Chapter map

1. 1. Target
2. 2. Coverage
3. 3. Candidate
4. 4. Residuals
5. 5. Squares
6. 6. Order statistics
7. 7. Trimmed sum
8. 8. Initial subset
9. 9. C-step
10. 10. Monotonicity
11. 11. FAST-LTS
12. 12. Breakdown
13. 13. Scale
14. 14. Diagnostics
15. 15. Uncertainty
16. 16. Claim

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1. Target

A small example makes the mechanism visible. Specify response, design matrix, p and intercept convention.

A concrete situation

Use n=5 and p=2.

What the mathematics does not license

Different p conventions change h formulas. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Define p. 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. Coverage

This is where modelling choices begin to affect evidence. Choose h before fitting.

A concrete situation

Use h=4 for the teaching case.

What the mathematics does not license

Choosing h after seeing residuals invites bias. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Predefine h. 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. Candidate

The central mathematical idea is concrete. Compute predictions for each candidate line.

A concrete situation

Compare y=1+x with y=3.

What the mathematics does not license

Candidate comparison is not global optimisation. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Keep scope. 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. Residuals

Here the geometry becomes operational. Calculate signed vertical residuals.

A concrete situation

A gives 0,.1,−.1,.2,15.

What the mathematics does not license

Vertical residuals depend on response choice. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

State direction. 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. Squares

This step deserves a careful pause. Square and retain observation IDs.

A concrete situation

A gives 0,.01,.01,.04,225.

What the mathematics does not license

Large values should remain in the data record. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

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

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6. Order statistics

The useful question is not only how to calculate, but what the calculation assumes. Sort squared residuals r²_(1) through r²_(n).

A concrete situation

Identify the first four.

What the mathematics does not license

Ties need deterministic handling. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

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

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7. Trimmed sum

A small example makes the mechanism visible. Add the first h squared residuals.

A concrete situation

A=.06 and B=6.26.

What the mathematics does not license

The omitted residual still exists. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Report both sets. 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. Initial subset

This is where modelling choices begin to affect evidence. Generate a p-subset or other reproducible start.

A concrete situation

Fit a candidate line.

What the mathematics does not license

A poor start may reach a poor local minimum. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use many starts. 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. C-step

The central mathematical idea is concrete. Refit on the h smallest residuals.

A concrete situation

Repeat ordering and least squares.

What the mathematics does not license

One C-step is not convergence. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Track objective. 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. Monotonicity

Here the geometry becomes operational. Check the objective never increases.

A concrete situation

Store every C-step sum.

What the mathematics does not license

An increase signals a coding or tie issue. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Test implementation. 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. FAST-LTS

This step deserves a careful pause. Use documented starts and nested search for large data.

A concrete situation

Set seed and trial count.

What the mathematics does not license

Fast algorithms may return an approximation. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Disclose settings. 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. Breakdown

The useful question is not only how to calculate, but what the calculation assumes. Relate h to contamination resistance.

A concrete situation

Smaller admissible h gives more protection.

What the mathematics does not license

Efficiency usually changes too. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Show trade-off. 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. Scale

A small example makes the mechanism visible. Estimate residual scale with the documented correction.

A concrete situation

Use the final raw or reweighted fit consistently.

What the mathematics does not license

Raw residual RMS is not automatically robust. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Name formula. 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. Diagnostics

This is where modelling choices begin to affect evidence. Plot predictor leverage and both kept and trimmed residuals.

A concrete situation

Look for a second coherent group.

What the mathematics does not license

Trimmed does not mean false. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Investigate mechanisms. 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. Uncertainty

The central mathematical idea is concrete. Resample the complete subset-selection pipeline.

A concrete situation

Refit every bootstrap sample.

What the mathematics does not license

Conditional OLS errors understate selection uncertainty. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

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

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16. Claim

Here the geometry becomes operational. State h, algorithm and candidate/global status.

A concrete situation

A has the smaller teaching LTS sum.

What the mathematics does not license

Do not guarantee unique truth. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Keep options open. 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

Least trimmed squares regression orders the squared residuals from smallest to largest and sums only the first h. The fitted coefficients minimise that trimmed sum. The coverage h controls the central subset that must be fitted: smaller admissible h increases resistance, while larger h uses more observations and can improve precision when the model is clean. The method therefore makes the robustness–efficiency trade-off visible in one integer.

For the five-point example and a two-parameter line with intercept, take the common high-breakdown teaching choice h=floor((n+p+1)/2)=4. Candidate A, y=1+x, has four smallest squared residuals 0,.01,.01,.04, so its LTS objective is .06. Candidate B, y=3, has four smallest values .01,.81,1.44,4, summing to 6.26. Candidate A wins this comparison, while the value 225 is trimmed from its objective. Trimming an objective is not the same as deleting a record from the dataset.

A concentration step starts from a candidate h-subset, fits least squares on it, calculates all residuals, keeps the h observations with the smallest squared residuals and refits. The LTS objective cannot increase under the standard C-step result. Repeating until the subset stabilises reaches a local solution. FAST-LTS uses many starts and additional search ideas because one C-step path need not locate the global minimum.

LTS can reveal a coherent majority pattern, but a real minority cluster may represent another process, a policy group or a new regime. High leverage can also be subtle. Preserve every observation, plot predictor space, compare several h values, inspect residual scale and use domain knowledge before attaching labels. Report h, p convention, number of starts, seed, C-step tolerance, final subset and the uncertainty method.

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Frequently asked questions

What does LTS minimise?

The sum of the h smallest ordered squared residuals.

What does h control?

How many observations must contribute to the objective and the resistance–efficiency trade-off.

What is a C-step?

Refit on the h smallest-residual observations and repeat; the standard objective cannot increase.

Does trimming delete data?

No. All observations remain available for diagnostics and interpretation.

Is FAST-LTS always exact?

It can be exact for small problems and uses a fast search for larger ones; settings and starts matter.

What must be reported?

h, p convention, starts, seed, C-steps, scale, final subset, reweighting and uncertainty.

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Primary source and fact-check note

Peter J. Rousseeuw and Katrien Van Driessen, “Computing LTS Regression for Large Data Sets,” Data Mining and Knowledge Discovery 12, 29–45 (2006), DOI 10.1007/s10618-005-0024-4. 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

• Least median of squares regression
• Tukey bisquare M-estimator
• How Robust Statistics Work
• 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 sampling units, response, predictors, intercept, dependence, missingness and the scientific target.

Residual audit

Show every prediction, signed residual, squared residual, order statistic, subset membership and original unit.

Robustness audit

Connect h, bounded loss, b, start and scale to a precise finite-sample or asymptotic claim.

Solver audit

Record starts, seed, subset path, C-steps, scale roots, IRLS updates, objective and stopping tolerance.

Leverage audit

Inspect predictor-space geometry separately from response residuals.

Uncertainty audit

Use covariance or resampling that repeats the estimator’s complete selection and scale pipeline.

Sensitivity audit

Change one justified h, loss, b, starting fit, scale correction, tuning value or cluster at a time.

Final claim

Describe the fitted majority pattern and limitations without turning robustness into a verdict about people or records.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Target

Begin with this mechanism: Specify response, design matrix, p and intercept convention. Recreate the concrete case: Use n=5 and p=2. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Different p conventions change h formulas. Complete the transfer step: Define p. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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: Coverage

Begin with this mechanism: Choose h before fitting. Recreate the concrete case: Use h=4 for the teaching case. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Choosing h after seeing residuals invites bias. Complete the transfer step: Predefine h. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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: Candidate

Begin with this mechanism: Compute predictions for each candidate line. Recreate the concrete case: Compare y=1+x with y=3. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Candidate comparison is not global optimisation. Complete the transfer step: Keep scope. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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: Residuals

Begin with this mechanism: Calculate signed vertical residuals. Recreate the concrete case: A gives 0,.1,−.1,.2,15. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Vertical residuals depend on response choice. Complete the transfer step: State direction. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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: Squares

Begin with this mechanism: Square and retain observation IDs. Recreate the concrete case: A gives 0,.01,.01,.04,225. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Large values should remain in the data record. Complete the transfer step: Preserve provenance. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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: Order statistics

Begin with this mechanism: Sort squared residuals r²_(1) through r²_(n). Recreate the concrete case: Identify the first four. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Ties need deterministic handling. Complete the transfer step: Record tie rule. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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: Trimmed sum

Begin with this mechanism: Add the first h squared residuals. Recreate the concrete case: A=.06 and B=6.26. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: The omitted residual still exists. Complete the transfer step: Report both sets. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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: Initial subset

Begin with this mechanism: Generate a p-subset or other reproducible start. Recreate the concrete case: Fit a candidate line. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A poor start may reach a poor local minimum. Complete the transfer step: Use many starts. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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: C-step

Begin with this mechanism: Refit on the h smallest residuals. Recreate the concrete case: Repeat ordering and least squares. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: One C-step is not convergence. Complete the transfer step: Track objective. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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: Monotonicity

Begin with this mechanism: Check the objective never increases. Recreate the concrete case: Store every C-step sum. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: An increase signals a coding or tie issue. Complete the transfer step: Test implementation. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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: FAST-LTS

Begin with this mechanism: Use documented starts and nested search for large data. Recreate the concrete case: Set seed and trial count. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Fast algorithms may return an approximation. Complete the transfer step: Disclose settings. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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: Breakdown

Begin with this mechanism: Relate h to contamination resistance. Recreate the concrete case: Smaller admissible h gives more protection. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Efficiency usually changes too. Complete the transfer step: Show trade-off. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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: Scale

Begin with this mechanism: Estimate residual scale with the documented correction. Recreate the concrete case: Use the final raw or reweighted fit consistently. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Raw residual RMS is not automatically robust. Complete the transfer step: Name formula. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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: Diagnostics

Begin with this mechanism: Plot predictor leverage and both kept and trimmed residuals. Recreate the concrete case: Look for a second coherent group. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Trimmed does not mean false. Complete the transfer step: Investigate mechanisms. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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: Uncertainty

Begin with this mechanism: Resample the complete subset-selection pipeline. Recreate the concrete case: Refit every bootstrap sample. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Conditional OLS errors understate selection uncertainty. Complete the transfer step: Mirror procedure. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. 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 h, algorithm and candidate/global status. Recreate the concrete case: A has the smaller teaching LTS sum. Write responses, predictors, predictions, ordered squared residuals, h values, subset indices, C-step objectives, coefficients, scales and original measurement units with units, signs, subset labels and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Do not guarantee unique truth. Complete the transfer step: Keep options open. Change exactly one response, predictor, sampling unit, h value, rho function, b constant, robust scale, starting subset, starting coefficients, tuning constant, leverage value, solver tolerance, resampling rule or covariance formula. Explain which objective changes, which observations influence the next step, which solution is selected, which conclusion survives and which claim must weaken. Keep design, predictions, residuals, ordering, subset selection, scale, score, weight, optimisation, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the full calculation table and once to a parent with the decision in original units. Add a counterexample in which a similar robust line comes from rounding, censoring, dependence, mixture structure, high leverage or a genuine minority process. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.