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eduKateSG · Why Mathematics? · Multi-Arm Multi-Stage Trials: Dropping Arms, Shared Controls and Efficient Comparisons
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 8–14
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 15–21
Check assumptions
See where design and calibration matter.Route 4 · Chapters 22–28
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 29–42
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when several treatments can be evaluated against one control and weak options can stop early? Multi-arm multi-stage trials share control information and apply planned interim boundaries so resources move toward promising comparisons without losing error control.
Did You Know? Three separate 1:1 trials needing 100 participants per arm use 300 control participants. One four-arm trial with three experimental arms and one shared control uses 100 controls in the simplest equal-size illustration, reducing the total from 600 to 400.
Full chapter index
Chapters 1–7 · Understand the question
Chapters 8–14 · Reproduce the example
Chapters 15–21 · Check assumptions
Chapters 22–28 · Explore applications
Chapters 29–42 · Study and verify
- Final decision
- A spreadsheet workflow
- A coding workflow
- Applications and careers
- A student investigation
- Common misconceptions
- Connection to recent mathematics articles
- Guidance for students and parents
- Practice questions
- Frequently asked questions
- A reporting checklist
- Correlation is an asset and a duty
- A final MAMS audit
- Useful next reading
Several treatments, doses or programmes may be ready at once. Comparing each with a common control under one coordinated design can reduce duplication and answer more questions.
Participants randomised to control can contribute to several contemporaneous treatment comparisons. The correlations this creates are useful and must appear in multiplicity calculations.
Three experimental groups of 100 each plus one shared control of 100 total 400. Three separate equal trials require 300 experimental and 300 control participants, totalling 600.
CHAPTER 4 OF 42 · Understand the question
4. Why the arithmetic is only illustrative
Optimal allocation may put more participants in control, outcomes can be delayed, and eligibility may differ. The valid comparison uses control participants who are eligible and concurrent with the relevant arm.
At each planned interim analysis, every active comparison is assessed against a lack-of-benefit, efficacy or safety boundary. The next stage retains the arms allowed by the rule.
| Design | Experimental participants | Control participants | Total |
|---|---|---|---|
| Three separate 1:1 trials | 300 | 300 | 600 |
| One four-arm trial | 300 | 100 | 400 |
Suppose stage one drops an arm below z=0.5, continues values from 0.5 to 2.5 and permits early efficacy above 2.5. Actual boundaries require design-specific calibration.
Arm A at z=0.4 drops for lack of benefit, B at 1.6 continues, and C at 2.7 crosses the illustrative efficacy boundary.
The rule says the arm lacks enough promise for this design and remaining resources. It does not establish equivalence or rule out every worthwhile effect.
| Arm | Z statistic | Rule |
|---|---|---|
| A | 0.4 | Drop for lack of benefit |
| B | 1.6 | Continue |
| C | 2.7 | Cross early efficacy boundary |
With several null hypotheses and repeated looks, unadjusted 0.05 tests inflate the chance of at least one false positive. Boundaries must protect the chosen family.
Comparisons sharing one control are positively correlated. Dunnett-type methods can use that correlation and be less conservative than treating tests as independent.
Control should hold across configurations where some treatments work and others do not, not only the global null where every arm is ineffective.
A resource cap may allow only the best one or two arms to continue. The selection rule and final critical value must be prespecified and simulated.
The MAMS R package paper describes group-sequential multi-arm designs, treatment selection and conditional-error modifications. Read the Journal of Statistical Software paper.
A short-term interim outcome can accelerate dropping, but its relationship to the definitive outcome must be credible. Otherwise a useful arm may be discarded.
Shared efficacy infrastructure does not mean shared safety profiles. Each treatment needs appropriate exposure, monitoring and adverse-event interpretation.
If control treatment evolves, comparisons across stages may no longer target the same effect. Versioning, concurrent comparison and sensitivity analysis become essential.
Equal arm sizes are not always optimal. More control participants can improve several contrasts simultaneously, while too few on an experimental arm harms its precision.
The surviving arm tends to look better because it was selected. Naive estimates and confidence intervals may be biased or have poor coverage.
Combination tests or conditional-error methods can support valid modifications when their stagewise assumptions and hypotheses are preserved.
Every interim cutoff should list outcomes available, pending and missing. Operational decisions must not selectively wait for favourable late results.
Simulate all null and alternative configurations, recruitment, shared control, correlation, stopping, selection, missingness and final estimation.
An effective design may keep the same maximum sample but use fewer participants on average by dropping weak arms. Show expected size by effect scenario.
A school network might compare three tutoring formats with usual support. A shared control saves pupils, but schools, terms and assessment timing must remain comparable.
An independent monitoring group should receive unblinded comparisons and implement decisions through a reproducible charter, limiting information released to sites.
Report arm entry, information time, z paths, boundaries, control overlap, correlation, selection probability, familywise error, power and expected sample.
Reproduce 600 versus 400, simulate correlated z statistics and verify rejection under global null and mixed-effect scenarios.
MAMS designs do not make treatments interchangeable, eliminate multiplicity or guarantee faster answers. Shared control is valuable only when scientifically comparable.
Publish every prespecified and actual arm decision, including negative arms. Avoid presenting a selected winner as if it came from a single fixed comparison.
Use MAMS when several mature questions share a credible control and common infrastructure. Complexity is justified only when efficiency and learning improve together.
Track arm, stage, information fraction, estimate, SE, boundary, control window, decision and final claim in one auditable matrix.
Use validated MAMS software and independent simulation. Test ties, arm drops, missing stages, correlation matrices and every mixed null configuration.
Clinical trialists, agricultural researchers, policy evaluators and product teams use multi-arm designs to compare several options efficiently.
Simulate three treatments and one shared control. Compare separate trials, fixed multi-arm and two-stage MAMS designs across mixed effects.
A shared control is not historical control, dropping is not equivalence, the winning arm is selected, and per-comparison error is not familywise error.
The alpha-spending guide covers repeated looks. The futility guide explains lack-of-benefit decisions.
Students should draw one control node connected to every active arm and label which controls are concurrent. Parents can ask whether school comparisons share the same term and eligibility.
- Reproduce the sample saving.
- Apply three boundaries.
- Add an arm-selection cap.
- Change the control standard.
- Simulate familywise error.
### Does every arm need its own control? A valid shared control can serve several comparisons.
### Is a dropped arm ineffective? Not proven.
### Can the best arm be selected? Yes, with a valid rule.
### Are ordinary intervals enough? Selection may require adjustment.
Report arms and hypotheses, shared-control eligibility, allocation, stages and boundaries, multiplicity, selection, interim cutoffs, simulations, all decisions and adjusted estimation.
Shared controls reduce duplication because comparisons share information; the same correlation must appear in error control and uncertainty.
Confirm every control participant belongs to the relevant comparison period, all stopping rules were followed and negative arms remain visible.
Continue through the Mathematics Learning Hub. Recalculate the central worked example, identify the assumptions and explain which claim the result can and cannot support.
Extended practice laboratory
Practice case 1: Count experimental arms
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 1. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 2: Define shared control
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 2. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 3: Set allocation ratio
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 3. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 4: Calculate total sample
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 4. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 5: Set stage-one boundary
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 5. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 6: Set final boundary
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 6. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 7: Drop futile arm
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 7. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 8: Continue promising arm
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 8. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 9: Stop successful arm
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 9. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 10: Protect familywise error
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 10. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 11: Use Dunnett correlation
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 11. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 12: Add arm selection
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 12. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 13: Cap survivors
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 13. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 14: Add outcome delay
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 14. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 15: Add safety stop
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 15. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 16: Change control standard
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 16. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 17: Calculate concurrent control
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 17. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 18: Estimate treatment contrast
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 18. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 19: Adjust confidence interval
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 19. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 20: Add missingness
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 20. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 21: Simulate global null
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 21. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 22: Simulate mixed effects
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 22. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 23: Audit arm decisions
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 23. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 24: Name limits
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 24. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 25: Write conclusion
Setup: Start from the article’s multi-arm multi-stage design mechanism for case 25. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.