Conditional-Power Criteria for Arm Continuation

Why Calculate Conditional Power?

At an interim analysis, an arm may have an encouraging score but little chance of meeting the trial’s final criterion, or a modest score with substantial information still to come. Conditional power quantifies the first question directly: given the interim data, how likely is this arm to meet its specified final criterion if its future outcomes follow a stated effect assumption? The answer can be used in a prespecified rule for continuing or dropping that arm.

Conditional power is not a probability that the treatment effect is true, and it is not determined by the interim score alone. It depends on the interim data, the amount of information remaining, an assumed future effect, and the final decision boundary. In particular, the assumed effect must be made explicit: different plausible effects can lead to different continuation decisions from the same interim data.

Model and Calculation

Consider one experimental arm and its control comparison, oriented so that larger values favor the experimental arm. Let \(I_1\) denote information available at the interim analysis and \(I_2\) the planned final information, with \(I_2>I_1\). Let \(Z_1\) be the interim standardized score, and write the corresponding unstandardized score as \(S_1=Z_1\sqrt{I_1}\). The parameter \(\delta\) is the assumed treatment effect used to describe future data on the score’s information scale.

For the calculation below, assume a normal independent-increments model: conditional on the interim data, the future score increment is normally distributed with mean \(\delta(I_2-I_1)\) and variance \(I_2-I_1\), and is independent of the interim score. This is the canonical large-sample model when information is correctly specified and the variance structure is known or adequately estimated. It is an approximation when those conditions hold only approximately.

Suppose the final criterion for this arm is a one-sided boundary \(Z_2\ge c\), where \(Z_2=S_2/\sqrt{I_2}\) is the final standardized score. Here \(c\) is the boundary the trial has specified for the final decision; it must reflect the actual trial design rather than being chosen automatically as a conventional single-test cutoff.

The final criterion \(Z_2\ge c\) is equivalent to \(S_2\ge c\sqrt{I_2}\). Conditional on the interim data, only the future increment is random. Standardizing that increment gives the conditional-power formula.

Here \(\Phi\) is the standard normal cumulative distribution function. The interim score contributes \(Z_1\sqrt{I_1}\); the assumed future effect contributes \(\delta(I_2-I_1)\); and the denominator is the standard deviation of the future score increment. For fixed information and boundary, a higher assumed benefit increases conditional power; the effect of increasing remaining information depends on the other inputs and should be assessed from the formula for the specified design.

Turning Conditional Power Into a Continuation Rule

A continuation rule needs both a calculation and a decision threshold. Before the interim analysis, specify the final criterion \(c\), the effect assumption \(\delta\), the planned final information \(I_2\), and the conditional-power threshold \(q\). For each eligible arm, calculate its conditional power using its own interim score and information. Retain the arm when \(\operatorname{CP}(\delta)\ge q\), and drop it otherwise.

The threshold \(q\) is a design choice, not a universal constant. A larger value requires stronger evidence that the arm can succeed under the specified future-effect scenario, so it tends to drop more arms. A smaller value tends to retain more. A rule based on conditional power does not, by itself, guarantee a particular familywise error rate or overall operating characteristic; those properties depend on the complete trial design and its final testing procedure.

Justification, Checks, and Limitations

The calculation follows directly from the future-increment distribution. Given the observed \(S_1\), the final score is the fixed interim contribution plus a normal random increment. Subtract the interim contribution from both sides of the final success inequality, then standardize the increment by its standard deviation. That yields the \(\Phi\) expression above. If no information remains, \(I_2=I_1\), this formula is not used because the increment has zero variance; the final score is already determined by the interim score.

Before applying the rule, verify that \(I_2>I_1\), the information values match the planned continuation, the score direction and assumed-effect direction agree, and the boundary \(c\) is the final criterion actually intended for the arm. The normal increment approximation should also be credible for the outcome and analysis. If variance is uncertain, the score is poorly approximated by a normal distribution, or the future sample size may change, a simple plug-in calculation can be misleading; use a model and operating-characteristic evaluation appropriate to those features.

A useful diagnostic is to recompute conditional power across a defensible range of future effects. If a small change in \(\delta\) flips the retain/drop decision, the rule is sensitive to its effect assumption and that fragility should be visible in design evaluation. Also check arithmetic by noting that, for fixed information and boundary, the formula increases with \(Z_1\) and \(\delta\): the coefficient of each is positive. A result that moves in the opposite direction signals a sign, boundary, or substitution error.

Conditional power differs from several common alternatives. A confidence-bound rule compares an interim estimate or interval with a chosen threshold; it need not calculate the chance of final success under future data. A rule based on an interim score alone ranks observed evidence without incorporating remaining information or a future-effect assumption. Conditional power makes both explicit, but that additional structure does not make the rule universally superior. It can be sensitive to the assumed effect, depends on a suitable probability model, and cannot replace evaluation of the full design’s error rates and selection behavior.

A disciplined implementation records the assumed effect, final boundary, information plan, and threshold before making decisions. Then calculate conditional power arm by arm, apply the same stated rule consistently (or prespecified arm-specific thresholds if the design calls for them), and document which assumption drove each decision. This makes the rule interpretable and exposes the assumptions on which retention depends.