Boundary-Aware Conditional-Power Calculations

Why the Final Boundary Matters

A conditional-power calculation asks how likely a future analysis is to reject \(H_0\), given the interim data and an assumption about the effect in the data yet to be collected. In a group sequential design, rejection at the final look is determined by that design’s final efficacy boundary. Substituting a familiar single-look critical value, such as the one-sided normal quantile for \(\alpha=0.025\), can give the wrong answer: the final boundary is calibrated as part of the group sequential design and can depend on the look schedule and the design’s error-spending rule.

This distinction matters directly in sample size re-estimation. Changing the planned final information can change the information fractions and the calibrated boundary. The revised information must therefore be assessed using the final boundary for the revised design, not automatically the boundary from the original design.

Model and Conditional-Power Formula

Use the canonical normal-score model. Let \(J_1\) denote the information at the interim look and \(D\) the additional information planned after that look, so the proposed final information is \(J_1+D\). Let \(S_1\) be the observed interim score on the scale where a score increment from information \(D\), under assumed effect \(\delta\), has distribution \(N(\delta D,D)\). Thus, conditional on the interim data, the future score increment \(S_F\) satisfies

The final score is \(S_1+S_F\), and the final standardized statistic is \((S_1+S_F)/\sqrt{J_1+D}\). Suppose the design rejects at the final look when this statistic exceeds the calibrated upper efficacy boundary \(c_K(J_1+D)\), where \(K\) labels the final look. The boundary is expressed on the standardized-statistic scale. Equivalently, the score must exceed \(c_K(J_1+D)\sqrt{J_1+D}\).

Assume that the interim data are observed without error for this conditional calculation, future score increments are independent of the observed interim score under the canonical model, and the future variance is correctly represented by the planned information. The assumed effect \(\delta\) is the effect used to project the future score; it need not equal the effect observed at the interim look. We calculate the probability of crossing the final efficacy boundary, conditional on continuing to the final look. If the design has a separate stopping rule, this formula alone does not include the probability of stopping before that look.

Derivation

For fixed interim score \(S_1\), the final score has conditional mean \(S_1+\delta D\) and conditional variance \(D\). Standardizing the score relative to its conditional standard deviation gives the following result.

where \(\Phi\) is the standard normal cumulative distribution function. To see this directly, subtract the conditional mean from the score and divide by \(\sqrt D\). The resulting random variable is standard normal, so the event of crossing the boundary has probability \(\Phi\) evaluated at the conditional mean’s standardized distance above the score-scale boundary.

The boundary appears multiplied by \(\sqrt{J_1+D}\), because it is a cutoff for the final standardized statistic, while the future conditional variability is \(\sqrt D\). These are different quantities. Using \(\sqrt D\) to scale the efficacy boundary would incorrectly treat the final statistic as though it were based only on future information.

How Boundary Recalibration Changes the Revision

A common planning shortcut is to solve for \(D\) while holding the original final boundary fixed. That can provide a rough candidate, but it does not necessarily solve the actual re-estimation problem: after the proposed final information changes, the design may have a different calibrated final boundary. The proper calculation uses the candidate plan’s boundary and checks whether the resulting conditional power meets the target.

For the hypothetical interim data above, take a target conditional power of \(80\%\), whose standard normal quantile is approximately \(0.8416\). If, only as a preliminary calculation, the final standardized boundary is held at \(2.10\), the target equation is

At \(D=183\), the left side minus the right side is

At \(D=184\), the corresponding difference is approximately \(0.025\). These signs locate a rough crossing near \(D=184\) for the fixed-boundary calculation. They do not establish that \(D=184\) is sufficient under a newly calibrated group sequential design. The boundary must be updated for the proposed design, and the target must then be checked again.

Using the Formula in Practice

The calculation is usually iterative because \(c_K(J_1+D)\) can change with the proposed final information. For each candidate \(D\), obtain the final boundary for the design that would use that information, calculate conditional power, and compare it with the target. If the target is missed, evaluate a larger candidate; if it is exceeded, determine whether a smaller candidate also suffices. For discrete sample sizes, check the adjacent feasible values rather than relying only on a continuous approximation.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to calculate the score-scale boundary and conditional mean separately before calculating the normal probability. This catches a frequent scale error: the boundary must be multiplied by \(\sqrt{J_1+D}\), while the conditional standard deviation in the denominator is \(\sqrt D\). Also verify that the supplied \(c_K\) is the final boundary for the proposed design and that its sidedness and efficacy direction match the test. A boundary for a different number of looks, timing schedule, or spending rule is not interchangeable merely because it is also a final-look cutoff.

The principal alternative is to use a single-look critical value, often the one-sided normal quantile associated with \(\alpha\). That is appropriate only if it is in fact the final boundary of the design being evaluated. Otherwise it ignores the group sequential calibration. Another shortcut is to retain the original design’s final boundary after changing the final information. This avoids recalibration but answers a different question and can misstate whether the conditional-power target is attained.

Boundary-aware conditional power is still conditional on the chosen effect assumption. A favorable interim score does not guarantee that the assumed effect will describe future data, and a change in the assumed effect can materially change the result. The calculation also relies on the canonical normal model and the specified future variance. If those assumptions are poor, the numerical conditional power may be misleading. It is a planning quantity, not by itself a guarantee of unconditional power or a substitute for checking the operating characteristics of the complete design.

Common mistake: “The target is \(80\%\), so I can use the ordinary one-sided \(2.5\%\) cutoff and solve once for the revised sample size.” The correction is to use the final efficacy boundary calibrated for each candidate group sequential design, recompute conditional power when that boundary changes, and verify the selected candidate against the target.