Target Conditional Power and Target Final Power

Why Two Power Targets Are Not the Same

At an interim analysis, conditional power asks a path-specific question: given the information observed so far, how likely is rejection at the end if a specified effect is assumed from now on? Final power asks a design-level question: before the trial begins, what is the probability that the complete design will reject under a specified true effect? The first conditions on interim data; the second averages over the interim data the design could produce.

This distinction matters when interim results determine the final sample size. A rule can be designed to attain a conditional-power target for a particular observed interim result, while the trial’s unconditional power remains below that target. Conversely, a design can meet its unconditional power target without attaining a chosen conditional-power target on every possible interim path.

Definition: Conditional power at an interim look is the probability of eventual rejection, conditional on the interim data, under a specified assumption about the effect after that look and under the design’s specified continuation and rejection rule. Unconditional final power under a true effect is the probability of rejection by the end of the trial, averaged over all possible interim data and all resulting design paths.

Model and Assumptions

For a precise comparison, consider a one-sided two-stage test with one interim analysis and a final analysis. Let $I_1$ be the interim information and $I_f$ the final information, with $I_f>I_1$. Let $Z_1$ be the standardized interim statistic. Use the canonical normal model: for an effect parameter $\theta$, the cumulative score has mean $\theta I$ and variance $I$ at information $I$, with independent increments. Thus, conditional on $Z_1=z$, the final standardized statistic has distribution

Z_f\mid Z_1=z \sim N\left( \sqrt{\frac{I_1}{I_f}}z+ \frac{\theta(I_f-I_1)}{\sqrt{I_f}}, \frac{I_f-I_1}{I_f} \right).

The effect $\theta$ is measured on the canonical scale: its contribution to the mean score over an information increment $\Delta I$ is $\theta\Delta I$. The formula assumes the interim statistic and future score increment follow the stated model, that the remaining information is positive, and that the specified information plan is implemented. If the final information $I_f$ is chosen using $Z_1$, use the selected value for that path.

Suppose the final rejection rule on the path is $Z_f>c$, where $c$ is the applicable final boundary. The boundary must be valid for the adaptive design’s type I error rate; one cannot automatically assume that a fixed-sample critical value remains valid after data-dependent sample-size changes. For the calculations below, the relevant boundary is treated as specified and properly calibrated.

Definition: If the continuation effect assumed for conditional power is $\theta^*$, then, for a final-only rejection boundary $c$, $$ CP_{\theta^*}(z;I_f)= \Phi\left( \frac{ \sqrt{I_1/I_f}\,z+\theta^*(I_f-I_1)/\sqrt{I_f}-c }{ \sqrt{(I_f-I_1)/I_f} } \right). $$ Here $\Phi$ is the standard normal cumulative distribution function. If the boundary depends on the interim result or selected information, use that path’s applicable boundary in place of $c$.

The choice of $\theta^*$ is substantive. It might be a prespecified planning effect, or an effect estimate informed by the interim data. Conditional power calculated using an estimated effect is a plug-in forecast, not a guarantee that this effect is true. The model and assumed continuation effect should therefore be stated whenever a conditional-power target guides a decision.

Worked Conditional-Power Calculation

Worked Example: Conditional Power on One Interim Path

Hypothetical design. Suppose $I_1=0.50$, the current interim statistic is $z=1.50$, and the planned final information on this path is $I_f=1.00$. Assume a continuation effect $\theta^*=2.50$ and a valid final boundary $c=1.96$. Calculate the conditional power for final rejection.

The conditional mean of $Z_f$ is $$ \sqrt{0.50/1.00}(1.50)+\frac{2.50(1.00-0.50)}{\sqrt{1.00}} =1.060660+1.250000=2.310660. $$ The conditional variance is $(1.00-0.50)/1.00=0.50$, so the conditional standard deviation is $\sqrt{0.50}=0.707107$.

Therefore, $$ CP_{\theta^*} =\Phi\left(\frac{2.310660-1.96}{0.707107}\right) =\Phi(0.4959) \approx 0.690. $$ Under the stated continuation effect and model, the chance of final rejection, conditional on this interim result, is about 69.0%. This is not the probability that the effect is true, nor is it the prospective power of the entire design.

From Conditional Power to Final Power

The connection between the two quantities follows from conditioning on the interim data. Assume, for clarity, that the design has no early efficacy rejection and every path reaches a final analysis. Let $I_f(z)$ denote the information selected by the adaptation rule after observing $Z_1=z$, and let $R$ denote final rejection. If conditional power is evaluated under the same true effect $\theta$ used to assess the design, then

P_\theta(R) =\int P_\theta(R\mid Z_1=z)\,f_\theta(z)\,dz =E_\theta\left[CP_\theta\{Z_1;I_f(Z_1)\}\right],

where $f_\theta$ is the density of the interim statistic under $\theta$. This is the law of total probability: partition the overall rejection probability by the interim result, calculate the conditional rejection probability on each path, and average those probabilities using how often the paths occur under the true effect.

A conditional-power target usually applies to a specific interim result, a region of interim results, or a rule for choosing information. It does not ordinarily say that all paths have the same conditional power. Even if a particular path attains the target, paths with lower conditional power may occur often enough to reduce the average. When conditional power uses $\theta^*\ne\theta$, the displayed averaging identity still applies to conditional rejection probabilities calculated under $\theta$, but not generally to the plug-in values $CP_{\theta^*}$.

If the design includes early rejection, overall power also includes the probability of early rejection. More generally, calculate rejection probabilities across all mutually exclusive paths, including early stopping and continuation, under the design’s actual rule. The central distinction remains: unconditional power is averaged over paths, not read off from one interim observation.

Worked Design-Level Power Calculation

Worked Example: Choosing Information for a Final-Power Target

Hypothetical fixed design. Suppose a one-sided test uses a valid boundary $c=1.96$, and the prespecified effect of interest is $\theta=0.40$. There is no sample-size adaptation. Find the total information needed for 80% power under the canonical normal model.

At total information $I$, the final statistic has distribution $Z_f\sim N(\theta\sqrt{I},1)$. Thus the power is $P_\theta(Z_f>1.96)=\Phi(\theta\sqrt{I}-1.96)$. Let $z_{0.80}\approx0.8416$. Setting this power equal to 0.80 gives $$ 0.40\sqrt{I}-1.96=0.8416, \qquad I=\left(\frac{1.96+0.8416}{0.40}\right)^2 \approx 49.06. $$

Checking the result, $0.40\sqrt{49.06}-1.96\approx0.8416$, so the power is $\Phi(0.8416)\approx0.800$. This is a prospective design-level target under the stated effect and model. It differs from selecting information to make conditional power equal 80% after observing a particular interim statistic.

Worked Example: Why a Path Target Does Not Set Overall Power

Worked Example: Averaging Across Interim Paths

Hypothetical design evaluation. Suppose an interim rule produces two possible path categories under the true effect. The lower-information category occurs with probability 0.40 and has conditional rejection probability 0.55. The increased-information category occurs with probability 0.60 and has conditional rejection probability 0.80. Assume these probabilities are calculated under the same true effect using a valid rejection rule.

The probability of final rejection is the sum of the path probabilities multiplied by their conditional rejection probabilities: $$ P_\theta(R)=(0.40)(0.55)+(0.60)(0.80) =0.22+0.48=0.70. $$

The second category attains an 80% conditional-power target, but the overall power is 70%, not 80%. The lower-power category contributes 22 percentage points to overall power, while the higher-information category contributes 48 percentage points. A design-level target of 80% would require evaluating and adjusting the full path distribution, not merely checking the conditional power in one category.

How the Targets Inform Design Choices

A conditional-power target is useful as a local decision criterion: it expresses how promising a particular interim path looks under a stated continuation effect and information plan. It can help define or assess a sample-size adaptation rule. A final-power target is a prospective operating characteristic: it is evaluated under a true-effect scenario by averaging over every possible interim result and the action taken in response.

The main alternatives are therefore complementary, not interchangeable. A designer can calculate conditional power to describe interim paths, then evaluate unconditional power for the full adaptation rule. Alternatively, the designer can choose a rule by directly optimizing or meeting unconditional power across specified effect scenarios. A fixed-sample design uses a simpler direct power calculation because its information does not depend on interim data; it does not answer how an adaptive design performs.

Quantity	Conditions on	Typical use	Key limitation
Conditional power	The observed interim data and a stated continuation-effect assumption	Assess or define an interim-path rule	Can be sensitive to the assumed effect and does not average over other paths
Unconditional final power	The specified true effect, before observing interim data	Compare or select complete designs	Requires accounting for the full adaptation and rejection rule
Fixed-design power	A single prespecified information plan	Plan a nonadaptive design under a target effect	Does not characterize data-dependent information choices

Diagnostics, Pitfalls, and Limitations

A power calculation is only as reliable as its model, effect assumption, and decision rule. Before using a target, check that the conditional distribution matches the endpoint and analysis, that the interim and final information are defined consistently, and that the final boundary is calibrated for the adaptive design. If information is selected from interim data, recalculating a fixed-design power formula separately for each selected sample size is not, by itself, an unconditional power calculation.

Confusing the two estimands. The wrong reasoning is: “The observed path has 80% conditional power, so the trial has 80% power.” The correction is to average conditional rejection probabilities over the interim-path distribution under the true effect.
Changing the effect assumption without saying so. A conditional power based on a planned effect and one based on an interim estimate answer different questions. Label the assumption explicitly.
Ignoring the rejection rule. A sample-size change may affect the null distribution of the final statistic. Verify the test’s type I error control rather than presuming the original boundary still applies.
Treating a target as a guarantee. A target is a design criterion conditional on assumptions. Model misspecification, a different true effect, or a different path distribution changes the achieved power.

The canonical normal formulation is an approximation or working model in many applications. Discrete outcomes, estimated nuisance parameters, information loss, or departures from independent increments can affect calculations. In such settings, design-level power may need to be evaluated with an appropriate model-based computation or simulation that reproduces the full adaptation and analysis. Neither a single conditional-power calculation nor a fixed-design formula captures those limitations automatically.

Key takeaway: Conditional power describes the chance of rejection from a specified interim path under a stated continuation assumption. Unconditional final power averages path-specific rejection probabilities over the interim outcomes generated by the design. A conditional-power target is not, by itself, a final-power guarantee.

Check Your Understanding

Answer each question using the distinction between a path-specific probability and a design-level probability.

If conditional power is 75% at the observed interim result, what additional information is needed to determine unconditional final power?
Under what condition can an 80% conditional-power target on every possible path imply unconditional power of at least 80%?
A conditional-power calculation uses a planning effect $\theta^*$, but the design is evaluated under a different true effect $\theta$. Can the plug-in conditional powers simply be averaged to obtain power under $\theta$?
In a two-category design, path probabilities are 0.30 and 0.70 and the corresponding conditional rejection probabilities under the true effect are 0.40 and 0.90. Calculate unconditional power.

Solutions. (1) The path probabilities under the true effect and the conditional rejection probabilities on the other paths are needed, together with any early-rejection probabilities. (2) If the conditional probabilities are computed under the true effect, and the design reaches a final decision on every path, then $P_\theta(R)=E_\theta[CP_\theta]$; if every path has conditional power at least 0.80, their average is at least 0.80. (3) No. Evaluate conditional rejection probabilities under $\theta$, or otherwise calculate the design’s power under $\theta$; plug-in probabilities based on $\theta^*$ target a different assumption. (4) $P_\theta(R)=(0.30)(0.40)+(0.70)(0.90)=0.12+0.63=0.75$, so unconditional power is 75%.