Conditional Power at an Interim Analysis

What Conditional Power Answers

At an interim analysis, the data already observed are fixed for the purpose of a conditional calculation. Conditional power asks: given those data, and given a stated model for the observations still to come, what is the probability of meeting the trial’s prespecified success criterion? Its value depends both on the interim evidence and on the assumption made about future outcomes.

That second dependence is essential. The observed interim effect describes the data collected so far; it does not determine what effect will occur in future data. A calculation may assume that the future effect is zero, equals the interim estimate, or equals a design value. These are different conditional-power calculations, even when the interim data and success boundary are identical.

Definition: Let $\mathcal{F}_t$ denote the information available at an interim analysis, and let $A$ be the prespecified event that constitutes final success. Conditional power under a specified model for future data is $$ \operatorname{CP}(t)=P(A\mid \mathcal{F}_t). $$ The probability is conditional on the observed interim data and is evaluated using the stated assumptions for data not yet observed. Those assumptions must be reported because they are part of the calculation.

Conditional power is not a probability that $H_0$ is true, nor is it a p-value. A p-value summarizes how incompatible the observed data are with a null model under a specified test. Conditional power instead concerns the probability of a future success event, conditional on what has already been observed and on a model for future data.

A One-Sided Normal Score Model

For a transparent calculation, consider a trial with a cumulative score $U(I)$ at information level $I$. Use the following working model: under an effect $\delta$, the score has mean $\delta I$ and variance $I$, and score increments over nonoverlapping information intervals are independent. Thus, for a remaining information increment $\Delta I$, the future score increment has distribution $N(\delta_f\Delta I,\Delta I)$, where $\delta_f$ is the effect assumed for future data. This model requires a valid information scale and the normal, independent-increment approximation; the examples below assume both are appropriate.

Suppose the final success event is crossing a fixed, one-sided score boundary $c$: $A=\{U(I_T)\ge c\}$, where $I_T$ is planned final information. At the interim, information is $I_t$, the observed score is $U_t$, and $\Delta I=I_T-I_t>0$. The observed score $U_t$ is held fixed in the conditional calculation. Only the future increment is modeled.

U(I_T)=U_t+\{U(I_T)-U_t\},\qquad U(I_T)-U_t\sim N(\delta_f\Delta I,\Delta I).

The final criterion is met if the future increment is at least $c-U_t$. Standardizing that increment gives the conditional-power expression.

\operatorname{CP}_{\delta_f}(t) =P\{U(I_T)\ge c\mid \mathcal{F}_t\} =1-\Phi\left(\frac{c-U_t-\delta_f\Delta I}{\sqrt{\Delta I}}\right).

Here $\Phi$ is the standard normal cumulative distribution function. The formula is appropriate only when the final event really is the stated one-sided boundary crossing and the future increment model applies. If the success criterion or information plan changes, the event and its probability model must be reconsidered rather than carried over mechanically.

Worked Conditional-Power Calculations

Worked Example: Zero Future Effect Versus a Positive Future Effect

Suppose a hypothetical trial has interim information $I_t=40$, planned final information $I_T=100$, and an observed standardized interim score $U_t/\sqrt{I_t}=1.00$. In this score model, $U_t=I_t\hat{\delta}_t$, so $\hat{\delta}_t=U_t/I_t=1/\sqrt{40}$ and $U_t=\sqrt{40}=6.3246$. The one-sided final boundary is $c=1.96\sqrt{100}=19.6$, and the remaining information is $\Delta I=100-40=60$. Assume the normal independent-increment model is suitable and that this boundary is the trial’s stated final success criterion.

If the future effect is assumed to be zero, then the remaining score has mean $0(60)=0$ and standard deviation $\sqrt{60}=7.7460$. Therefore,

\operatorname{CP}_{0}(t) =1-\Phi\left(\frac{19.6-\sqrt{40}}{\sqrt{60}}\right) =1-\Phi(1.7139) \approx 0.0433.

Under the zero-future-effect assumption, the modeled chance of crossing the final boundary is about 4.33%. If instead the future effect is assumed to be $\delta_f=0.20$, the mean future score is $0.20(60)=12$, with the same standard deviation:

\operatorname{CP}_{0.20}(t) =1-\Phi\left(\frac{19.6-\sqrt{40}-12}{\sqrt{60}}\right) =1-\Phi(0.1647) \approx 0.4346.

Under this positive future-effect assumption, conditional power is about 43.46%. The interim score has not changed between these calculations; only the assumed future effect has changed.

Worked Example: An Interim Score Above Zero With No Future Effect

In another hypothetical trial, let $I_t=60$, $I_T=100$, and let the observed interim score be $U_t=1.50\sqrt{60}=11.619$. The final boundary remains $c=19.6$, and the remaining information is $40$. Assume that the same normal independent-increment model applies, that the trial continues to the planned final look, and that the future effect is zero. The interim score is evidence already accumulated; zero here describes only the assumed mean of the future increment.

The remaining score has mean zero and standard deviation $\sqrt{40}=6.3246$. Substitution gives

\operatorname{CP}_{0}(t) =1-\Phi\left(\frac{19.6-11.619}{\sqrt{40}}\right) =1-\Phi(1.262) \approx 0.1035.

Conditional on the observed interim score and the zero-future-effect assumption, the modeled chance of crossing the final boundary is about 10.35%. This differs from the previous example’s 4.33% because the interim score and the remaining information differ.

Worked Example: Plugging in the Interim Effect for Future Data

Use the interim data from the preceding example: $I_t=60$, $U_t=11.619$, and $\Delta I=40$. The interim estimate on the score-model effect scale is $\hat{\delta}_t=U_t/I_t=11.619/60=0.19365$. Suppose, as an explicit modeling choice, that the effect in future data will equal this interim estimate. Retain the same normal independent-increment model and final boundary $c=19.6$.

Under this plug-in assumption, the future score increment has mean $0.19365(40)=7.746$ and standard deviation $\sqrt{40}=6.3246$. Therefore,

\operatorname{CP}_{\hat{\delta}_t}(t) =1-\Phi\left(\frac{19.6-11.619-0.19365(40)}{\sqrt{40}}\right) =1-\Phi(0.0372) \approx 0.4852.

Under this plug-in assumption, the modeled chance of final success is about 48.52%. It is not a data-determined probability: it relies on treating the interim estimate as the future effect. In particular, the calculation does not account for uncertainty in that estimate by averaging over possible effects.

Worked Example: A Prespecified Design Effect for Future Data

Consider a hypothetical trial with $I_t=60$, $I_T=100$, final boundary $c=19.6$, and observed interim score $U_t=0.50\sqrt{60}=3.8730$. Assume a future effect of $\delta_f=0.25$, chosen as the design effect for this calculation. The normal independent-increment model and the fixed final criterion are assumed to apply.

The remaining information is $40$, so the assumed future mean score is $0.25(40)=10$, and its standard deviation is $6.3246$. Thus,

\operatorname{CP}_{0.25}(t) =1-\Phi\left(\frac{19.6-0.50\sqrt{60}-0.25(40)}{\sqrt{40}}\right) =1-\Phi(0.9055) \approx 0.1826.

The modeled chance of meeting the final boundary is about 18.26% under the specified design-effect assumption. The result is lower than in the plug-in example because both the interim score and the assumed future effect differ.

Why the Conditioning and Assumption Matter

A short justification of the formula makes its interpretation clear. Given the interim information, $U_t$ is already observed. Final success requires $U_t+\{U(I_T)-U_t\}\ge c$, or equivalently that the future increment be at least $c-U_t$. Under the stated model, subtracting its mean $\delta_f\Delta I$ and dividing by its standard deviation $\sqrt{\Delta I}$ converts this requirement to a standard normal tail probability. This yields the conditional-power expression above; no probability distribution is assigned to the already observed score in this calculation.

Unconditional power answers a different question: before observing interim data, what is the probability of success under a specified effect and design? Conditional power updates that question by fixing the interim data that actually occurred. A plug-in conditional-power calculation uses the observed interim estimate as the future effect, whereas a design-effect calculation uses a prespecified value. Neither choice is automatically correct in every setting; the choice should match the purpose of the analysis and be stated.

A predictive probability is also conceptually different: it averages over uncertainty about the future effect rather than conditioning on one fixed assumed effect. The conditional-power calculations here do not perform that averaging. Similarly, a high conditional power does not establish that the assumed future effect is true, and a low value does not change the prespecified significance level or success criterion.

Diagnostics, Failure Modes, and Limitations

Before interpreting a reported value, check that the calculation corresponds to the trial as actually planned and analyzed. The following questions are useful:

Is the success event stated exactly? The examples use one fixed final one-sided boundary. If success instead involves several possible looks, a two-sided rule, or another decision criterion, the event in the probability must reflect that rule.
Are the information quantities coherent? The remaining information must be positive and consistent with the final plan: $\Delta I=I_T-I_t>0$. An incorrect information scale or a mismatched boundary can produce a plausible-looking but irrelevant probability.
Is the future-effect assumption explicit? Report whether it is zero, a design value, the interim estimate, or another value. A large change in conditional power across reasonable assumptions is a warning that the result is assumption-sensitive.
Does the increment model fit the design and data? Dependence between increments, a poor normal approximation, or an unmodeled change to the sampling plan can invalidate the calculation. Reassess the model if information is estimated unreliably or the endpoint has important discreteness.
Was the trial stopped or the rule changed? If the trial cannot continue as assumed, or the success criterion or planned information has changed, the original calculation no longer answers the relevant question. Recompute for the actual decision event and future-data model.

Conditional power is a model-based forecast, not a guarantee of success and not a replacement for prespecifying monitoring rules. Its main limitation is dependence on assumptions about future effects and the distribution of future data. It is most informative when those assumptions are transparent and the calculation is presented as a sensitivity analysis when more than one future-effect assumption is scientifically plausible.

A Common Mistake

Common mistake: “The interim estimate is $0.20$, so the future effect must be $0.20$; therefore the conditional power is the one calculated using $\delta_f=0.20$.”

Correction: The interim estimate is observed evidence about the effect, not a known value for future observations. A calculation using $\delta_f=0.20$ is a plug-in scenario only if that assumption is deliberately chosen. State it, and consider how the result changes under other defensible future-effect assumptions.

Check Your Understanding

Answer each question using the definition and assumptions above. Brief solutions follow the questions.

In one sentence, distinguish the observed interim score from the assumed future effect in conditional power.
With $U_t=8$, $c=20$, $\Delta I=36$, and assumed future effect $\delta_f=0$, write the conditional-power calculation.
Why is a conditional-power calculation using the interim estimate as $\delta_f$ not the same as a probability that the true effect equals that estimate?
What happens to the conditional-power question if the planned final success boundary is changed after the interim analysis?
Name one reason why the normal independent-increment calculation might be unreliable.

Solutions. (1) The interim score is fixed observed information, whereas $\delta_f$ specifies the assumed mean effect for unobserved future data. (2) $\operatorname{CP}_0(t)=1-\Phi((20-8)/\sqrt{36})=1-\Phi(2)\approx0.0228$. (3) Plugging in an estimate fixes a future-effect assumption; it neither assigns a probability distribution to the true effect nor averages over its uncertainty. (4) The event $A$ has changed, so conditional power must be recalculated using the revised criterion if that is now the relevant decision event. (5) For example, dependent score increments would violate the stated independent-increment assumption.