No-Change Regions and No-Increase Rules

Why the Interim Rule Needs Explicit Regions

An unblinded sample-size re-estimation rule does more than return a revised sample size. For some interim results, the trial stops for futility; for others, it continues with the original planned size; and for still others, it continues with an increased size. Writing these outcomes as explicit regions makes the rule auditable and makes its operating consequences calculable.

Here, conditional power is the probability of meeting the final rejection criterion, conditional on the interim data and on a specified assumption about the treatment effect. Write this quantity as \(C\). It is a function of the interim result, not a fixed property of the design. The rule below uses \(C\) to identify regions; it does not imply that every conditional-power rule should increase sample size as \(C\) increases. In a common promising-zone arrangement, very low conditional power leads to futility, intermediate conditional power prompts an increase, and sufficiently high conditional power requires no increase.

The rule must be specified before its interim result is observed. In particular, specify the conditional-power calculation and effect assumption, the cutoffs, the sample-size limits, and what happens at a cutoff. Otherwise, identical interim results could lead to different actions depending on discretionary choices made after unblinding.

Formalizing the Rule and Its Assumptions

For a simple rule that increases to a fixed permitted maximum, define the total sample size selected by the interim result as \(N(C)\). If \(n_I\) participants have already contributed interim data, then a futility stop has total enrollment \(n_I\); a continuation has the stated total sample size. The rule is operational only if \(n_I\leq n_0\leq N_{\max}\), and if the planned increase is feasible for the trial. If the increase is instead computed from a sample-size mapping, the increase region consists of interim results for which that mapping selects a total size greater than \(n_0\). The mapping, rounding, and cap must all be prespecified.

This version assumes one interim decision, a fixed interim enrollment \(n_I\), and one prespecified conditional-power calculation. The statistical model used to calculate \(C\) must be appropriate for the endpoint and trial design, and its assumptions must be stated. For example, a normal-model calculation needs the specified variance and the assumed treatment effect; if these are not known, the procedure must say how they are supplied or estimated. The conditional-power rule also does not, by itself, ensure that the final hypothesis test controls its Type I error. That must be checked for the complete adaptive procedure, including the final test.

The three regions partition the possible values of \(C\): every interim result falls into exactly one of them. Their probabilities depend on the interim-data distribution, and therefore on the true treatment effect and other model parameters. A cutoff alone does not tell us how often a region will be entered.

Deriving Expected Sample Size and Overall Rejection Probability

Let \(R_F\), \(R_I\), and \(R_N\) denote the events that the interim result falls in the futility, increase, and no-change regions, respectively. Write their probabilities under a particular true treatment effect as \(p_F\), \(p_I\), and \(p_N\). Since the regions partition all possible interim results, \(p_F+p_I+p_N=1\). If the trial enrolls \(n_I\) participants before the decision, stops in the futility region, and otherwise reaches its selected total sample size, the law of total expectation gives:

This is an expectation under the specified true effect and interim-data model. It is not generally the arithmetic mean of the possible sample sizes unless the region probabilities happen to give that weighting. If the increase region uses a range of revised sizes rather than one fixed size, its term is instead the expected sample size conditional on entering that region, multiplied by \(p_I\).

A similar decomposition evaluates the probability of final rejection. Let \(q_F\), \(q_I\), and \(q_N\) be the probabilities of final rejection conditional on entering each respective region, under the same true effect. For a futility stop with no rejection, \(q_F=0\). Then:

This follows by conditioning on which region occurs. Under an alternative, this probability is the design's power for that effect. Under \(H_0\), it is the Type I error probability. The region probabilities and conditional rejection probabilities must be calculated for the procedure actually used; treating them as independent, or borrowing values from a different sample-size rule, can give an incorrect answer.

Consequences, Diagnostics, and Comparisons

The regions clarify the trade-offs in the design. A larger futility region may reduce expected enrollment by stopping more often, but can reduce power if trials that would eventually reject are stopped. An increase region can recover power for some interim results, but increases expected sample size when entered and does not necessarily improve power for every treatment effect. A no-change region saves resources relative to increasing for all continuing trials, but leaves the original sample size in place even when the interim evidence is less favorable than the design's target.

A useful diagnostic is to calculate the probabilities of all three regions across a range of plausible true effects, not just at the effect used to define conditional power. Also calculate expected sample size and final rejection probability for each scenario. Check that the region probabilities sum to one, that every result maps to exactly one action, and that the selected total size never exceeds \(N_{\max}\) or falls below the enrollment already observed. If a boundary is left undefined, or if a mapped sample size conflicts with a cap, the algorithm is not fully specified.

A fixed-sample design avoids interim sample-size changes and has no rule-induced region probabilities, but does not adapt enrollment to interim evidence. A futility-only design can stop early but does not increase sample size to recover power. An always-increase design has no no-change region and can be costly. The three-region design combines stopping and selective increase, but its operating characteristics depend on the entire rule and its assumptions, not just on the nominal cutoff values.

• Model diagnostic: check that the conditional-power model, effect assumption, and interim information match the planned analysis. A conditional-power value calculated under an unsuitable model can place results in misleading regions.
• Rule diagnostic: verify equality conventions, the increase mapping, and the maximum-size cap. A vague instruction such as “increase if promising” is not an executable rule.
• Operating-characteristic diagnostic: assess power, expected sample size, and Type I error under relevant true effects. A gain in power under one alternative does not establish acceptable error control elsewhere.

A common mistake is to assume that increasing sample size whenever conditional power is low must improve the design without cost. The correction is to distinguish the three actions and evaluate their probabilities and consequences. In a promising-zone rule, very low conditional power can lead to stopping, intermediate conditional power can trigger an increase, and high conditional power can lead to no increase. Moreover, neither this pattern nor a promising power calculation automatically guarantees that the final test retains its intended significance level.