Conditional Power as a Population-Selection Quantity
At an interim analysis, a trial may have evidence about treatment effects in both a broad population and a biomarker-defined population. A population-selection rule can use that interim evidence to decide whether recruitment should remain broad or become restricted to a candidate subgroup. Conditional power quantifies one input to this decision: the chance of crossing the planned final efficacy boundary, conditional on the interim results and on specified assumptions about future data.
The calculation is not a forecast independent of assumptions. It requires a model for the remaining trial, a planned final analysis, and an assumed effect in each candidate population. Prespecifying how the resulting values map to enrollment decisions is essential; the introductory material on prespecifying interim enrichment decision rules establishes that decision-rule principle.
The result depends on the candidate population’s interim statistic, its information available at interim, the information expected by the final analysis, the assumed effect, and the final rejection boundary. These inputs should be defined on the same estimand and analysis scale. For example, a standardized statistic for a biomarker-positive subgroup must be paired with a planning effect and information appropriate to that subgroup, not borrowed uncritically from the all-comer population.
Canonical Normal Calculation
Suppose a one-sided test uses a standardized statistic \(Z\), with success at the final analysis defined as \(Z_F>c\), where \(c\) is the prespecified critical value. Let \(t=I_1/I_F\) be the information fraction at interim, with \(0<t<1\). Here \(I_1\) is information accrued by interim and \(I_F\) is the planned total information. Let \(z_1\) be the observed interim statistic. For a specified planning effect, write \(\theta\) for its standardized drift at the final information level.
Under the canonical normal model, the statistic has independent information increments, and under the assumed effect its mean at information fraction \(u\) is \(\theta\sqrt{u}\). The interim and final statistics are linked by those shared increments. Conditional on \(Z_1=z_1\), the final statistic therefore has distribution:
The second argument of \(N\) is the variance, so the conditional standard deviation is \(\sqrt{1-t}\). The one-sided conditional power is consequently:
where \(\Phi\) is the standard normal cumulative distribution function. The formula combines the interim evidence with the assumed drift in the information yet to be accrued. As \(t\) approaches 1, little information remains and the conditional final statistic is increasingly determined by the interim result. At smaller \(t\), the assumed future effect contributes more to the calculation.
Why the Conditional Formula Holds
The canonical model represents the information-time statistic as \(Z(u)=\theta\sqrt{u}+W(u)/\sqrt{u}\), where \(W(u)\) is Brownian motion with independent increments and variance \(u\). At interim time \(t\), condition on the observed value \(Z(t)=z_1\). Since \(W(t)=\sqrt{t}(z_1-\theta\sqrt{t})\), the Brownian increment from \(t\) to 1 is independent of the observed interim value, has mean zero, and has variance \(1-t\). Thus \(W(1)=W(t)+[W(1)-W(t)]\), and:
The last term is conditionally normal with mean zero and variance \(1-t\). This gives the stated conditional distribution. Taking the probability that this normal variable exceeds \(c\) and standardizing by its standard deviation gives the conditional-power formula.
Worked Example: Broad Enrollment Falls Below Its Threshold
Consider a hypothetical trial with one-sided final boundary \(c=1.960\). At interim, the all-comer statistic is \(z_1=1.20\), and the information fraction is \(t=0.40\). Under the prespecified design alternative for the all-comer estimand, the final standardized drift is \(\theta=2.50\). Suppose the enrollment rule is: continue broad enrollment if all-comer conditional power is at least 0.70; otherwise assess the biomarker-positive option.
The normal approximation and information-increment assumptions are taken as adequate for this hypothetical design; the interim statistic, effect assumption, and information fraction all refer to the all-comer analysis. The conditional mean is:
The standardized distance above the final boundary is:
Therefore \(CP=\Phi(0.3859)\approx0.650\), or about 65.0%. This is below the 70% continuation threshold, so the specified rule does not continue broad enrollment on this criterion. It does not, by itself, imply that the treatment is ineffective; it says that success is not sufficiently likely under this particular conditional calculation and assumed effect.
Applying Thresholds Across Candidate Populations
For each candidate population, calculate conditional power using its own interim statistic, information fraction, effect assumption, and final boundary. Then apply the prespecified rule in its specified order. Thresholds are decision cutoffs chosen for the design; they are not universal constants. A rule might first retain broad enrollment if its conditional power is adequate, then enrich if the subgroup conditional power meets its own threshold, and otherwise stop for futility. The order matters because it defines how competing options are handled.
Worked Example: Enriching After the Broad Option Misses Its Cutoff
Continue the hypothetical setting above. Suppose the prespecified rule next considers biomarker-positive enrollment, with an enrichment threshold of 0.60. For this candidate population, assume \(t=0.40\), \(z_1=1.50\), and planning drift \(\theta=3.00\), with the same final one-sided boundary \(c=1.960\). Assume the subgroup-specific statistic and effect are appropriately modeled and that the remaining subgroup information matches the stated information fraction.
The conditional mean and standard deviation are:
Thus:
The subgroup conditional power is about 84.6%, above its 60% threshold. Under the stated ordered rule, the design selects biomarker-positive enrollment: broad enrollment missed its 70% cutoff, while the enriched option met its 60% cutoff. This decision is conditional on both assumed effects and the prespecified enrollment rule; it is not evidence that the subgroup effect has an 84.6% probability of being beneficial.
Worked Example: A Low Subgroup Conditional Power
In another hypothetical trial, a candidate subgroup has interim statistic \(z_1=0.50\), information fraction \(t=0.50\), and planning drift \(\theta=2.00\). The final one-sided boundary is \(c=1.960\). Suppose the rule requires conditional power of at least 0.60 to enrich. Assume the normal model and information fraction are valid for this subgroup.
The conditional mean is \(\sqrt{0.50}(0.50)+0.50(2.00)=0.3536+1.000=1.3536\), and the conditional standard deviation is \(\sqrt{0.50}=0.7071\). Hence:
The subgroup has approximately 19.6% conditional power under the specified alternative, below the 60% threshold. The rule therefore does not enrich on this criterion. A low value can result from weak interim evidence, a modest assumed future effect, or both; it is not a posterior probability that the subgroup is ineffective.
Alternative Boundaries and Practical Checks
If final success requires a two-sided test, success is \(|Z_F|>c\), not simply \(Z_F>c\). Given conditional mean \(m=\sqrt{t}z_1+(1-t)\theta\) and standard deviation \(s=\sqrt{1-t}\), the corresponding conditional power is:
The two tail probabilities must both be included, even if the lower tail is negligible under a positive assumed effect. The boundary must match the actual final test; using a one-sided boundary for a two-sided test will misstate power.
Before applying a population-selection rule, verify the following:
- Population and estimand: the interim statistic and planning effect describe the same candidate population and treatment contrast.
- Information: the information fraction reflects the actual analysis scale and the planned final information; unequal accrual or changing subgroup availability can make a simple fraction misleading.
- Boundary: the critical value matches the planned final test, including its sidedness and any design-specific boundary.
- Assumed effect: the effect used to project future evidence is prespecified or otherwise justified, not chosen after inspecting results to obtain a desired decision.
- Selection rule: thresholds, order of evaluation, and actions for all possible outcomes are specified before the interim results are examined.
When candidate populations overlap, their statistics are generally correlated. The conditional-power calculation for each candidate can still be made separately if its marginal model is appropriate, but the values should not be treated as independent evidence. Moreover, choosing among populations based on interim data raises inferential questions about final claims and error rates. A conditional-power threshold does not itself solve those questions; the final analysis and its error control must follow the trial’s prespecified design.
Comparison, Diagnostics, and Limitations
Conditional power differs from unconditional design power. Unconditional power averages over possible interim outcomes as well as future outcomes under a specified effect. Conditional power instead fixes the observed interim result and asks about the remaining trial. It also differs from a simple interim \(p\)-value: the \(p\)-value measures extremeness of the interim data under a null hypothesis, while conditional power combines those data with a specified future-effect assumption and the final boundary.
A useful diagnostic is sensitivity to the assumed effect. If a small change in \(\theta\) moves conditional power across an enrichment threshold, the action is fragile and should be recognized as such during design and operational review. Also check that information is still available to accrue in the selected population. A subgroup estimate with very little information may be unstable, and a planned future sample that cannot realistically be recruited makes the calculated conditional power operationally implausible.
The method has limitations. Normal approximations can be poor with sparse outcomes or very small subgroups; information can be misestimated when follow-up, prevalence, or measurement availability differs from projections; and the assumed effect may not represent future effects. If the final boundary changes with the adaptation or if the analysis is not well described by canonical independent increments, use a calculation derived for that design. Conditional power is a transparent decision aid, not a substitute for checking the statistical and operational model.
Common Mistake
A frequent error is to say, “The subgroup has 85% conditional power, so there is an 85% chance that treatment works in the subgroup.” That interpretation is wrong. The 85% is the chance of crossing a specified final boundary, conditional on the interim data and under the assumed effect and model. It does not give the probability that the effect is true. A second error is to use one candidate population’s interim statistic or planning effect for another population. Recalculate with population-matched inputs, then follow the prespecified rule.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In the one-sided formula, what does \(\theta\) represent, and why is it not estimated simply by relabeling the interim statistic?
- If the broad-population conditional power is below its threshold but the enriched-population conditional power exceeds its threshold, what determines the enrollment action?
- For a two-sided final test, why is the upper-tail one-sided formula insufficient?
- Explain why a conditional power of 0.80 is not an 80% probability that the treatment effect is real.
- Name two checks that could reveal a conditional-power calculation is not suitable for a candidate population.
Solutions: (1) \(\theta\) is the assumed standardized treatment-effect drift at the planned final information level. It specifies the future-effect scenario; the observed interim statistic is data, not that planning assumption. (2) The prespecified rule, including its thresholds and order of evaluation, determines the action; conditional-power values alone do not dictate a universal choice. (3) A two-sided test succeeds in either tail, so power is the sum of the lower- and upper-tail probabilities beyond their respective boundaries. (4) It is the modeled chance of crossing the final success boundary conditional on the interim data and assumed future effect, not a probability assigned to the truth of the effect. (5) Examples include mismatched population and estimand, an implausible information fraction, a poor normal approximation for sparse data, and an assumed effect that is unjustified or inconsistent with the planned analysis.