Conditional-Power Rules Within Error-Controlled Designs

Conditional Power Guides the Adaptation; It Does Not Calibrate the Test

A conditional-power rule asks how likely the trial is to cross a specified success boundary, given the data observed so far and an assumed effect for planning. That calculation can help decide how much information to collect in stage two. It does not, by itself, determine a valid rejection rule. In an error-controlled adaptive design, the sample-size choice and the Type I error calibration have distinct jobs: conditional power guides the former, while a combination test or a conditional-error procedure governs the latter.

The distinction matters because a sample size selected after seeing interim data changes the sampling plan. Applying an ordinary fixed-design cutoff to a final statistic as though the sample size had been fixed in advance need not control the probability of a false positive. Here we use a one-sided normal model to make both calculations explicit. The same logic is useful more broadly, but the formulas below rely on the stated normal independent-increments assumptions.

Model and assumptions: Let \(\theta\) be the treatment effect on a scale where the accumulated information is \(I\). At the interim analysis, information is \(I_1\) and the observed stage-one statistic is \(Z_1=z_1\). For a chosen stage-two information \(I_2>0\), the stage-two statistic \(Z_2\), conditional on the first-stage data and the selected \(I_2\), has distribution \(N(\theta\sqrt{I_2},1)\). Under the null hypothesis \(H_0:\theta=0\), this is \(N(0,1)\). We assume a prespecified one-sided test, independent stagewise increments, a correctly specified normal model, and a stage-two test that remains valid conditional on the first-stage data and the sample-size choice. The planning effect \(\theta^*\) used for conditional power is an assumption, not an estimate known to be true.

For a canonical cumulative statistic, the final statistic at total information \(I_1+I_2\) is

Z_{\mathrm{final}} = \frac{\sqrt{I_1}Z_1+\sqrt{I_2}Z_2}{\sqrt{I_1+I_2}}.

Suppose the reference design's success boundary is \(Z_{\mathrm{final}}>c\). Conditional power is the probability of crossing that boundary, given \(Z_1=z_1\), if the true effect were the planning value \(\theta^*\). It follows by solving the boundary inequality for \(Z_2\) that

\operatorname{CP}(I_2;\theta^*,z_1) = 1-\Phi\!\left( \frac{c\sqrt{I_1+I_2}-\sqrt{I_1}z_1-\theta^* I_2} {\sqrt{I_2}} \right).

This is a planning probability under the specified effect and model. It is not the probability that \(H_a\) is true, and it is not the probability of Type I error. In an adaptive design, the cutoff used to control Type I error must still be justified after the information choice.

Worked Example: Conditional Power at a Chosen Information Level

Suppose that, after the interim, the relevant stage-two rejection rule is \(Z_2>1.570\). The planning effect is \(\theta^*=0.25\), and the proposed stage-two information is \(I_2=16\). Calculate the conditional power for this stage-two rule.

Under the model, \(Z_2\) has mean \(\theta^*\sqrt{I_2}=0.25(4)=1\) and standard deviation 1. Therefore,

\operatorname{CP} = P_{\theta^*}(Z_2>1.570) = 1-\Phi(1.570-1) = 1-\Phi(0.570) = \Phi(-0.570) \approx 0.2843.

Thus, under the assumed effect of 0.25, the probability of passing this stage-two boundary is about 28.4% at information 16. The calculation uses the stated normal model and the same boundary as the stage-two rule; it does not establish that the rule controls Type I error.

Turning a Conditional-Power Target Into an Information Rule

A simple rule chooses the smallest stage-two information that reaches a specified conditional-power target, subject to operational limits. For a fixed stage-two boundary \(b\), the conditional-power expression is \(1-\Phi(b-\theta^*\sqrt{I_2})\). Because \(\Phi\) is increasing, reaching target \(\gamma\) is equivalent to

1-\Phi(b-\theta^*\sqrt{I_2})\geq\gamma \quad\Longleftrightarrow\quad \theta^*\sqrt{I_2}\geq b+\Phi^{-1}(\gamma), \qquad \theta^*>0.

If the right-hand side is positive, this gives \(I_2\geq[(b+\Phi^{-1}(\gamma))/\theta^*]^2\). A lower bound, an upper cap, and any rounding rule should be stated before the interim. If the required information exceeds the cap, the target is unattainable under that rule; it is not appropriate to report the capped value as if it achieved the target.

Worked Example: Solving for Information to Reach a Target

Continue with the boundary \(b=1.570\) and planning effect \(\theta^*=0.25\). Find the information needed for conditional power of at least 80%, and check the earlier proposal \(I_2=16\).

For \(\gamma=0.80\), \(\Phi^{-1}(0.80)\approx0.8416\). The information condition is

I_2 \geq \left(\frac{1.570+0.8416}{0.25}\right)^2 = \left(\frac{2.4116}{0.25}\right)^2 = 9.6464^2 \approx93.05.

If information must be an integer number of units, choose at least 94 units. At \(I_2=16\), the conditional power was about 0.2843, well below 0.80. The target calculation is consistent with the formula: more information raises the assumed-effect mean \(\theta^*\sqrt{I_2}\), making the fixed boundary more likely to be crossed. This rule presumes that the boundary \(1.570\) is the applicable calibrated stage-two boundary; the power calculation does not supply that calibration.

Embedding the Rule in a Conditional-Error Procedure

A conditional-error procedure starts with an original design and calculates its conditional Type I error after the interim data. Write \(A(z_1)\) for the probability, under \(H_0\), that the original design would reject, conditional on the observed first-stage data. For a one-sided stage-two statistic with a conditionally standard normal null distribution, a replacement procedure can preserve that conditional error by rejecting when

P_{H_0}(Z_2>c_2\mid\text{first-stage data})=A(z_1), \qquad c_2=\Phi^{-1}\!\left(1-A(z_1)\right).

The sample-size rule may choose \(I_2\) using the first-stage data and a planning effect, while the stage-two rejection probability under the null remains \(A(z_1)\). This preservation requires the conditional validity assumption in the model box: in particular, the stage-two p-value must remain valid after the choice. The overall error control follows by averaging the conditional rejection probability over the interim data, as the next justification shows.

Justification: Let \(R\) denote rejection by the adaptive procedure. If, for every interim outcome, \(P_{H_0}(R\mid\text{first-stage data})=A(z_1)\), then the law of total probability gives \(P_{H_0}(R)=E_{H_0}\{A(Z_1)\}\). By definition, \(A(Z_1)\) is the original design's conditional rejection probability, so this expectation is the original design's Type I error, at most \(\alpha\). The sample-size choice can affect power, but it has not increased Type I error under these conditions.

Worked Example: Reusing Conditional Error After Increasing Information

An original one-sided normal design has \(I_1=25\), planned stage-two information 25, and rejects if its final statistic exceeds 1.96. The interim statistic is \(z_1=1.00\). Calculate the original conditional error and state the stage-two rule if the new information is chosen as \(I_2=36\).

Under the original design, rejection requires

\frac{5(1.00)+5Z_2}{\sqrt{50}}>1.96 \quad\Longleftrightarrow\quad Z_2> \frac{1.96\sqrt{50}-5}{5} \approx1.772.

Thus \(A(1.00)=1-\Phi(1.772)\approx0.0382\). To preserve this conditional error with the adapted design, the stage-two test must have conditional null rejection probability 0.0382. Since \(Z_2\sim N(0,1)\) under \(H_0\), use \(Z_2>1.772\), equivalently a stage-two one-sided p-value at most 0.0382. This is not a new ordinary final-statistic cutoff; it is the stage-two rule calibrated to the original conditional error.

For illustration, under \(\theta^*=0.25\) and \(I_2=36\), the stage-two mean is \(0.25\sqrt{36}=1.5\), so the conditional power for this rule is \(1-\Phi(1.772-1.5)=1-\Phi(0.272)\approx0.3929\). This power calculation concerns the planning effect; it does not change the conditional null rejection probability of 0.0382.

Embedding the Rule in an Inverse-Normal Combination Test

An alternative is to combine stagewise one-sided p-values. Define \(Z_j=\Phi^{-1}(1-p_j)\) for stage \(j\), and choose weights \(w_1,w_2\) in advance with \(w_1^2+w_2^2=1\). The inverse-normal combination statistic is \(Z_{\mathrm{comb}}=w_1Z_1+w_2Z_2\), with rejection for \(Z_{\mathrm{comb}}>z_{1-\alpha}\). The weights are fixed rather than recalculated from the selected information.

Under \(H_0\), \(Z_1\) is standard normal and, conditional on the stage-one data and the selected information, \(Z_2\) is standard normal and independent of stage one. Consequently \(Z_2\) has its required null distribution despite the information choice. With fixed weights, the combination statistic is standard normal under the null, so the stated cutoff controls the one-sided Type I error. This reasoning depends on valid stagewise p-values and the specified independence; the power-based sample-size rule is not a substitute for either condition.

Worked Example: Testing After a Conditional-Power-Based Adaptation

Suppose the prespecified information rule selects \(I_2=94\) after the interim, based on a conditional-power calculation. The design uses fixed weights \(w_1=0.5\) and \(w_2=\sqrt{0.75}\approx0.8660\), and the observed stagewise statistics are \(Z_1=1.20\) and \(Z_2=2.00\). Test at one-sided \(\alpha=0.025\).

The combined statistic is

Z_{\mathrm{comb}} = 0.5(1.20)+0.8660(2.00) = 2.332.

The critical value is \(z_{0.975}\approx1.960\). Since \(2.332>1.960\), reject \(H_0\) using the combination test; the corresponding one-sided p-value is approximately \(1-\Phi(2.332)=0.0099\). This conclusion follows from the prespecified combination rule, not from applying a fixed-design cutoff to a pooled statistic. Under the stated assumptions, selection of \(I_2\) using first-stage data does not invalidate the stage-two null p-value.

Diagnostics, Alternatives, and Limitations

Before implementing a conditional-power rule, check that the effect scale and information used in its formula match the stagewise test, that the planning effect and target were specified clearly, and that the chosen information respects the protocol's minimum and maximum. Recalculate the conditional power using the actual boundary that the error-control procedure will use. If the boundary changes but the calculation still uses an obsolete cutoff, the reported conditional power does not describe the implemented design.

The two error-control approaches have different practical emphases. A conditional-error procedure preserves the original design's conditional rejection probability and can express the new rule directly as a stage-two p-value threshold. A combination test specifies how stagewise evidence is combined, often with fixed weights. Both can accommodate a sample-size choice based on interim data under their validity conditions. Neither makes the conditional-power target an error-control guarantee.

Common failure modes include a stage-two test whose null distribution depends on the interim data in an unaccounted-for way, changing weights based on the selected sample size without a valid calibration, and silently changing the second-stage test. These situations break assumptions used in the arguments above and require an appropriate recalibration. A capped sample size may also fail to reach the target conditional power. Finally, conditional power is sensitive to \(\theta^*\): a value chosen optimistically can produce a reassuring number that is not representative of power under smaller effects. Report the assumed effect, information constraints, and resulting probability, rather than presenting conditional power as a guarantee of success.

Common mistake: “The conditional-power rule chose the new sample size, so I can use the usual fixed-design final cutoff.” The correction is that conditional power only guides the information choice. Control Type I error with a valid combination test or by preserving the original conditional error; do not assume an ordinary fixed-design cutoff remains valid after adaptation.

Check Your Understanding

Answer each question, then compare with the solutions below.

With stage-two boundary 1.50, planning effect 0.20, and information \(I_2=25\), what is the conditional power?
For boundary 1.50 and planning effect 0.20, what information is needed for conditional power of at least 80%?
Why does a conditional-power target not itself control Type I error?
If an original design's conditional error is \(A(z_1)=0.04\), what stage-two one-sided null rejection probability preserves it?
Name two conditions needed for the fixed-weight combination-test argument after information re-estimation.

Solutions. (1) The stage-two mean is \(0.20\sqrt{25}=1\), so power is \(1-\Phi(1.50-1)=1-\Phi(0.50)=\Phi(-0.50)\approx0.3085\). (2) The required information is \([(1.50+0.8416)/0.20]^2=(11.708)^2\approx137.1\), so at least 138 integer information units are needed. (3) It is a probability of crossing a boundary under a specified planning effect, not a null rejection probability; Type I error requires null calibration. (4) The stage-two null rejection probability must be 0.04, giving cutoff \(Z_2>\Phi^{-1}(0.96)\approx1.751\). (5) For example, weights are fixed in advance and stage two supplies a valid null p-value conditional on the interim data and selected information; the stagewise independence assumptions must also hold.