Promising-Zone Sample Size Adaptation

Why a Promising Zone Uses Conditional Power

An interim result can be neither clearly discouraging nor already sufficiently encouraging. A promising-zone rule formalizes this middle ground: it specifies a range of interim conditional-power values within which the trial may increase its planned sample size. The boundaries and the maximum increase are set before the interim data are examined. They are not chosen afterward to make a particular result look favorable.

Conditional power is calculated relative to a specified effect assumption. It answers a conditional planning question: if the interim data are as observed and the assumed effect governs the remaining data, what is the probability of crossing the final efficacy boundary? A promising-zone rule uses this quantity to decide whether, and by how much, to increase the planned information. It does not make conditional power a probability that the treatment works.

The rule distinguishes three quantities that should not be conflated. The lower and upper boundaries decide whether the interim result is in the zone. The target conditional power determines the desired degree of assurance after an increase. The maximum sample size restricts how far the trial can grow. A trial may be in the zone but unable to reach its target before the cap.

Model and Conditional-Power Calculation

For a transparent derivation, consider a one-sided normal test at significance level \(\alpha=0.025\), with final critical value \(c=1.96\). Let \(I_1\) be the interim information, \(I\) the proposed total information, and \(Z_1=z_1\) the interim standardized statistic. Assume \(I>I_1\). Under a fixed assumed treatment effect \(\theta\), represent the accumulated score at information \(I\) as \(S(I)\), with independent increments such that the future score increment over \(\Delta I=I-I_1\) is normally distributed with mean \(\theta\Delta I\) and variance \(\Delta I\). The interim score is \(S(I_1)=z_1\sqrt{I_1}\). This normal-increments model is appropriate, for example, for a normal endpoint with known variance and a fixed analysis plan.

Under these assumptions, the final score is the observed interim score plus the future increment, and the final statistic is that score divided by \(\sqrt I\). Conditional power is the probability that the final statistic exceeds \(c\), given the interim result and the assumed effect:

Here \(\Phi\) is the standard normal cumulative distribution function. The denominator is the standard deviation of the future score increment, not the standard deviation of the final statistic. Equivalently, conditional on the interim result, the final statistic has mean \(\{z_1\sqrt{I_1}+\theta(I-I_1)\}/\sqrt I\) and standard deviation \(\sqrt{(I-I_1)/I}\). Subtracting the critical value and dividing by that conditional standard deviation gives the same formula.

The equation follows directly from the normal distribution of the future increment. Given the interim score \(z_1\sqrt{I_1}\), reaching the final boundary requires the future increment to exceed \(c\sqrt I-z_1\sqrt{I_1}\). Standardizing that increment gives:

This conditional probability depends on the assumed effect as well as the observed interim statistic. A promising-zone policy therefore must state which effect assumption defines its boundaries. In the examples below, the assumed effect is \(\theta=0.20\). The numerical information units are illustrative; when information is proportional to total sample size, the proportional conversion is stated explicitly.

Worked Examples of Zone Classification and Adaptation

Writing the Rule Before the Interim

A reproducible promising-zone rule needs more than two boundaries. Its protocol or statistical analysis plan should specify the conditional-power effect assumption, the interim analysis and its information, \(L\) and \(U\), the target power, the permitted sample-size totals or increments, and the maximum total. It should also say what happens below the zone, above the zone, and when no permitted total up to the cap reaches the target.

One possible policy is to leave the original total unchanged outside the zone. Within the zone, it selects the smallest permitted total that reaches the target, or the maximum if none does. This is only one policy; other prespecified actions are possible. For example, a separate futility rule could apply below the zone. The boundaries and actions should be treated as a single decision rule, not improvised separately after the interim result is known.

Diagnostics, Alternatives, and Limitations

Several checks help identify a faulty calculation or an unsuitable rule. Confirm that \(I>I_1\); at \(I=I_1\), there is no future information and the formula is undefined. Check that the observed statistic, information, assumed effect, and final critical value all come from the planned analysis. Recompute conditional power at the original total before evaluating increases, and inspect the full permitted grid rather than assuming an increase will meet the target. Because the formula can depend nonlinearly on \(I\), do not assume that an unverified interpolation or a rounded intermediate value identifies the first qualifying total.

Conditional power is also sensitive to the effect assumption. A rule using a larger assumed effect can produce higher conditional power and a different sample-size decision than a rule using a smaller, clinically relevant effect. This is why the assumed effect is part of the prespecified rule. An effect assumption is not a claim that the effect is known, and conditional power is not the unconditional probability that the trial will succeed.

A fixed-sample design avoids a sample-size decision based on interim conditional power, but it cannot respond to an interim result that is promising yet short of the desired target. Other re-estimation approaches can use nuisance-parameter estimates to update information requirements rather than using an unblinded treatment-effect statistic to define a promising zone. Each approach answers a different design question and has its own assumptions and error-control considerations. A conditional-power calculation by itself does not establish that the final adaptive test controls type I error; that property must be evaluated for the complete design and analysis procedure.

Common Pitfalls

• Using the observed interim effect as if it were the future effect assumption. The observed statistic is already represented by \(z_1\); the future score in this calculation uses the prespecified \(\theta\). Substituting an unplanned effect assumption changes the rule.
• Calling conditional power the probability that the null hypothesis is false. It is a probability of crossing the final boundary conditional on the interim data and a specified model assumption. It is not a probability about whether a hypothesis is true.
• Confusing zone boundaries with the target. A result can be in the zone without reaching the target at the maximum. State both the classification and the achieved power after adaptation.
• Ignoring the cap or reporting an unattainable target as achieved. If no permitted total reaches the target, apply the prespecified capped action and report the shortfall accurately.