Power At Each Event-Driven Look

What Power at One Look Measures

In an event-driven group sequential trial, a logrank statistic is evaluated when specified amounts of event information have accumulated. At a particular look, its marginal power is the probability, under a specified alternative, that the statistic exceeds the efficacy boundary assigned to that look. This is a single-look probability: it does not account for whether the trial would already have stopped at an earlier look.

That distinction matters. A statistic can exceed the boundary at look \(k\) in the probability model even on paths that would have crossed an earlier boundary, so the marginal power at look \(k\) is not the probability that look \(k\) is the first crossing. Calculating power for the full stopping rule requires the joint distribution across looks. Here we calculate the probability for one specified look.

The calculation uses the canonical normal approximation for sequential logrank statistics and a proportional-hazards alternative. It is most useful when the assumed treatment effect, the accumulated information, and the efficacy boundary are all stated on compatible scales.

Model and Per-Look Power

The information \(I_k\) is the information for the log hazard-ratio contrast used by the standardized statistic; it is not necessarily the event count itself. The event count may be used to plan or approximate information, but the power calculation uses \(I_k\). In this parameterization, the alternative's drift, or mean of \(Z_k\), is \(\theta\sqrt{I_k}\). The variance of the standardized statistic is one.

For this approximation to be appropriate, the logrank statistic must be oriented consistently with the definition of \(\theta\), and the proportional-hazards alternative treats the hazard ratio as constant over follow-up. The usual large-sample logrank approximation is also assumed, along with suitable censoring and risk-set behavior for the logrank score and its information estimate. The canonical model treats the information at the look as specified (or uses a planned/realized information value as an approximation). If the effect changes over time, or information is substantially different from what the model assumes, the calculation may not describe the actual distribution well.

The standardization in the second line is valid because, under the assumed alternative, subtracting the mean \(\theta\sqrt{I_k}\) gives a standard normal variable. The last expression is often convenient: calculate the alternative drift, subtract the efficacy boundary, and evaluate the standard normal distribution function.

Worked Per-Look Calculations

Why the Formula Works—and What It Does Not Say

The calculation follows directly from the assumed distribution. If \(Z_k\sim N(\theta\sqrt{I_k},1)\), then \(Z_k-\theta\sqrt{I_k}\sim N(0,1)\). The event \(Z_k\ge c_k\) is equivalent to \(Z_k-\theta\sqrt{I_k}\ge c_k-\theta\sqrt{I_k}\). By the definition of the standard normal distribution function, the probability of being at least \(a\) is \(1-\Phi(a)\). This yields the stated power formula without needing a separate approximation for the tail.

The interpretation is conditional on the model inputs: the assumed effect, information, boundary, and normal approximation. It is not a probability that the hazard ratio is true, and it does not guarantee that the trial will reach that look. If an earlier efficacy boundary is crossed, the actual design stops; the marginal calculation at a later look still describes the distributional event at that look, not the operational chance of observing it after continuing the trial.

Diagnostics, Alternatives, and Limitations

Several checks help catch errors before interpreting the result:

• Check the direction. With the stated convention, a beneficial treatment has \(\theta>0\) and raises the mean of \(Z_k\). Reversing the log hazard-ratio contrast without reversing the statistic changes the interpretation and can produce a nonsensical power calculation.
• Check the scale. The drift uses \(\theta\sqrt{I_k}\), not \(\theta I_k\). A larger information value increases the drift for a fixed positive effect, but it does not change the variance of the standardized \(Z_k\) from one.
• Check the boundary and tail. An upper efficacy boundary uses \(P(Z_k\ge c_k)\). A lower or two-sided rule requires a different event definition; do not substitute an absolute-value rule for a one-sided benefit boundary.
• Check the resulting probability. It must lie between zero and one. With a positive upper boundary and no effect (\(\theta=0\)), the formula gives the one-look null exceedance probability \(1-\Phi(c_k)\). For a fixed boundary, increasing a beneficial effect or information should increase the calculated power.

A single-look normal approximation is simpler than evaluating the joint sequential distribution, but it cannot answer how likely the design is to cross at least one boundary or to cross first at a particular look. Those are overall sequential power questions. A simulation based on the intended event process and monitoring rule can be useful when the normal approximation, constant hazard ratio, or planned information values are questionable. Neither method removes the need to specify a realistic treatment effect and appropriate censoring and event assumptions.

A frequent mistake is to calculate \(P(Z_k\ge c_k)\) and call it the probability that the trial rejects by look \(k\). That reasoning ignores earlier looks: a path may have crossed an earlier boundary, and a later look's statistic is correlated with earlier statistics. The correction is to call the single-look result marginal power at look \(k\). To obtain rejection probability under the complete sequential design, one must account for the joint distribution and stopping rule, not add the per-look probabilities.