WHY MATHEMATICS?

Why Mathematics? | Cuzick–Edwards Nearest-Neighbour Tests: Case–Control Labels, Tₖ Statistics, Inhomogeneous Risk and Permutations

How Cuzick–Edwards statistics detect case clustering against sampled population controls while keeping k choice and design validity visible.

Why mathematics is important becomes easier to see when a real decision depends on it. This guide follows the mechanism from definitions and dimensions to a worked calculation, validation, practical learning and responsible use. It keeps factual evidence separate from interpretation and treats every illustrative scenario as an example rather than a promise of results.

Choose a reading route

Sample cases and representative controls from the risk population
Build nearest-neighbour orders on the combined coordinates
Count case neighbours around each case to obtain Tₖ
Calibrate labels, combine scales carefully and validate

Chapter map

1. Case definition
2. Control sampling
3. Combined coordinates
4. Distance matrix
5. Neighbour order
6. Case-centred count
7. T k statistic
8. Conditional expectation
9. Observed contrast
10. Label permutation
11. Exact enumeration
12. Choice of k
13. Combined T
14. Control ratio
15. Boundary coverage
16. Sensitivity and follow-up

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

1. Case definition

Here the geometry becomes operational. Cases follow one eligibility rule.

A concrete situation

Three mapped children meet the case definition.

What the mathematics does not license

Changing definitions changes clustering. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Freeze criteria. 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.

2. Control sampling

This step deserves a careful pause. Controls represent the population at risk.

A concrete situation

Three controls are sampled from the same region.

What the mathematics does not license

Convenience controls distort density. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Document 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.

3. Combined coordinates

The useful question is not only how to calculate, but what the calculation assumes. Cases and controls share one metric space.

A concrete situation

Six points use metres.

What the mathematics does not license

Different geocoding quality biases neighbours. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Harmonize 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.

4. Distance matrix

A small example makes the mechanism visible. Compute all nonself separations.

A concrete situation

Each row ranks five candidates.

What the mathematics does not license

Barriers can invalidate Euclidean distance. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use a relevant 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.

Did You Know? In this chapter, the most important number is useful only together with its construction rule. Changing scaling, sample membership or validation design can change the numerical answer without changing the arithmetic.

5. Neighbour order

This is where modelling choices begin to affect evidence. Sort other points by distance.

A concrete situation

The two closest define k=2.

What the mathematics does not license

Ties need deterministic handling. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record tie rules. 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.

6. Case-centred count

The central mathematical idea is concrete. For each case count case labels among k neighbours.

A concrete situation

Counts are 1,1,0.

What the mathematics does not license

One neighbour can serve several cases. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Retain dependence. 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.

7. T k statistic

Here the geometry becomes operational. Sum the case-centred counts.

A concrete situation

T2 equals 2.

What the mathematics does not license

The statistic depends on k. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Report k. 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.

8. Conditional expectation

This step deserves a careful pause. Under random labels E[Tk]=c k(c−1)/(n−1).

A concrete situation

3×2×2/5=2.4.

What the mathematics does not license

Matched designs change the null. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use the correct conditioning. 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.

9. Observed contrast

The useful question is not only how to calculate, but what the calculation assumes. Compare T2 with its centre.

A concrete situation

Two is below 2.4.

What the mathematics does not license

Difference is not a p-value. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Calculate the distribution. 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.

10. Label permutation

A small example makes the mechanism visible. Reassign case labels under design constraints.

A concrete situation

Choose three of six points as cases.

What the mathematics does not license

Unrestricted shuffling breaks matching. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Permute within strata. 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.

11. Exact enumeration

This is where modelling choices begin to affect evidence. Small samples allow every allocation.

A concrete situation

There are 20 unstratified label sets.

What the mathematics does not license

Discrete distributions limit resolution. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

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

12. Choice of k

The central mathematical idea is concrete. k controls spatial scale.

A concrete situation

Evaluate k=1,2,3 by plan.

What the mathematics does not license

Choosing the best after inspection inflates type I error. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use a combined statistic. 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.

13. Combined T

Here the geometry becomes operational. Covariance-aware weights can combine several T k values.

A concrete situation

T1,T2,T3 contribute jointly.

What the mathematics does not license

Naive summation double counts related evidence. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use published 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.

14. Control ratio

This step deserves a careful pause. More controls may improve representation and power.

A concrete situation

A 1:4 ratio samples local risk density better.

What the mathematics does not license

Extra biased controls do not help. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Prioritize validity. 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.

15. Boundary coverage

The useful question is not only how to calculate, but what the calculation assumes. Cases near study edges need controls beyond their nearest range.

A concrete situation

A border case’s neighbours lie outside the map.

What the mathematics does not license

Truncation mimics isolation. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Expand or correct coverage. 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.

16. Sensitivity and follow-up

A small example makes the mechanism visible. Repeat control samples, metrics and independent periods.

A concrete situation

Clustering persists across valid control draws.

What the mathematics does not license

Persistence remains association. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Investigate exposures. 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.

A seven-day practice lab

Day 1: define the decision, observation unit, X variables and Y or reference condition.
Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
Day 3: build a transparent baseline and write what it assumes.
Day 4: fit the main method inside a training-only pipeline; record every parameter.
Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
Day 6: compare validation metrics with the practical workload created by errors.
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 Cuzick–Edwards test asks whether cases have more cases among their nearest neighbours than expected after accounting for an inhomogeneous population through sampled controls. Cases and controls share the same coordinate system. For each case, the method examines its k nearest neighbours and counts how many are also cases; the total is T_k.

Conditioning on the observed case and control counts makes label permutation a natural reference when labels are exchangeable under the design. Because one event can be a neighbour of several cases, contributions are dependent. Exact covariance formulas or permutations handle that structure. Several k values may be combined, but choosing k after inspecting the most favourable result inflates evidence.

Suppose three cases and three controls are mapped, with k=2. If the three cases have one, one and zero case neighbours, T_2=2. Under simple random labelling the expectation is c k(c−1)/(n−1)=3×2×2/5=2.4. The observed count is below that centre in this toy example; significance requires the full conditional distribution.

The method supports cluster detection where population density varies, including epidemiology and environmental studies. Its validity depends on how controls represent the population at risk, matching, geocoding, duplicate sites and boundary coverage. Responsible work records sampling fractions, repeats scientifically meaningful k values, evaluates matched or stratified permutations and distinguishes clustering from exposure, ascertainment and mechanism.

Frequently asked questions

What is the Cuzick–Edwards test?

It counts cases among the k nearest neighbours of each case in a combined case–control point pattern.

Why use controls?

Controls represent the inhomogeneous population at risk, preventing population density alone from looking like case clustering.

What is Tₖ?

Tₖ is the sum, over cases, of the number of case labels among each case’s k nearest neighbours.

How is significance assessed?

Use the conditional distribution of labels under the case–control design, through theory or valid permutations.

How should k be chosen?

Use scientifically meaningful scales or a predeclared combined test; selecting the strongest k is biased.

Does clustering identify a cause?

No. Exposure, ascertainment, control sampling and geocoding remain possible explanations.

Primary source and fact-check note

Jack Cuzick and Robert Edwards, “Spatial Clustering for Inhomogeneous Populations,” Journal of the Royal Statistical Society Series B 52(1), 73–96 (1990), DOI 10.1111/j.2517-6161.1990.tb01773.x. 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

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 IDs, units, precision, missingness and duplicates.

Geometry audit

Publish windows, metrics, thresholds and edge rules.

Calculation audit

Reproduce a hand example and software output.

Null audit

State exchangeability, conditioning and simulations.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Multiplicity audit

Control threshold grids, matrix tests or surfaces.

Sensitivity audit

Repeat scientifically plausible scales and designs.

Transfer audit

Reserve independent regions, periods or samples.

Final claim

Report interaction or association without automatic causal claims.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Case definition

Begin with this mechanism: Cases follow one eligibility rule. Recreate the concrete case: Three mapped children meet the case definition. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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 definitions changes clustering. Complete the transfer step: Freeze criteria. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Workshop 2: Control sampling

Begin with this mechanism: Controls represent the population at risk. Recreate the concrete case: Three controls are sampled from the same region. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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: Convenience controls distort density. Complete the transfer step: Document design. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Workshop 3: Combined coordinates

Begin with this mechanism: Cases and controls share one metric space. Recreate the concrete case: Six points use metres. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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 geocoding quality biases neighbours. Complete the transfer step: Harmonize precision. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Workshop 4: Distance matrix

Begin with this mechanism: Compute all nonself separations. Recreate the concrete case: Each row ranks five candidates. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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: Barriers can invalidate Euclidean distance. Complete the transfer step: Use a relevant metric. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 5: Neighbour order

Begin with this mechanism: Sort other points by distance. Recreate the concrete case: The two closest define k=2. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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 deterministic handling. Complete the transfer step: Record tie rules. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Workshop 6: Case-centred count

Begin with this mechanism: For each case count case labels among k neighbours. Recreate the concrete case: Counts are 1,1,0. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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 neighbour can serve several cases. Complete the transfer step: Retain dependence. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Workshop 7: T k statistic

Begin with this mechanism: Sum the case-centred counts. Recreate the concrete case: T2 equals 2. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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: The statistic depends on k. Complete the transfer step: Report k. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Workshop 8: Conditional expectation

Begin with this mechanism: Under random labels E[Tk]=c k(c−1)/(n−1). Recreate the concrete case: 3×2×2/5=2.4. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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: Matched designs change the null. Complete the transfer step: Use the correct conditioning. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 9: Observed contrast

Begin with this mechanism: Compare T2 with its centre. Recreate the concrete case: Two is below 2.4. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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: Difference is not a p-value. Complete the transfer step: Calculate the distribution. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Workshop 10: Label permutation

Begin with this mechanism: Reassign case labels under design constraints. Recreate the concrete case: Choose three of six points as cases. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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: Unrestricted shuffling breaks matching. Complete the transfer step: Permute within strata. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Workshop 11: Exact enumeration

Begin with this mechanism: Small samples allow every allocation. Recreate the concrete case: There are 20 unstratified label sets. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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: Discrete distributions limit resolution. Complete the transfer step: Report attainable p-values. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Workshop 12: Choice of k

Begin with this mechanism: k controls spatial scale. Recreate the concrete case: Evaluate k=1,2,3 by plan. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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 best after inspection inflates type I error. Complete the transfer step: Use a combined statistic. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 13: Combined T

Begin with this mechanism: Covariance-aware weights can combine several T k values. Recreate the concrete case: T1,T2,T3 contribute jointly. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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: Naive summation double counts related evidence. Complete the transfer step: Use published covariance. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Workshop 14: Control ratio

Begin with this mechanism: More controls may improve representation and power. Recreate the concrete case: A 1:4 ratio samples local risk density better. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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: Extra biased controls do not help. Complete the transfer step: Prioritize validity. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Workshop 15: Boundary coverage

Begin with this mechanism: Cases near study edges need controls beyond their nearest range. Recreate the concrete case: A border case’s neighbours lie outside the map. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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: Truncation mimics isolation. Complete the transfer step: Expand or correct coverage. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.

Workshop 16: Sensitivity and follow-up

Begin with this mechanism: Repeat control samples, metrics and independent periods. Recreate the concrete case: Clustering persists across valid control draws. Write cases, controls, coordinates, distances, neighbour ranks, case-neighbour counts, T k values, expectations, covariance and permutation 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: Persistence remains association. Complete the transfer step: Investigate exposures. Change exactly one event, case label, control, coordinate, timestamp, distance metric, threshold, k value, matrix transformation, permutation restriction, window boundary, edge correction, intensity model, simulation count, alternative hypothesis or random seed. 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 samples outside tuning when testing transfer.