Chapter map

1. 1. Null model
2. 2. Fit nuisance parameters
3. 3. Transform
4. 4. Basis
5. 5. Projection
6. 6. Component sums
7. 7. Square components
8. 8. Cumulative statistic
9. 9. Candidate ceiling
10. 10. Schwarz penalty
11. 11. Penalized path
12. 12. Select dimension
13. 13. Selected statistic
14. 14. Null calibration
15. 15. Interpretation
16. 16. Claim

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1. Null model

The central mathematical idea is concrete. Specify the composite distribution family.

A concrete situation

Normality may include unknown mean and variance.

What the mathematics does not license

An unspecified null makes scores meaningless. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Write parameters. 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. Fit nuisance parameters

Here the geometry becomes operational. Estimate parameters under the null.

A concrete situation

Store fitted mean and variance.

What the mathematics does not license

Fitting consumes directions of variation. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Keep method. 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. Transform

This step deserves a careful pause. Map observations to fitted-null probabilities or residual coordinates.

A concrete situation

Obtain a common reference scale.

What the mathematics does not license

Poor residual construction creates false departure. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Audit transformation. 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. Basis

The useful question is not only how to calculate, but what the calculation assumes. Choose orthonormal score functions.

A concrete situation

Each coordinate represents a distinct departure direction.

What the mathematics does not license

An unnormalised basis distorts component sizes. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Verify inner products. 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. Projection

A small example makes the mechanism visible. Remove nuisance-score directions for the composite null.

A concrete situation

Work with post-projection components.

What the mathematics does not license

Ignoring estimation double counts location or scale. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use paper construction. 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. Component sums

This is where modelling choices begin to affect evidence. Average or sum each score over observations with the prescribed scaling.

A concrete situation

Illustrative coordinates are 2.0,0.5,3.0,0.4.

What the mathematics does not license

These teaching numbers are not raw moments. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Label illustration. 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. Square components

The central mathematical idea is concrete. Compute z_j².

A concrete situation

Values are 4.00,0.25,9.00,0.16.

What the mathematics does not license

Squaring loses sign but measures magnitude. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Keep signed scores too. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Cumulative statistic

Here the geometry becomes operational. Form T_k=Σ_{j≤k}z_j².

A concrete situation

T values are 4.00,4.25,13.25,13.41.

What the mathematics does not license

Later dimensions include earlier evidence. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Show path. 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. Candidate ceiling

This step deserves a careful pause. Set d(n) before analysis.

A concrete situation

Here examine k=1 through 4.

What the mathematics does not license

Searching too many components increases noise. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Predefine ceiling. 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. Schwarz penalty

The useful question is not only how to calculate, but what the calculation assumes. Subtract k log n in the teaching form.

A concrete situation

log(100)≈4.6052.

What the mathematics does not license

The exact rule must match the cited procedure. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Name convention. 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. Penalized path

A small example makes the mechanism visible. Compare T_k−k log n.

A concrete situation

Values are about -0.605,-4.960,-0.566,-5.011.

What the mathematics does not license

Rounding can change a narrow winner. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Keep digits. 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. Select dimension

This is where modelling choices begin to affect evidence. Choose the maximizing dimension under the specified rule.

A concrete situation

k=3 narrowly wins.

What the mathematics does not license

Selection is part of the statistic. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record full path. 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. Selected statistic

The central mathematical idea is concrete. Report T_S with S.

A concrete situation

Here T_3=13.25.

What the mathematics does not license

A fixed-k chi-square table is not automatically valid. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use selected-null law. 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. Null calibration

Here the geometry becomes operational. Repeat fitting, projection and selection in every simulation.

A concrete situation

Build the selected statistic distribution.

What the mathematics does not license

Calibrating only T_3 conditions on the winner. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Mirror pipeline. 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. Interpretation

This step deserves a careful pause. Inspect which component caused selection.

A concrete situation

The third coordinate contributes 9.00.

What the mathematics does not license

A basis coordinate is not a unique physical cause. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Translate cautiously. 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. Claim

The useful question is not only how to calculate, but what the calculation assumes. State adaptive smooth-fit evidence.

A concrete situation

Some modeled score direction conflicts with the null.

What the mathematics does not license

The method does not choose a replacement model. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use follow-up plots. 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

Kallenberg and Ledwina address a central weakness of smooth goodness-of-fit tests: a fixed number of score components may be too small for a complex departure or too large for a simple one. Their data-driven construction expands departures in orthonormal directions, forms cumulative squared-score statistics and uses a Schwarz-style penalty to select the dimension before reporting the selected smooth statistic.

Consider an illustrative post-projection score vector (2.0,0.5,3.0,0.4) for n=100. The cumulative squared scores are 4.00,4.25,13.25 and 13.41. Using the simple teaching form T_k−k log n, the penalized values are about -0.605,-4.960,-0.566 and -5.011. The third dimension narrowly wins because the third coordinate adds enough evidence to pay its complexity penalty.

This arithmetic is illustrative rather than a substitute for the paper’s full composite-hypothesis construction. When mean, variance or other parameters are estimated, nuisance directions must be projected out or otherwise adjusted. The basis, maximum dimension d(n), selection rule and calibration all belong to the definition of the test; changing one produces a different procedure.

Adaptivity is useful because different alternatives appear in different score coordinates, yet it is not free exploration. Schwarz selection regularizes the search, and the null distribution must include the same selection step. A chi-square table for a fixed k cannot simply be attached after k was chosen from the data. Simulation or published critical values must reproduce estimation, projection and selection together.

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Frequently asked questions

What is a smooth test?

A goodness-of-fit test that measures departure through a finite set of orthonormal score components.

Why not fix the number of components?

The best dimension depends on the alternative; too few miss structure and too many add noise.

What does the Schwarz rule do?

It rewards cumulative fit evidence while penalizing every additional component.

What changes for a composite null?

Estimated nuisance parameters require projection or adjustment of score directions.

Can a fixed-k chi-square table follow data-driven selection?

Not automatically; the null calibration must repeat selection.

What should be reported?

The basis, maximum dimension, component scores, penalized path, selected dimension and calibration method.

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Primary source and fact-check note

Wilbert C. M. Kallenberg and Teresa Ledwina, “Data-Driven Smooth Tests When the Hypothesis Is Composite,” Journal of the American Statistical Association 92(439), 1094–1104 (1997), DOI 10.1080/01621459.1997.10474065. 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

• Desgagné–Lafaye de Micheaux Rao score test
• Bontemps–Meddahi GMM normality 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

Design audit

Freeze observations, chronology, target residuals, dependence and inclusion rules.

Transformation audit

Reproduce ordering, plotting positions, normal scores, fitted-null probabilities or standardized moments.

Statistic audit

Show every gap, projection, component, dependence factor, penalty and coefficient.

Calibration audit

Match sample size, parameter estimation, dependence, selection and statistic version.

Graph audit

Preserve Q–Q views, score paths, ACF displays, influence plots and component maps.

Magnitude audit

Explain consequences in original units and decision context.

Sensitivity audit

Use only prespecified validation, model and measurement checks.

Final claim

Keep the conclusion inside the measured population, time window and model.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Null model

Begin with this mechanism: Specify the composite distribution family. Recreate the concrete case: Normality may include unknown mean and variance. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: An unspecified null makes scores meaningless. Complete the transfer step: Write parameters. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 2: Fit nuisance parameters

Begin with this mechanism: Estimate parameters under the null. Recreate the concrete case: Store fitted mean and variance. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Fitting consumes directions of variation. Complete the transfer step: Keep method. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 3: Transform

Begin with this mechanism: Map observations to fitted-null probabilities or residual coordinates. Recreate the concrete case: Obtain a common reference scale. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Poor residual construction creates false departure. Complete the transfer step: Audit transformation. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 4: Basis

Begin with this mechanism: Choose orthonormal score functions. Recreate the concrete case: Each coordinate represents a distinct departure direction. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: An unnormalised basis distorts component sizes. Complete the transfer step: Verify inner products. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 5: Projection

Begin with this mechanism: Remove nuisance-score directions for the composite null. Recreate the concrete case: Work with post-projection components. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Ignoring estimation double counts location or scale. Complete the transfer step: Use paper construction. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 6: Component sums

Begin with this mechanism: Average or sum each score over observations with the prescribed scaling. Recreate the concrete case: Illustrative coordinates are 2.0,0.5,3.0,0.4. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: These teaching numbers are not raw moments. Complete the transfer step: Label illustration. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 7: Square components

Begin with this mechanism: Compute z_j². Recreate the concrete case: Values are 4.00,0.25,9.00,0.16. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Squaring loses sign but measures magnitude. Complete the transfer step: Keep signed scores too. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 8: Cumulative statistic

Begin with this mechanism: Form T_k=Σ_{j≤k}z_j². Recreate the concrete case: T values are 4.00,4.25,13.25,13.41. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Later dimensions include earlier evidence. Complete the transfer step: Show path. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 9: Candidate ceiling

Begin with this mechanism: Set d(n) before analysis. Recreate the concrete case: Here examine k=1 through 4. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Searching too many components increases noise. Complete the transfer step: Predefine ceiling. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 10: Schwarz penalty

Begin with this mechanism: Subtract k log n in the teaching form. Recreate the concrete case: log(100)≈4.6052. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: The exact rule must match the cited procedure. Complete the transfer step: Name convention. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 11: Penalized path

Begin with this mechanism: Compare T_k−k log n. Recreate the concrete case: Values are about -0.605,-4.960,-0.566,-5.011. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Rounding can change a narrow winner. Complete the transfer step: Keep digits. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 12: Select dimension

Begin with this mechanism: Choose the maximizing dimension under the specified rule. Recreate the concrete case: k=3 narrowly wins. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Selection is part of the statistic. Complete the transfer step: Record full path. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 13: Selected statistic

Begin with this mechanism: Report T_S with S. Recreate the concrete case: Here T_3=13.25. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A fixed-k chi-square table is not automatically valid. Complete the transfer step: Use selected-null law. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 14: Null calibration

Begin with this mechanism: Repeat fitting, projection and selection in every simulation. Recreate the concrete case: Build the selected statistic distribution. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Calibrating only T_3 conditions on the winner. Complete the transfer step: Mirror pipeline. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 15: Interpretation

Begin with this mechanism: Inspect which component caused selection. Recreate the concrete case: The third coordinate contributes 9.00. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A basis coordinate is not a unique physical cause. Complete the transfer step: Translate cautiously. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 16: Claim

Begin with this mechanism: State adaptive smooth-fit evidence. Recreate the concrete case: Some modeled score direction conflicts with the null. Write fitted-null parameters, transformed observations, basis functions, nuisance projections, component scores, cumulative statistics, penalties, selected dimensions and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: The method does not choose a replacement model. Complete the transfer step: Use follow-up plots. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.