Conditional Power Under a Prespecified Effect

Why Condition on Interim Results?

At an interim analysis, the trial has accumulated evidence, but the final efficacy result is not yet known. A useful planning quantity is the probability that the planned final analysis will cross its efficacy boundary, conditional on the interim data already observed. This probability depends on what effect is assumed for the observations still to come and on how much information the planned remaining sample will provide.

This tutorial uses a prespecified effect: a treatment effect chosen for planning, rather than an effect estimated from the interim treatment comparison. Conditional power is therefore a scenario-based calculation. It answers, “If the prespecified effect governs the future data, how likely is this design to meet its final efficacy criterion, given what has been observed so far?”

Definition: Conditional power is the probability, under a specified model and a specified treatment effect, that the final test statistic crosses its final efficacy boundary, conditional on the interim data observed and on the planned remaining sample size or information. In this tutorial, the final test is one-sided, with efficacy declared when \(Z_2 \ge c_2\), where \(c_2\) is the final upper boundary.

Model and Calculation

For a concrete formula, suppose the test statistic follows a canonical normal information model with independent increments. Let \(I_1\) be the information available at the interim analysis and \(I_2\) the planned total information at the final analysis, where \(0<I_1<I_2\). The observed interim statistic is \(Z_1=z_1\), and \(t=I_1/I_2\) is the information fraction. Let \(\theta\) be the prespecified standardized treatment effect per square root of information. Under this model, the score accumulated over an information increment \(I\) has mean \(\theta I\) and variance \(I\).

The independent-increments assumption means that, conditional on the interim data, the future score increment has mean \(\theta(I_2-I_1)\), variance \(I_2-I_1\), and is independent of the interim score. The variance is treated as known or adequately represented by the model. The design, information timing, effect scale, and final boundary must be specified consistently. The calculation also assumes the planned future information is actually obtained and that the final analysis uses the stated test and boundary.

Z_2 \mid Z_1=z_1 \sim N\left( \sqrt{t}\,z_1+\theta\sqrt{I_2}(1-t),\;1-t \right)

The second argument of \(N\) is the variance. Thus the conditional standard deviation is \(\sqrt{1-t}\). Since success is the event \(Z_2\ge c_2\), standardizing this conditional normal distribution gives:

\operatorname{CP} = P(Z_2\ge c_2\mid Z_1=z_1;\theta,I_2) = 1-\Phi\left( \frac{c_2-\sqrt{t}\,z_1-\theta\sqrt{I_2}(1-t)} {\sqrt{1-t}} \right)

Here \(\Phi\) is the cumulative distribution function of a standard normal random variable. The prespecified effect contributes only through the future increment: the observed interim score is already fixed by conditioning on \(Z_1=z_1\). The final boundary \(c_2\) must be the boundary appropriate to the planned design, including any group sequential monitoring that affects the final test.

Justification From the Future Score Increment

The formula follows by writing the final score as the interim score plus the future increment. At the interim analysis, the score is \(Z_1\sqrt{I_1}=z_1\sqrt{I_1}\). Under the specified effect, the future increment is normal with mean \(\theta(I_2-I_1)\) and variance \(I_2-I_1\). Dividing their sum by \(\sqrt{I_2}\) gives the conditional distribution of \(Z_2\):

Z_2 = \frac{z_1\sqrt{I_1}+\text{future score increment}}{\sqrt{I_2}}, \qquad E(Z_2\mid Z_1=z_1) = \sqrt{t}\,z_1+\theta\sqrt{I_2}(1-t)

The conditional variance is \((I_2-I_1)/I_2=1-t\). To obtain the probability of crossing the boundary, subtract the conditional mean from both sides of \(Z_2\ge c_2\) and divide by \(\sqrt{1-t}\), which is positive because \(I_1<I_2\). The resulting standardized threshold is the argument of \(\Phi\) in the conditional-power formula. This also shows why an interim statistic alone is not enough: the result depends on the assumed effect, planned final information, information fraction, and final boundary.

Worked Example: A Prespecified Effect With Modest Interim Evidence

Suppose a hypothetical trial has \(I_1=100\) and planned \(I_2=200\), so \(t=0.5\). The interim statistic is \(z_1=0.50\), the one-sided final boundary is \(c_2=1.96\), and the prespecified effect is such that \(\theta\sqrt{I_2}=1.8428\). Calculate conditional power.

First calculate the conditional mean and standard deviation of the final statistic:

\mu_{2\mid1} = \sqrt{0.5}(0.50)+1.8428(1-0.5) = 0.3536+0.9214 = 1.2750, \qquad SD(Z_2\mid Z_1)=\sqrt{0.5}=0.7071

The standardized distance from the conditional mean to the boundary is:

\frac{1.96-1.2750}{0.7071}=0.9688

Therefore,

\operatorname{CP} = 1-\Phi(0.9688) \approx 1-0.8337 = 0.1663

Under the stated effect and planned information, the chance of crossing the final boundary, conditional on \(Z_1=0.50\), is about 16.6%. This is not the probability that the treatment effect is true, nor is it a probability statement about the null hypothesis.

Worked Example: Stronger Interim Evidence

Consider another hypothetical calculation with information fraction \(t=0.60\), interim statistic \(z_1=1.80\), prespecified future-effect scale \(\theta\sqrt{I_2}=1.00\), and final upper boundary \(c_2=2.00\). The planned remaining information is \(0.40I_2\), so the future contribution to the conditional mean is \(1.00(0.40)=0.40\).

The conditional mean and standard deviation are:

\mu_{2\mid1} = \sqrt{0.60}(1.80)+0.40 = 1.3943+0.4000 = 1.7943, \qquad SD(Z_2\mid Z_1)=\sqrt{0.40}=0.6325

The boundary is \((2.00-1.7943)/0.6325=0.3253\) conditional standard deviations above the mean. Hence:

\operatorname{CP} = 1-\Phi(0.3253) \approx 1-0.6275 = 0.3725

The conditional probability of final efficacy is about 37.3%. Although the interim statistic is relatively high, it does not make success certain: the final boundary remains above the conditional mean, and future sampling variability remains substantial.

Worked Example: How Planned Information Changes the Calculation

Suppose a hypothetical trial has interim information \(I_1=100\), observed statistic \(z_1=1.20\), prespecified effect \(\theta=0.10\), and final boundary \(c_2=1.96\). Compare two possible planned final information levels, \(I_2=200\) and \(I_2=300\). For each calculation, assume the stated final boundary is the correct boundary for that plan.

With \(I_2=200\), \(t=100/200=0.5\) and \(\theta\sqrt{I_2}=0.10\sqrt{200}=1.4142\). Thus:

\mu_{2\mid1} = \sqrt{0.5}(1.20)+1.4142(0.5) = 0.8485+0.7071 = 1.5556, \qquad SD=\sqrt{0.5}=0.7071

The standardized distance is \((1.96-1.5556)/0.7071=0.5719\), giving:

\operatorname{CP}_{I_2=200} = 1-\Phi(0.5719) \approx 0.2837

With \(I_2=300\), \(t=100/300=1/3\) and \(\theta\sqrt{I_2}=0.10\sqrt{300}=1.7321\). Then:

\mu_{2\mid1} = \sqrt{\frac{1}{3}}(1.20)+1.7321\left(\frac{2}{3}\right) = 0.6928+1.1547 = 1.8475, \qquad SD=\sqrt{\frac{2}{3}}=0.8165

The standardized distance is \((1.96-1.8475)/0.8165=0.1378\), so:

\operatorname{CP}_{I_2=300} = 1-\Phi(0.1378) \approx 0.4452

For these assumptions, planning more information raises conditional power from about 28.4% to 44.5%. The comparison is specific to the assumed effect and model; it does not establish that increasing sample size is automatically appropriate or that the final boundary can be left unchanged in every design.

Interpretation, Diagnostics, and Alternatives

Conditional power is a conditional operating characteristic. It describes future success probabilities across hypothetical future data sets, holding the observed interim information fixed and assuming the stated effect and design. It is not the probability of success conditional only on a treatment-effect hypothesis being true. It is also not unconditional power, which averages success over possible interim outcomes as well as future outcomes.

The result is sensitive to the effect specified for future data. A larger assumed beneficial effect generally increases the conditional mean and conditional power; a smaller effect may produce low conditional power even when the interim statistic is favorable. Because the effect here is prespecified rather than estimated from interim treatment differences, the calculation does not substitute an interim effect estimate into the future model. That distinction is central when interpreting the number.

Check the following before relying on a result:

Effect scale: Confirm that \(\theta\) is expressed on the model’s standardized information scale and that \(\theta\sqrt{I_2}\) is calculated for the planned total information.
Information and timing: Verify \(0<I_1<I_2\), use the planned final information, and ensure that the interim statistic and information fraction refer to the same analysis.
Boundary: Use the final efficacy boundary for the actual monitoring design. A nominal one-sided cutoff cannot replace a group sequential boundary when earlier efficacy looks affect the design.
Model fit: Check whether the normal approximation, variance assumptions, and independent-increments representation are reasonable for the outcome and analysis.
Adaptation: If the future sample size, analysis method, or boundary changes in response to interim data, the calculation must represent that actual rule. A calculation using a fixed \(I_2\) and \(c_2\) does not automatically describe an adaptive design.

A common mistake is to treat a high conditional power as proof that the treatment works, or a low value as evidence that it does not. The correction is to keep the conditioning and assumption explicit: conditional power is the probability of crossing a specified final boundary, given the interim data, if the prespecified effect and model describe the future data. It does not establish the effect’s truth. Another mistake is to report a number without stating the planned information or boundary; because both directly enter the calculation, such a number is not reproducible or interpretable.

Conditional power differs from predictive power, which averages over uncertainty about the treatment effect rather than fixing it at one prespecified value. It also differs from using the interim effect estimate as the assumed future effect. Those alternatives answer different questions and require different assumptions. The prespecified-effect calculation is transparent and useful for evaluating a defined planning scenario, but it can be misleading if the chosen effect is implausible, the normal model is poor, or the design changes without recalibration.

Key takeaway: Under the canonical normal model, conditional power is the upper-tail probability for the final statistic’s conditional distribution. Always report the interim statistic, prespecified effect, planned final information, and final boundary alongside the result.

Check Your Understanding

Answer each question using the definitions and model above. The solutions follow the questions.

In the normal information model, what does conditioning on \(Z_1=z_1\) hold fixed?
For \(t=0.5\), \(z_1=1.0\), \(\theta\sqrt{I_2}=1.0\), and \(c_2=2.0\), calculate the conditional mean, standard deviation, and conditional power.
Why is conditional power not the probability that \(H_0\) is false?
Name two design or model checks needed before interpreting a calculated conditional power.
How does a prespecified-effect calculation differ from one that uses the interim treatment-effect estimate as the assumed future effect?

Solutions. (1) It holds the observed interim data, represented here by \(z_1\), fixed while future increments remain random. (2) The mean is \(\sqrt{0.5}(1.0)+1.0(0.5)=1.2071\), and the standard deviation is \(\sqrt{0.5}=0.7071\). The standardized distance is \((2.0-1.2071)/0.7071=1.1213\); therefore conditional power is \(1-\Phi(1.1213)\approx 0.1311\). (3) It is a probability of a future data event under a specified effect and model, not a probability assigned to a hypothesis. (4) Examples include checking that the boundary matches the monitoring design, that the planned information and interim timing are consistent, and that the normal independent-increments model is reasonable. (5) The prespecified-effect calculation fixes the future effect at a planning value; the estimate-based version uses the interim estimate as its assumed future effect and therefore answers a different, data-dependent question.

Conditional Power Under a Prespecified Effect

What You'll Learn