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eduKateSG · Why Mathematics? · Response-Adaptive Randomisation: Allocation Probabilities, Learning and Valid Inference
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 randomisation probabilities change while a study is running? Response-adaptive randomisation uses accumulated outcomes to alter future allocation. It may learn while assigning more participants to promising options, but delayed outcomes, time trends and valid inference require careful design.
Did You Know? A rule that sends more participants to the currently better-looking arm can reduce the amount of direct comparison information. Ethical appeal and statistical power do not automatically move in the same direction.
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
- Communication
- 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
- The probability trace is evidence
- Final adaptive-randomisation audit
- Useful next reading
Fixed randomisation keeps probabilities constant, such as one half to each arm. Response-adaptive randomisation updates them from accrued outcomes according to a prespecified algorithm.
An equal-allocation burn-in gives every arm initial information and reduces extreme reactions to the first few outcomes. Suppose 20 participants complete outcomes in each of two arms.
For a binary success probability, start each arm with Beta(1,1). With 12 successes among 20 on arm A, the posterior is Beta(13,9).
With 8 successes among 20 on arm B, its posterior is Beta(9,13). These posteriors describe uncertainty in each success probability under the model and prior.
The means are 13/22=0.5909 for A and 9/22=0.4091 for B. They are smoothed estimates, not guarantees that future participants benefit at those exact rates.
For illustration only, allocate in proportion to posterior means. Other rules use the probability of being best, a tempered power, variance targets or patient-benefit criteria.
| Arm | Successes/20 | Posterior | Posterior mean |
|---|---|---|---|
| A | 12/20 | Beta(13,9) | 0.5909 |
| B | 8/20 | Beta(9,13) | 0.4091 |
Because the two posterior means sum to 1 here, the next allocation probabilities are 0.5909 to A and 0.4091 to B. A random draw still determines the actual assignment.
Proportional posterior means may behave poorly in other settings and is not a recommendation. A clinical design needs calibrated operating characteristics, control protection and disease-specific justification.
| Rule | Probability to A | Probability to B |
|---|---|---|
| Proportional posterior mean | 0.5909 | 0.4091 |
A floor such as 20% prevents an arm from receiving almost no new data after an unlucky start. Floors improve estimation, preserve comparison and provide resilience to model error.
CHAPTER 10 OF 42 · Reproduce the example
10. Maximum allocation prevents determinism
A ceiling such as 80% maintains unpredictability and avoids effectively closing an arm without a formal dropping rule. Concealment remains essential.
If outcomes take months, many participants can be assigned before earlier responses are known. The algorithm may then adapt to a small, unrepresentative subset or barely adapt at all.
Changing standard care, season, site mix or learning can make later participants differ from earlier ones. Because allocation also changes over time, treatment and calendar effects can become entangled.
Adaptive probabilities can create chance imbalance in age, baseline risk or school. Stratification, covariate-adjusted models and simulation should reflect the intended implementation.
Multi-arm trials may maintain a minimum control allocation so every comparison has contemporaneous evidence. Sending too few participants to control can weaken all treatment estimates.
CHAPTER 15 OF 42 · Check assumptions
15. Randomisation probabilities must be recorded
Store each participant’s eligible arms, probability vector, decision time and random draw. Those values support audit, diagnostics and analyses using the known design.
Naive estimators can be biased or have incorrect standard errors under some adaptive rules. Randomisation-based tests, weighted estimators or model-based methods may be needed.
FDA adaptive-design guidance discusses response-adaptive randomisation, risks from time trends and delayed outcomes, and the importance of simulations, confidentiality and interpretable inference. Read the FDA guidance.
The complete algorithm, test and stopping rule must control false positives. Simulation should include the null, secular trends, delayed responses and protocol deviations.
More allocation to a promising arm does not guarantee greater power for the treatment contrast. A highly unbalanced comparison can have a larger standard error than an equal design.
Expected success during the trial is one possible ethical metric, but uncertainty, safety, learning for future patients and access after the trial also matter.
A few favourable early outcomes can push allocation toward one arm, producing more data there and less opportunity for the other arm to recover. Tempering and floors reduce this feedback.
Efficacy and safety may mature at different rates. An efficacy-driven allocation rule should not quietly override independent harm monitoring or contraindications.
If missingness differs by arm or time, the response stream used for adaptation can be biased. The algorithm and analysis need rules for unavailable and late outcomes.
Adding, dropping or graduating arms changes eligibility sets and shared-control comparisons. Calendar-period adjustment and multiplicity become central.
An adaptive tutoring study might send more students to a promising app, but school term, teacher support and test timing can trend. High-stakes allocation needs consent, equity review and stable measurement.
Plot probabilities over time, outcomes by calendar period, covariate balance, effective sample size, control allocation, missingness and the frequency of floors and ceilings.
Reproduce the Beta updates and 59.09% allocation, then change one success at a time. Simulate fixed and adaptive designs with immediate outcomes, delays and time trends.
Response adaptation does not identify the best arm instantly, guarantee participant benefit or remove confounding. It can amplify early noise when design safeguards are weak.
Report the exact algorithm, burn-in, update frequency, floors, ceilings, outcome delay, realised allocations and analysis method, not merely that the trial was adaptive.
Use rows for enrolment order, calendar time, available outcomes, posterior parameters, probability vector, floor or ceiling activation, assigned arm and observed outcome.
Simulate enrolment and outcome queues, not just instantaneous responses. Unit-test probability normalisation, unavailable arms, seeds, boundaries and the final inferential procedure.
Adaptive randomisation appears in clinical trials, online services, manufacturing trials and education experiments where learning and allocation happen together.
Run 2,000 two-arm trials under equal and unequal success rates. Compare power, bias, allocation imbalance, participant successes and sensitivity to a linear time trend.
Posterior probability is not the randomisation probability unless the rule makes it so, more assignment to a winner does not guarantee more power, and adaptive does not mean uncontrolled.
The Type I and Type II error guide frames errors. The alpha-spending guide explains repeated interim decisions.
Students should separate estimation, assignment and final testing as three mathematical layers. Parents can ask how an adaptive school experiment protects students from early noise and unequal opportunity.
- Reproduce both posteriors.
- Calculate the posterior means.
- Add a 30% floor.
- Delay ten outcomes.
- Simulate a time trend.
### Is the better-looking arm always chosen? No; allocation remains random.
### Why use a burn-in? To obtain initial information and limit early volatility.
### Does adaptation increase power? Not necessarily.
### Are delayed outcomes harmless? No.
Report the model and prior, burn-in, update schedule, allocation function, floors and ceilings, delay handling, probability records, covariate and time adjustments, simulations, safety rules and final inference.
A plot of realised allocation probabilities reveals early lock-in, floor activation and calendar changes that a final arm count hides.
Confirm concealed assignment, faithful probability logging, delay-aware simulation, protected contemporaneous control and inference tested under the full algorithm.
Continue through the Mathematics Learning Hub. Recalculate the central worked example, name the assumptions that make it valid and explain which conclusion the result can and cannot support.
Extended practice laboratory
Practice case 1: Set burn-in
Setup: Start from the article’s response-adaptive randomisation 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: Set Beta priors
Setup: Start from the article’s response-adaptive randomisation 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: Update arm A
Setup: Start from the article’s response-adaptive randomisation 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: Update arm B
Setup: Start from the article’s response-adaptive randomisation 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: Calculate posterior means
Setup: Start from the article’s response-adaptive randomisation 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 allocation rule
Setup: Start from the article’s response-adaptive randomisation 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: Calculate allocation probability
Setup: Start from the article’s response-adaptive randomisation 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: Add minimum allocation
Setup: Start from the article’s response-adaptive randomisation 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: Add maximum allocation
Setup: Start from the article’s response-adaptive randomisation 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: Add outcome delay
Setup: Start from the article’s response-adaptive randomisation 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: Add time trend
Setup: Start from the article’s response-adaptive randomisation 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 covariate adjustment
Setup: Start from the article’s response-adaptive randomisation 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: Add control protection
Setup: Start from the article’s response-adaptive randomisation 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: Update after success
Setup: Start from the article’s response-adaptive randomisation 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: Update after failure
Setup: Start from the article’s response-adaptive randomisation 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: Compare fixed randomisation
Setup: Start from the article’s response-adaptive randomisation 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: Estimate treatment effect
Setup: Start from the article’s response-adaptive randomisation 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: Use randomisation probabilities
Setup: Start from the article’s response-adaptive randomisation 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: Check Type I error
Setup: Start from the article’s response-adaptive randomisation 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: Check power
Setup: Start from the article’s response-adaptive randomisation 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: Check patient benefit
Setup: Start from the article’s response-adaptive randomisation 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: Add missing outcomes
Setup: Start from the article’s response-adaptive randomisation 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 concealment
Setup: Start from the article’s response-adaptive randomisation 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 response-adaptive randomisation 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 response-adaptive randomisation 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.