Conditional Power for Weighted Logrank Statistics

Why Conditional Power Depends on the Future Effect Assumption

At an interim analysis, the weighted logrank score summarizes the evidence accrued so far. Conditional power asks a forward-looking question: given that observed score, how likely is the final analysis to cross its efficacy boundary? The answer cannot be determined from the interim statistic alone. It also depends on how much information remains and what future treatment effect is assumed.

This dependence matters when effects are delayed. If the treatment has little or no effect early but a benefit later, carrying the interim average effect forward can give a misleading prediction. For a weighted logrank statistic, the assumed future effect must be translated into expected future score drift using the chosen weights and the anticipated event and risk-set patterns.

The calculations below use a benefit-oriented score: positive values favor treatment. A different sign convention is possible, but the score, assumed drift, and efficacy boundary must all use the same orientation. The introductory material on independent increments and information time supplies the monitoring framework used here; the focus is the conditional calculation under a specified future effect model.

Definition and Working Model

Let \(U_1\) be the observed weighted logrank score at the interim analysis, \(I_1\) its information, and \(I_F\) the planned information at the final analysis. Write \(I_2=I_F-I_1\) for the planned future information. Under the independent-increments normal approximation, the future score increment has variance approximately \(I_2\). Conditional on the interim data, represent its mean under the assumed future effect pattern by \(\mu_2\):

Here \(N(\mu_2,I_2)\) uses \(I_2\) as a variance. For a one-sided final efficacy boundary \(Z_F\ge c\), where \(Z_F=U_F/\sqrt{I_F}\), the equivalent score criterion is \(U_F\ge c\sqrt{I_F}\). Thus:

The approximation assumes that the future score increment is approximately normal, has the stated mean and variance, and is independent of the observed increment conditional on the design and interim history. It also assumes the weighted logrank weights and the final boundary are specified appropriately. Information must remain: \(I_2>0\). If the score is oriented so that negative values favor treatment, reverse the signs consistently rather than applying this formula unchanged.

Why the Formula Has This Form

Conditional on the interim data, \(U_1\) is fixed. Under the working model, the final score is therefore approximately normal with mean \(U_1+\mu_2\) and variance \(I_2\). Standardizing the final score threshold gives the formula directly:

The last equality uses the symmetry of the standard normal distribution. The numerator compares the expected final score under the assumed future model with the required final score; the denominator measures the standard deviation of the future increment. A larger assumed positive drift raises conditional power, while a higher boundary or less favorable interim score lowers it.

Worked Example: Conditional Power From an Interim Score

Turning a Delayed-Effect Assumption Into Future Drift

For a weighted logrank score, expected future drift is determined by the assumed hazards, weights, and risk sets over the remaining follow-up. One useful approximation divides future follow-up into intervals. Let \(Y_T\) and \(Y_C\) be the treatment and control numbers at risk, \(\lambda_T\) and \(\lambda_C\) their event hazards, and \(w(t)\) the prespecified weight. For a benefit-oriented score, the approximate expected score contribution in a short interval of duration \(dt\) is:

A treatment benefit, \(\lambda_T<\lambda_C\), gives positive expected drift under this convention. The approximate information contribution is:

where \(\bar{\lambda}(t)\) is the event hazard used for the pooled event-rate approximation. In practice, the planned future mean and information may be obtained by integrating a specified survival model or by design simulation. These expressions make the essential point: a delayed effect changes the expected score contributions in the future intervals where the hazards differ, while the weight function changes how strongly those contributions count. The mean drift \(\mu_2\) is not generally obtained by multiplying the interim score by a simple time ratio.

Diagnostics, Alternatives, and Limitations

A conditional-power calculation is only as credible as its future information and effect projections. Before interpreting it, check that the score sign and efficacy boundary agree; the remaining information is positive; the information scale is appropriate for the chosen weights; and the assumed treatment effect is expressed over the future follow-up rather than copied mechanically from the observed average. For a delayed-effect setting, inspect the predicted event and risk-set patterns around the assumed onset. If the timing of onset is uncertain, calculate conditional power under several plausible onset assumptions rather than presenting one value as definitive.

The displayed formula gives the chance of crossing a specified final boundary. It does not, by itself, give the probability of crossing one or more efficacy boundaries at several future looks. When future monitoring includes additional looks, the future increments are jointly dependent through their cumulative scores; the probability of crossing any boundary requires the full sequential boundary and joint distribution, often evaluated numerically or by simulation. A final-boundary calculation also does not account for stopping for futility unless that stopping rule is incorporated into the probability calculation.

Conditional power differs from unconditional power: unconditional power averages over the distribution of interim as well as future data under a design alternative, whereas conditional power fixes the observed interim data and models only the future. It also differs from a predictive probability that averages over uncertainty in the future effect model. A single assumed drift is useful for a transparent scenario calculation, but it does not represent uncertainty about when a delayed effect begins or how large it is.

The normal approximation may be poor with sparse events, highly irregular information accrual, or unstable risk sets. Estimated future information may also be wrong if event rates, censoring, or enrollment differ from projections. Finally, weights chosen to emphasize a particular effect pattern can make conditional power especially sensitive to misspecification of that pattern. These limitations do not make the calculation useless; they clarify that it is a conditional forecast, not a guaranteed operating characteristic.