Adaptive sample-size re-estimation
Simulate a two-stage trial that may increase its second-stage sample size using interim conditional power.
Concepts and reasoning
Learning during a trial without changing the question
An adaptive sample-size design prespecifies how interim information may change later enrollment. Its aim can be to preserve a useful chance of success when planning assumptions were uncertain. Adaptation is not permission to keep adding participants until a conventional test becomes significant.
Blinded re-estimation may update a nuisance parameter such as variance without examining comparative efficacy. Unblinded re-estimation uses treatment-effect information and raises different statistical and operational issues. The valid procedure depends on exactly what is observed, how the rule uses it, and how the final test is constructed.
Conditional power is the probability of eventual rejection given the interim data and an assumed effect for future observations. It is not a posterior probability that the treatment works. It can be calculated using the originally planned effect, the observed interim effect, or another specified value; these choices can give very different recommendations.
Why the interim estimate is uncertain
Early effects are noisy. A rule that treats the interim estimate as the truth may request a large increase because of random fluctuation, or terminate a useful treatment prematurely. A maximum sample-size cap limits resources but means the requested conditional-power target may remain unattainable.
A design therefore needs operating-characteristic evaluation across scenarios, not just an appealing interim rule. The FDA adaptive-design guidance describes the role of prespecified adaptation rules and simulation. Maintaining trial integrity also requires controlling who sees interim results.
Mathematical foundations
Combining independent stages
Suppose each stage provides a valid one-sided p value, p₁ and p₂, with independent stage evidence under the null. A fixed-weight inverse-normal combination statistic is
The positive weights are specified before seeing outcomes. Under the required null conditions, the combination is standard normal, so rejection can use the prespecified one-sided cutoff. Stage-two sample size may depend on stage-one data if the stage-two test remains conditionally valid under the adaptation.
For stage Z statistics, the formula becomes w₁Z₁ + w₂Z₂. Given Z₁ = z₁ and future Z₂ distributed as N(δ√I₂, 1), conditional power is
The assumed future effect δ enters explicitly. Changing it changes CP even with identical observed interim data. Recalculating the weights after seeing a favorable or unfavorable interim result generally changes the test and cannot be justified by this fixed-weight argument.
Worked example
Worked example: what the second stage must achieve
Suppose the original plan gives equal information to two stages, so w₁ = w₂ = √0.5 ≈ 0.7071. Use a one-sided alpha of 0.025, with a normal cutoff near 1.96.
At interim, Z₁ = 1.0. The second stage must satisfy
If the assumed stage-two Z mean is 2, conditional power is 1 − Φ(1.772 − 2) ≈ 0.590. If stage-two information doubles with the same effect and variance, that mean grows by √2 to 2.828, and CP becomes approximately 0.855.
These are conditional calculations under the assumed future effect, not unconditional power for the full adaptive design. The latter averages over all possible interim results and their consequent adaptations.
Suppose stage-one enrollment was 100 and stage two grows from 100 to 200. The original combination weights remain 0.7071 each. Using weights based on the realized 100:200 ratio after the adaptation creates a different procedure. It might be possible to design another valid method, but it requires its own justification.
A cap at only 120 stage-two participants would allow much less extra information. The design should record the chance of reaching that cap and the power achieved there rather than reporting the desired CP as if it had been attained.
Interpreting the results
Conditional power depends on the assumed future effect
Conditional power is the chance of eventual rejection given interim data and an assumption about future observations. It is not the posterior probability that treatment works. If the future effect is estimated from noisy interim data, the resulting recommended sample size can vary greatly between otherwise similar trials.
Adaptive sample-size re-estimation is a complete policy: the interim rule, minimum and maximum second-stage sizes, futility rule, and final test jointly determine performance. A higher conditional-power target does not guarantee the same unconditional power, because some trials stop and others reach the cap.
Expected total N summarizes resource use across simulated paths. Increase and cap probabilities explain how often the design changes and how often its requested information cannot be obtained. Read these alongside null rejection and alternative success rather than judging the design by average sample size alone.
Assumptions and limitations
Preserve the validity of the final test
Using an observed effect to enlarge a study changes the sampling procedure. A final test must account for the adaptation. In a fixed-weight inverse-normal combination design, independent stage statistics are combined using weights specified from the original plan; recomputing weights after observing the selected sample size generally changes the test.
The simple design assumes normally distributed known-variance outcomes, equal allocation, and independent stage information. Participants or outcomes reused between stage statistics violate that independence. Recruitment and outcome delays can also complicate the practical stage boundary.
The cap, futility rule, target, and effect-estimation rule should be prespecified. Adaptation cannot solve uninformative endpoints, bias, or unrealistic recruitment. Operational safeguards are needed so knowledge of interim results does not alter trial conduct in ways the simulation omits.
Diagnostics and sensitivity analysis
Explore the entire adaptation rule
Evaluate null rejection, power, expected enrollment, futility, increases, and cap use over a range of true effects. High cap use suggests that the desired conditional power often exceeds feasible recruitment. Compare the adaptive policy with its original fixed design on both inferential and operational outcomes.
Examine sensitivity to a noisy or optimistic interim effect estimate. A small first stage can produce volatile adaptations; a large first stage leaves less flexibility. Increasing the cap changes the policy and must be reevaluated under the null as well as the alternative.
Use enough simulation replications to distinguish real performance differences from Monte Carlo variation. Retain the original combination weights and report the actual rule, not only its preferred scenario. FDA's adaptive-design guidance discusses simulation-based evaluation and prespecification.
References and other material
Read the methodological references and supporting tutorials below. The examples in this lesson are original teaching calculations, not findings from these sources.
- FDA: adaptive designs for clinical trials of drugs and biologics — Planning adaptations and evaluating operating characteristics through simulation.