Chapter map

1. 1. Area table
2. 2. Common-rate baseline
3. 3. Case target k
4. 4. Candidate centre
5. 5. Distance order
6. 6. Window expansion
7. 7. Local population
8. 8. Expected count
9. 9. Poisson upper tail
10. 10. Local extent l
11. 11. Repeat centres
12. 12. Global count R
13. 13. Monte Carlo calibration
14. 14. Scale sensitivity
15. 15. Geographic audit
16. 16. Follow-up investigation

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. Area table

The central mathematical idea is concrete. Each region has a centre, observed cases and population at risk.

A concrete situation

Five neighbouring areas are mapped.

What the mathematics does not license

Centroids simplify irregular geography. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Publish boundaries and denominators. 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. Common-rate baseline

Here the geometry becomes operational. Total cases divided by total population gives a crude rate.

A concrete situation

20 divided by 10000 equals .002.

What the mathematics does not license

Age structure can invalidate a crude rate. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use justified expected counts. 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. Case target k

This step deserves a careful pause. Choose the number of cases defining a window.

A concrete situation

Set k=4 before viewing centres.

What the mathematics does not license

Post-hoc k selection favours chance clusters. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Predeclare or adjust a scale set. 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. Candidate centre

The useful question is not only how to calculate, but what the calculation assumes. Begin at an area where a case is observed or every eligible centre.

A concrete situation

Area A is the first centre.

What the mathematics does not license

Centre rules affect multiplicity. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

State eligibility. 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. Distance order

A small example makes the mechanism visible. Rank surrounding areas by centre distance or adjacency.

A concrete situation

B, C and D follow A.

What the mathematics does not license

Centroid distance can cross barriers. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare adjacency and travel distance. 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. Window expansion

This is where modelling choices begin to affect evidence. Add areas until cumulative cases reach k.

A concrete situation

A plus B plus C contain four cases.

What the mathematics does not license

The last area can overshoot k. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Keep the published stopping rule. 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. Local population

The central mathematical idea is concrete. Sum the risk population inside the stopping window.

A concrete situation

The window contains 1000 people.

What the mathematics does not license

Wrong denominators create false excess. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Audit census and time alignment. 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. Expected count

Here the geometry becomes operational. Multiply the background rate by local population.

A concrete situation

.002 times 1000 gives μ=2.

What the mathematics does not license

Expected counts may require indirect standardization. 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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9. Poisson upper tail

This step deserves a careful pause. Calculate P(X at least k) under mean μ.

A concrete situation

P(X≥4|μ=2) is about .143.

What the mathematics does not license

Poisson approximation may miss finite-population conditioning. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Compare exact or simulated nulls. 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. Local extent l

The useful question is not only how to calculate, but what the calculation assumes. Record how many neighbouring regions were required.

A concrete situation

Two neighbours plus the centre reached k.

What the mathematics does not license

Region count ignores population differences. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report both l and population. 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. Repeat centres

A small example makes the mechanism visible. Run the same procedure from every eligible centre.

A concrete situation

Nearby centres produce overlapping windows.

What the mathematics does not license

Duplicates are not independent clusters. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Merge or report overlap transparently. 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. Global count R

This is where modelling choices begin to affect evidence. Count centres meeting the declared local criterion.

A concrete situation

Three centres pass alpha .05.

What the mathematics does not license

R needs a search-wide reference. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Simulate the whole procedure. 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 calibration

The central mathematical idea is concrete. Generate cases under expected risk and rerun all centres.

A concrete situation

999 null maps give an R distribution.

What the mathematics does not license

Conditional and Poisson simulations answer different nulls. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Match the 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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14. Scale sensitivity

Here the geometry becomes operational. Repeat only predeclared k values.

A concrete situation

k=3 and k=8 target different extents.

What the mathematics does not license

Choosing the best result multiplies tests. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use adjusted inference. 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. Geographic audit

This step deserves a careful pause. Inspect shapes, islands and boundary centres.

A concrete situation

One window crosses water.

What the mathematics does not license

Circular proximity may be unrealistic. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use defensible neighbourhoods. 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. Follow-up investigation

The useful question is not only how to calculate, but what the calculation assumes. Check later data, exposure and ascertainment.

A concrete situation

A flagged area persists after denominator revision.

What the mathematics does not license

Persistence is not causation. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Investigate with domain experts. 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

The Besag–Newell method searches for small geographic clusters while accounting for an uneven population at risk. The analyst chooses a case target k. Starting from a candidate centre, neighbouring areas are added in distance order until the accumulated observed cases reach k. The local statistic records how few areas or how little population were needed. A compact window containing k cases may be unusual if its expected case count is small.

The null model assumes a common risk after allowing for population size or supplied expected counts. For a candidate window with expected count μ, a Poisson upper tail can summarize how surprising at least k cases would be. Because many centres are examined and their windows overlap, isolated local p-values are not a complete global test. Monte Carlo randomization or another multiplicity-aware calibration is needed for the number of flagged centres.

Suppose a study region has 20 cases among 10,000 people, giving a crude rate of 0.002. Around one candidate centre, the smallest window containing k=4 cases includes 1,000 people, so μ=2. The Poisson probability of four or more is about 0.143: one minus e to the minus two times the sum of the masses for zero through three. This toy window is not locally significant at 0.05, and it must still be judged within the full search.

Choosing k sets the cluster scale: a small k is sensitive to small clusters, while a larger k looks for broader accumulations. Centroid distance, adjacency, irregular geography, population denominators, tied distances and duplicated detections all matter. The output is a screening map for follow-up, not a declaration that every overlapping circle is a separate real outbreak or that a nearby feature caused the excess.

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Frequently asked questions

What does the Besag–Newell method detect?

It screens candidate centres for compact collections of a predeclared k cases relative to the local population at risk.

Why fix the number of cases?

Fixing k lets the required geographic or population extent reveal how concentrated those cases are.

What is the local null?

Cases follow a common background rate after accounting for population or supplied expected counts.

Why is one local p-value insufficient?

Many overlapping centres are searched, creating dependent repeated tests.

How does k affect the answer?

Small k targets smaller clusters; larger k targets broader accumulations.

Does a flagged window identify a cause?

No. It is a screening region requiring denominator, ascertainment, exposure and independent follow-up.

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Primary source and fact-check note

Julian Besag and James Newell, “The Detection of Clusters in Rare Diseases,” Journal of the Royal Statistical Society Series A 154(1), 143–155 (1991), DOI 10.2307/2982708. 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

• Kulldorff spatial scan statistics
• Cuzick–Edwards nearest-neighbour tests
• 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

Denominator audit

Version population-at-risk and expected-count inputs.

Centre audit

State which locations can start a window.

Order audit

Compare centroid, adjacency and travel distance.

Tail audit

Reproduce Poisson or exact probabilities by hand.

Multiplicity audit

Calibrate the complete set of overlapping centres.

Scale audit

Predeclare k or adjust across k values.

Transfer audit

Repeat with later denominators and events.

Final claim

A flagged window is a screening result for investigation.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Area table

Begin with this mechanism: Each region has a centre, observed cases and population at risk. Recreate the concrete case: Five neighbouring areas are mapped. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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 geography. Complete the transfer step: Publish boundaries and denominators. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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: Common-rate baseline

Begin with this mechanism: Total cases divided by total population gives a crude rate. Recreate the concrete case: 20 divided by 10000 equals .002. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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: Age structure can invalidate a crude rate. Complete the transfer step: Use justified expected counts. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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: Case target k

Begin with this mechanism: Choose the number of cases defining a window. Recreate the concrete case: Set k=4 before viewing centres. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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 k selection favours chance clusters. Complete the transfer step: Predeclare or adjust a scale set. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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: Candidate centre

Begin with this mechanism: Begin at an area where a case is observed or every eligible centre. Recreate the concrete case: Area A is the first centre. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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: Centre rules affect multiplicity. Complete the transfer step: State eligibility. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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: Distance order

Begin with this mechanism: Rank surrounding areas by centre distance or adjacency. Recreate the concrete case: B, C and D follow A. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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: Centroid distance can cross barriers. Complete the transfer step: Compare adjacency and travel distance. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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: Window expansion

Begin with this mechanism: Add areas until cumulative cases reach k. Recreate the concrete case: A plus B plus C contain four cases. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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: The last area can overshoot k. Complete the transfer step: Keep the published stopping rule. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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: Local population

Begin with this mechanism: Sum the risk population inside the stopping window. Recreate the concrete case: The window contains 1000 people. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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 denominators create false excess. Complete the transfer step: Audit census and time alignment. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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: Expected count

Begin with this mechanism: Multiply the background rate by local population. Recreate the concrete case: .002 times 1000 gives μ=2. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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 counts may require indirect standardization. Complete the transfer step: Version the model. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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: Poisson upper tail

Begin with this mechanism: Calculate P(X at least k) under mean μ. Recreate the concrete case: P(X≥4|μ=2) is about .143. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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: Poisson approximation may miss finite-population conditioning. Complete the transfer step: Compare exact or simulated nulls. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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: Local extent l

Begin with this mechanism: Record how many neighbouring regions were required. Recreate the concrete case: Two neighbours plus the centre reached k. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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: Region count ignores population differences. Complete the transfer step: Report both l and population. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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: Repeat centres

Begin with this mechanism: Run the same procedure from every eligible centre. Recreate the concrete case: Nearby centres produce overlapping windows. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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: Duplicates are not independent clusters. Complete the transfer step: Merge or report overlap transparently. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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: Global count R

Begin with this mechanism: Count centres meeting the declared local criterion. Recreate the concrete case: Three centres pass alpha .05. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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: R needs a search-wide reference. Complete the transfer step: Simulate the whole procedure. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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 calibration

Begin with this mechanism: Generate cases under expected risk and rerun all centres. Recreate the concrete case: 999 null maps give an R distribution. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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: Conditional and Poisson simulations answer different nulls. Complete the transfer step: Match the design. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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: Scale sensitivity

Begin with this mechanism: Repeat only predeclared k values. Recreate the concrete case: k=3 and k=8 target different extents. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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 best result multiplies tests. Complete the transfer step: Use adjusted inference. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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: Geographic audit

Begin with this mechanism: Inspect shapes, islands and boundary centres. Recreate the concrete case: One window crosses water. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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: Circular proximity may be unrealistic. Complete the transfer step: Use defensible neighbourhoods. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, 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: Follow-up investigation

Begin with this mechanism: Check later data, exposure and ascertainment. Recreate the concrete case: A flagged area persists after denominator revision. Write area IDs, centroids, case counts, risk populations, expected counts, distance orders, cumulative cases, cumulative population, stopping extents, Poisson masses, local tails and global Monte Carlo counts 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: Persistence is not causation. Complete the transfer step: Investigate with domain experts. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.