Chapter map

1. 1. Observed counts
2. 2. Expected counts
3. 3. Raw SMR
4. 4. Ensemble mean
5. 5. Dispersion
6. 6. Gamma prior
7. 7. Posterior mean
8. 8. Reliability weight
9. 9. Large-area contrast
10. 10. Global shrinkage
11. 11. Local mean
12. 12. Iterative estimation
13. 13. Predictive check
14. 14. Map comparison
15. 15. Sensitivity analysis
16. 16. External validation

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. Observed counts

This step deserves a careful pause. Fix Yi using one case definition and period.

A concrete situation

A small area has Y=4.

What the mathematics does not license

Late reporting changes ratios. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Reconcile records. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. Expected counts

The useful question is not only how to calculate, but what the calculation assumes. Calculate Ei from population and risk strata.

A concrete situation

The area has E=2 after age standardization.

What the mathematics does not license

Wrong standards bias every estimate. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Publish the standard. 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. Raw SMR

A small example makes the mechanism visible. Compute Yi/Ei.

A concrete situation

4/2=2.0.

What the mathematics does not license

Small E makes the ratio volatile. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Show counts beside ratios. 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. Ensemble mean

This is where modelling choices begin to affect evidence. Estimate the overall relative-risk centre.

A concrete situation

The fitted prior mean is 1.1.

What the mathematics does not license

One mean may hide spatial regimes. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Check heterogeneity. 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. Dispersion

The central mathematical idea is concrete. Estimate how much risks vary beyond Poisson noise.

A concrete situation

The ensemble supports finite variance.

What the mathematics does not license

Poor estimates affect all shrinkage. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use diagnostics and sensitivity. 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. Gamma prior

Here the geometry becomes operational. Represent θi with shape ν and rate α.

A concrete situation

α=8 and ν=8.8 give mean 1.1.

What the mathematics does not license

Parameter conventions differ. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

State shape and rate. 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. Posterior mean

This step deserves a careful pause. Compute (Yi+ν)/(Ei+α).

A concrete situation

(4+8.8)/(2+8)=1.28.

What the mathematics does not license

A point summary hides spread. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report intervals. 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. Reliability weight

The useful question is not only how to calculate, but what the calculation assumes. Express the posterior as data weight plus prior weight.

A concrete situation

E/(E+α)=.2 for the small area.

What the mathematics does not license

Weight depends on model assumptions. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Show it explicitly. 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. Large-area contrast

A small example makes the mechanism visible. Repeat for a better-informed area.

A concrete situation

Y=24,E=20 gives about 1.171.

What the mathematics does not license

Large E does not cure bias. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Audit 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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10. Global shrinkage

This is where modelling choices begin to affect evidence. Pull toward one ensemble mean.

A concrete situation

Every area uses centre 1.1.

What the mathematics does not license

It ignores spatial configuration. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare local smoothing. 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. Local mean

The central mathematical idea is concrete. Estimate a neighbourhood centre.

A concrete situation

Adjacent areas suggest mean 1.3.

What the mathematics does not license

Neighbour selection can import bias. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Publish the graph. 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. Iterative estimation

Here the geometry becomes operational. Update ν and α from current regional estimates.

A concrete situation

Iterations converge to stable parameters.

What the mathematics does not license

Local optima or coding errors can mislead. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Check convergence. 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. Predictive check

This step deserves a careful pause. Simulate counts under the fitted ensemble.

A concrete situation

Replicates should match zeros and extremes.

What the mathematics does not license

Good averages can hide tail failure. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare full distributions. 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. Map comparison

The useful question is not only how to calculate, but what the calculation assumes. Display raw SMR and EB estimates together.

A concrete situation

The small-area spike contracts visibly.

What the mathematics does not license

Colour scales can exaggerate change. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use common legends. 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. Sensitivity analysis

A small example makes the mechanism visible. Change standard, prior family and local graph.

A concrete situation

The area remains moderately elevated.

What the mathematics does not license

Selecting one favourable map is biased. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Predeclare variants. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. External validation

This is where modelling choices begin to affect evidence. Apply fitted rules to another period.

A concrete situation

Later counts test the stabilized pattern.

What the mathematics does not license

Prediction does not prove mechanism. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Investigate exposures separately. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

Clayton–Kaldor empirical Bayes mapping starts from a practical problem: a raw standardized mortality or incidence ratio can be extremely noisy when a region has a small expected count. If one area observes four events when two were expected, its SMR is 2.0, yet that estimate contains far less information than an SMR based on hundreds of expected events. The method uses the complete regional ensemble to estimate how much genuine variation exists and how strongly each raw ratio should be shrunk.

In a gamma–Poisson version, Yi conditional on relative risk θi follows Poisson(Eiθi), while θi follows a gamma distribution with shape ν and rate α. The posterior mean is (Yi+ν)/(Ei+α). Because the prior mean is ν/α, this expression is a reliability-weighted compromise between Yi/Ei and the ensemble mean. Clayton and Kaldor also describe spatially informed compromise with a local mean from nearby areas.

Let the fitted prior mean be 1.1 and α=8, so ν=8.8. A small area with Y=4 and E=2 has raw SMR 2.0 but posterior mean (4+8.8)/(2+8)=1.28. A larger area with Y=24 and E=20 has raw SMR 1.2 and posterior mean (24+8.8)/(20+8)≈1.171. The small area moves more because its denominator provides less information. These numbers are illustrative; empirical Bayes estimates α and ν from all areas.

Shrinkage does not certify that the ensemble model is correct. A global mean can blur real spatial structure, while a local mean can import bias from a poor neighbour set. Expected counts can also be wrong. Good mapping shows raw and smoothed risks, their information levels, uncertainty or predictive checks, sensitivity to global versus local shrinkage and clear limits on causal interpretation.

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Frequently asked questions

What is an SMR?

It is the observed count divided by the expected count after the chosen standardization.

Why shrink a raw SMR?

A ratio based on a small expected count has high sampling variation and should carry less weight than a ratio supported by more information.

What makes the method empirical Bayes?

The prior or ensemble parameters are estimated from the same collection of regional data before posterior means are calculated.

What is the gamma–Poisson posterior mean?

With shape ν and rate α, it is (Yi+ν)/(Ei+α), a compromise between Yi/Ei and ν/α.

What is local shrinkage?

It pulls an area toward a mean informed by nearby areas rather than only one overall mean.

Does smoothing remove uncertainty?

No. It changes estimation and should be accompanied by reliability, intervals, predictive checks and raw counts.

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Primary source and fact-check note

David Clayton and John Kaldor, “Empirical Bayes estimates of age-standardized relative risks for use in disease mapping,” Biometrics 43(3), 671–681 (1987), DOI 10.2307/2532003. 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–York–Mollié disease mapping
• Oden’s Ipop 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

Count audit

Verify observed and expected counts before any ratio.

Reliability audit

Show how denominator size changes the data weight.

Prior audit

Reproduce empirical ν and α estimates and convergence.

Shrinkage audit

Compare raw, global and local estimates.

Uncertainty audit

Report intervals or predictive variation.

Mapping audit

Use common scales and display counts.

Transfer audit

Apply frozen rules to a later period.

Final claim

Describe stabilized risk without hiding denominator or causal limits.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Observed counts

Begin with this mechanism: Fix Yi using one case definition and period. Recreate the concrete case: A small area has Y=4. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: Late reporting changes ratios. Complete the transfer step: Reconcile records. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or 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: Expected counts

Begin with this mechanism: Calculate Ei from population and risk strata. Recreate the concrete case: The area has E=2 after age standardization. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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 standards bias every estimate. Complete the transfer step: Publish the standard. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or 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: Raw SMR

Begin with this mechanism: Compute Yi/Ei. Recreate the concrete case: 4/2=2.0. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: Small E makes the ratio volatile. Complete the transfer step: Show counts beside ratios. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or 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: Ensemble mean

Begin with this mechanism: Estimate the overall relative-risk centre. Recreate the concrete case: The fitted prior mean is 1.1. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: One mean may hide spatial regimes. Complete the transfer step: Check heterogeneity. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 5: Dispersion

Begin with this mechanism: Estimate how much risks vary beyond Poisson noise. Recreate the concrete case: The ensemble supports finite variance. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: Poor estimates affect all shrinkage. Complete the transfer step: Use diagnostics and sensitivity. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or 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: Gamma prior

Begin with this mechanism: Represent θi with shape ν and rate α. Recreate the concrete case: α=8 and ν=8.8 give mean 1.1. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: Parameter conventions differ. Complete the transfer step: State shape and rate. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or 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: Posterior mean

Begin with this mechanism: Compute (Yi+ν)/(Ei+α). Recreate the concrete case: (4+8.8)/(2+8)=1.28. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: A point summary hides spread. Complete the transfer step: Report intervals. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or 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: Reliability weight

Begin with this mechanism: Express the posterior as data weight plus prior weight. Recreate the concrete case: E/(E+α)=.2 for the small area. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: Weight depends on model assumptions. Complete the transfer step: Show it explicitly. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 9: Large-area contrast

Begin with this mechanism: Repeat for a better-informed area. Recreate the concrete case: Y=24,E=20 gives about 1.171. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: Large E does not cure bias. Complete the transfer step: Audit expected counts. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or 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: Global shrinkage

Begin with this mechanism: Pull toward one ensemble mean. Recreate the concrete case: Every area uses centre 1.1. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: It ignores spatial configuration. Complete the transfer step: Compare local smoothing. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or 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: Local mean

Begin with this mechanism: Estimate a neighbourhood centre. Recreate the concrete case: Adjacent areas suggest mean 1.3. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: Neighbour selection can import bias. Complete the transfer step: Publish the graph. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or 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: Iterative estimation

Begin with this mechanism: Update ν and α from current regional estimates. Recreate the concrete case: Iterations converge to stable parameters. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: Local optima or coding errors can mislead. Complete the transfer step: Check convergence. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.

Workshop 13: Predictive check

Begin with this mechanism: Simulate counts under the fitted ensemble. Recreate the concrete case: Replicates should match zeros and extremes. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: Good averages can hide tail failure. Complete the transfer step: Compare full distributions. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or 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: Map comparison

Begin with this mechanism: Display raw SMR and EB estimates together. Recreate the concrete case: The small-area spike contracts visibly. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: Colour scales can exaggerate change. Complete the transfer step: Use common legends. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or 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: Sensitivity analysis

Begin with this mechanism: Change standard, prior family and local graph. Recreate the concrete case: The area remains moderately elevated. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: Selecting one favourable map is biased. Complete the transfer step: Predeclare variants. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or 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: External validation

Begin with this mechanism: Apply fitted rules to another period. Recreate the concrete case: Later counts test the stabilized pattern. Write observed counts, expected counts, raw SMRs, ensemble means, dispersion parameters, gamma shapes, gamma rates, reliability weights, posterior means, local means, intervals and predictive 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: Prediction does not prove mechanism. Complete the transfer step: Investigate exposures separately. Change exactly one observed count, expected count, population standard, gamma parameter, reliability weight, neighbour edge, rho value, precision, constraint, time period, interaction type, coordinate, triangle, gradient threshold, connection rule, null model, replicate count, random seed, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, likelihood, prior or null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or samples outside tuning when testing transfer.