrete situation

Ratios are 4,2.5,1.67,1.5.

What the mathematics does not license

A large first ratio may be one unstable area. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Report intervals and influence. 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. Monotone risks

The useful question is not only how to calculate, but what the calculation assumes. Fit risks constrained not to increase with distance.

A concrete situation

Near bands can share pooled fitted levels.

What the mathematics does not license

True plumes may be nonmonotone. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Check alternative shapes. 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. Likelihood contrast

A small example makes the mechanism visible. Compare declining-risk and constant-risk fits.

A concrete situation

The alternative improves fit near the source.

What the mathematics does not license

Parameter boundaries complicate asymptotics. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use the published calibration. 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. Bootstrap null

This is where modelling choices begin to affect evidence. Simulate counts under constant relative risk.

A concrete situation

Every replicate recomputes the maximum.

What the mathematics does not license

Conditioning choices change the null. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

State Poisson or fixed-total design. 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. Multiple sources

The central mathematical idea is concrete. Predeclare how several foci are handled.

A concrete situation

Two plants require a combined or adjusted analysis.

What the mathematics does not license

Trying each source multiplies tests. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use search-wide control. 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. Directional sensitivity

Here the geometry becomes operational. Compare radial and pathway models.

A concrete situation

Downwind distance fits better than Euclidean distance.

What the mathematics does not license

Post-hoc direction choice is biased. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Reserve confirmation data. 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. Influence audit

This step deserves a careful pause. Remove one area only as a diagnostic.

A concrete situation

The nearest area creates most evidence.

What the mathematics does not license

Deletion cannot erase inconvenient data. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Explain quality findings. 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 evidence

The useful question is not only how to calculate, but what the calculation assumes. Validate in later periods and exposure measurements.

A concrete situation

A later cohort shows the same decline.

What the mathematics does not license

Replication remains association. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Investigate mechanism. 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

Stone’s focused test begins with a location specified before the outcome map is inspected: for example, an industrial site, emission source or other scientifically motivated focus. Geographic areas are ordered by distance from that point. The null states that relative risk is constant after expected counts are considered. The alternative asks whether risk is highest near the focus and declines as distance increases.

A common implementation compares the constant-risk model with a distance-ordered alternative and uses a maximum likelihood or cumulative risk statistic. One transparent diagnostic sorts bands by distance and calculates the cumulative observed-to-expected ratio for every prefix. The largest prefix ratio shows where the most concentrated near-source excess occurs, while valid significance comes from the full test and its simulated or bootstrap reference.

Suppose distance-ordered areas have observed counts 4,1,0,1 and expected counts 1,1,1,1. The cumulative observed-to-expected ratios are 4/1=4, 5/2=2.5, 5/3≈1.67 and 6/4=1.5. The largest near-source ratio is four at the first area. That descriptive maximum must be calibrated because the prefix was selected by the same data.

Focused testing is stronger than drawing a circle after seeing a high rate because the focus and decline direction are predeclared. It still treats distance as a proxy for exposure. Wind, water flow, commuting, barriers, age structure and geocoding can break radial decline. A responsible analysis audits expected counts, distance bands, monotonicity, multiple sources and independent exposure evidence before making any causal claim.

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Frequently asked questions

What is a focused cluster test?

It evaluates a pattern around a location specified independently of the observed outcome excess.

What is Stone’s alternative?

Relative risk is allowed to be highest near the focus and decline as distance increases.

Why sort areas by distance?

The ordering turns a geographic hypothesis into nested near-source prefixes or constrained risks.

Why bootstrap?

The maximum or likelihood ratio has a selection distribution that is not the distribution of one fixed band.

Can distance stand in for exposure?

Only imperfectly; transport pathways, barriers and individual movement can make radial distance inadequate.

Does rejection prove the source caused disease?

No. It supports the declared distance-risk pattern under the null and data model.

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Primary source and fact-check note

Richard A. Stone, “Investigations of excess environmental risks around putative sources: statistical problems and a proposed test,” Statistics in Medicine 7(6), 649–660 (1988), DOI 10.1002/sim.4780070604. 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

• Besag–Newell cluster detection
• Kulldorff spatial scan statistics
• 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: Prespecified focus

Begin with this mechanism: Name the source without using the outcome map. Recreate the concrete case: A registered industrial stack defines the origin. Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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: Choosing the highest-rate area as focus is circular. Complete the transfer step: Document provenance. 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: Area geometry

Begin with this mechanism: Store boundaries and centroids in one coordinate system. Recreate the concrete case: Four census areas surround the focus. Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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 may misrepresent large regions. Complete the transfer step: Compare boundary distances. 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: Distance measure

Begin with this mechanism: Calculate separation or another exposure proxy. Recreate the concrete case: Areas are ordered 1,2,3,4 km away. Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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: Wind may make direction more important. Complete the transfer step: Test a transport-informed metric. 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: Observed counts

Begin with this mechanism: Count eligible events in a fixed period. Recreate the concrete case: O=(4,1,0,1). Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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: Reporting completeness may vary. Complete the transfer step: Audit ascertainment. 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: Expected counts

Begin with this mechanism: Model baseline burden from population and covariates. Recreate the concrete case: E=(1,1,1,1). Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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: Crude expectations ignore age structure. Complete the transfer step: Standardize appropriately. 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 ordering

Begin with this mechanism: Sort area pairs by exposure distance. Recreate the concrete case: Area A is nearest, D farthest. Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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: Ties need a rule. Complete the transfer step: Publish ordering and ties. 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: Cumulative observed

Begin with this mechanism: Add counts from nearest outward. Recreate the concrete case: Cumulative O is 4,5,5,6. Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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 prefix is selected after inspection. Complete the transfer step: Use a maximum reference. 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: Cumulative expected

Begin with this mechanism: Add expected counts in the same prefixes. Recreate the concrete case: Cumulative E is 1,2,3,4. Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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: Expected values may be estimated. Complete the transfer step: Refit them in simulation. 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: Cumulative ratio

Begin with this mechanism: Divide cumulative O by cumulative E. Recreate the concrete case: Ratios are 4,2.5,1.67,1.5. Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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 large first ratio may be one unstable area. Complete the transfer step: Report intervals and influence. 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: Monotone risks

Begin with this mechanism: Fit risks constrained not to increase with distance. Recreate the concrete case: Near bands can share pooled fitted levels. Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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: True plumes may be nonmonotone. Complete the transfer step: Check alternative shapes. 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: Likelihood contrast

Begin with this mechanism: Compare declining-risk and constant-risk fits. Recreate the concrete case: The alternative improves fit near the source. Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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: Parameter boundaries complicate asymptotics. Complete the transfer step: Use the published calibration. 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: Bootstrap null

Begin with this mechanism: Simulate counts under constant relative risk. Recreate the concrete case: Every replicate recomputes the maximum. Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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: Conditioning choices change the null. Complete the transfer step: State Poisson or fixed-total design. 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: Multiple sources

Begin with this mechanism: Predeclare how several foci are handled. Recreate the concrete case: Two plants require a combined or adjusted analysis. Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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: Trying each source multiplies tests. Complete the transfer step: Use search-wide control. 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: Directional sensitivity

Begin with this mechanism: Compare radial and pathway models. Recreate the concrete case: Downwind distance fits better than Euclidean distance. Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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: Post-hoc direction choice is biased. Complete the transfer step: Reserve confirmation data. 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: Influence audit

Begin with this mechanism: Remove one area only as a diagnostic. Recreate the concrete case: The nearest area creates most evidence. Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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: Deletion cannot erase inconvenient data. Complete the transfer step: Explain quality findings. 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 evidence

Begin with this mechanism: Validate in later periods and exposure measurements. Recreate the concrete case: A later cohort shows the same decline. Write focus coordinates, area distances, observed counts, expected counts, cumulative counts, cumulative expectations, prefix ratios, constrained risks, likelihood values, bootstrap maxima and tail 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: Replication remains association. Complete the transfer step: Investigate mechanism. 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.