A Worked Conditional-Power Calculation

Why Calculate Conditional Power From Interim Data?

At an interim analysis, a trial team may ask how likely the trial is to meet its final success criterion if it continues as planned. Conditional power answers this question by conditioning on the data already observed and specifying an effect size for the future-data model. It is not one fixed property of the trial: the answer can change substantially depending on which effect size is assumed.

This tutorial works through a two-stage normal trial with one final efficacy test. It calculates conditional power in two common ways: using the interim effect estimate as the assumed effect, and using a prespecified design effect. The first reflects the interim estimate; the second asks what success probability remains if a chosen effect, often the effect used in the design, is the true effect.

Two-Stage Normal Model and Calculation

Let \(I_1\) be the information at the interim analysis and \(I_2\) the planned total information at the final analysis, with \(0<I_1<I_2\). Let \(\theta\) be the treatment effect, with positive values favoring treatment. Information is measured in inverse squared effect units. In the canonical normal model, the cumulative score \(S_k\) at information \(I_k\) has mean \(\theta I_k\) and variance \(I_k\), and score increments from disjoint information intervals are independent. The standardized statistic is \(Z_k=S_k/\sqrt{I_k}\).

For the calculations below, assume the effect is constant across the trial, the information values are fixed and known, and the normal independent-increments model is adequate. Suppose the final test is one-sided, rejects \(H_0:\theta=0\) in favor of \(H_a:\theta>0\) when \(Z_2>c\), and has significance level \(\alpha=0.025\). Thus \(c=z_{1-\alpha}\), approximately \(1.960\). If the interim result is \(Z_1=z_1\), the conditional power under an assumed effect \(\theta_A\) is calculated from the conditional distribution of \(Z_2\).

The future increment is independent of the observed interim score \(S_1=\sqrt{I_1}z_1\). Dividing the sum by \(\sqrt{I_2}\) gives the conditional mean and variance of the final statistic.

The second argument of \(N\) is the variance. Therefore the conditional standard deviation is \(\sqrt{1-I_1/I_2}\), and the one-sided conditional power is

where \(\Phi\) is the standard normal cumulative distribution function. The numerator is the conditional mean of \(Z_2\) minus the final boundary; the denominator expresses the remaining uncertainty from future data.

Worked Calculations

Why the Two Calculations Differ

The distinction can be seen directly from the assumed effect. The plug-in approach uses \(\hat{\theta}_1=z_1/\sqrt{I_1}\), the effect estimate from the interim data. The design-effect approach uses a fixed \(\theta_D\), chosen before or independently of the interim estimate. For fixed \(z_1\), the conditional mean increases with \(\theta_A\) when \(I_2>I_1\), so conditional power increases with the assumed effect. A larger plug-in result than design-effect result is therefore expected when \(\hat{\theta}_1>\theta_D\), and the reverse is expected when \(\hat{\theta}_1<\theta_D\).

As a check, substituting the plug-in estimate into the conditional mean gives \(\sqrt{I_2/I_1}\,z_1\). This follows because

This simplified expression is useful for checking arithmetic, but it does not remove the underlying assumption that the interim effect estimate is the effect to use for future increments.

Conditional power differs from unconditional power, which averages over possible interim results before the trial begins. It also differs from a p-value: a p-value is calculated from observed data under \(H_0\), whereas conditional power is a future-success probability conditional on the current data and an assumed effect. It is not a probability that \(H_0\) is true, nor is it a guarantee that the final test will succeed.

Assumptions, Diagnostics, and Limitations

Before using the calculation, confirm that the quantities have the intended meaning. Information must be cumulative, so \(I_2>I_1>0\); if \(I_2=I_1\), there is no future information and the formula's conditional variance is zero. The interim statistic and information must correspond to the same analysis and scale. The direction of \(Z_1\), \(\theta_A\), and the rejection boundary must also agree: positive values in these examples favor treatment.

The normal model assumes independent future increments with the stated mean and variance. Dependence between increments, an effect that changes over time, or a substantial departure from the normal approximation can make the calculation unreliable. Information may also differ from the planned amount if recruitment, measurement, or follow-up changes. Replacing \(I_2\) with a revised value is not just an arithmetic update unless the revised design and its assumptions are justified.

A useful diagnostic is to check the conditional standard deviation and the direction of the result. As \(I_1/I_2\) approaches one, little information remains and the conditional variance approaches zero; a result far from the final boundary will then imply a conditional power near zero or one. If \(\theta_A\) is increased while the other inputs remain fixed, conditional power for this positive one-sided test should not decrease. A violation usually signals a sign, boundary, or substitution error.

Plug-in conditional power can be unstable because the interim estimate is noisy, particularly when \(I_1\) is small. Design-effect conditional power avoids using that estimate as the future effect, but its answer depends on whether the chosen effect is plausible. Both are conditional calculations under a fixed effect, not methods for averaging over uncertainty about the effect. If the variance is estimated, information is uncertain, or the future design changes, the simple known-information normal calculation may require modification.

Common Pitfall