Conditional Power Based on Intermediate Outcomes

Why Condition on Intermediate Evidence?

In a multi-arm, multi-stage trial, an intermediate outcome can be available before enough events have accrued for the final time-to-event analysis. Investigators may use that evidence to assess the chance that a treatment arm will eventually meet its final event-endpoint criterion. Conditional power expresses that chance under a specified statistical model and a specified assumption about the treatment effect. It is a planning or monitoring quantity, not a conclusion that the treatment works.

The intermediate measure need not be the final endpoint. For example, it might be a landmark outcome or an early response measure. The earlier material on translating intermediate contrasts into final-endpoint predictions motivates modeling their relationship. Here, the task is to use such a model to calculate the probability of crossing the final event-test boundary, conditional on the intermediate evidence already observed.

Definition: Let \(Z_M\) be an interim statistic based on the intermediate outcome and \(Z_F\) the final statistic for the event endpoint. If final success requires \(Z_F>c\), the conditional power at observed interim value \(Z_M=z_M\) is \(P(Z_F>c\mid Z_M=z_M)\), evaluated under a specified joint distribution for the statistics. The boundary \(c\), model parameters, and planned final information must be stated.

This definition is conditional on the observed interim evidence, but it is not a Bayesian probability that the null hypothesis is false or that the treatment effect exceeds a threshold. It is a frequentist probability calculated under the assumed data-generating model. Different assumptions about the future event effect can therefore produce different conditional-power values from the same interim data.

Conditional Normal Model and Calculation

A useful approximation is that the interim and final statistics have a joint normal distribution. Write their means as \(\mu_M\) and \(\mu_F\), their standard deviations as \(\sigma_M\) and \(\sigma_F\), and their correlation as \(\rho\). The model is \((Z_M,Z_F)\) jointly normal, with \(-1<\rho<1\). The means and dependence must represent the assumed treatment-effect scenario, the design, and the planned final event information. Standardized statistics often have standard deviations close to one, but the general formula does not require that.

Conditional normal result: Under the joint normal model, \(Z_F\mid Z_M=z_M\) is normal with conditional mean \(\mu_{F\mid M}=\mu_F+\rho(\sigma_F/\sigma_M)(z_M-\mu_M)\) and conditional standard deviation \(\sigma_{F\mid M}=\sigma_F\sqrt{1-\rho^2}\). For an upper-boundary final test, conditional power is \(1-\Phi\!\left((c-\mu_{F\mid M})/\sigma_{F\mid M}\right)\), where \(\Phi\) is the standard normal cumulative distribution function.

The observed intermediate statistic contributes through its deviation \(z_M-\mu_M\) from the mean under the chosen scenario. If it is higher than expected, and \(\rho>0\), the conditional mean of the final statistic shifts upward. The conditional standard deviation reflects the uncertainty in the final statistic that remains after conditioning on the interim statistic.

Why the Formula Holds

The result follows by removing from \(Z_F\) the part linearly associated with \(Z_M\). Define the residual \(R=Z_F-\mu_F-\rho(\sigma_F/\sigma_M)(Z_M-\mu_M)\). Its covariance with \(Z_M\) is zero:

\operatorname{Cov}(R,Z_M) =\operatorname{Cov}(Z_F,Z_M) -\rho\frac{\sigma_F}{\sigma_M}\operatorname{Var}(Z_M) =\rho\sigma_F\sigma_M-\rho\frac{\sigma_F}{\sigma_M}\sigma_M^2 =0.

Because \(R\) and \(Z_M\) are jointly normal, zero covariance implies independence. Its variance is \[ \operatorname{Var}(R) =\sigma_F^2+\rho^2\sigma_F^2-2\rho^2\sigma_F^2 =\sigma_F^2(1-\rho^2). \] Thus, given \(Z_M=z_M\), the final statistic is the stated conditional mean plus an independent normal residual with that variance. The upper-tail probability at boundary \(c\) then gives the conditional-power formula.

Worked Example: Updating the Final Statistic

Suppose a hypothetical trial uses standardized intermediate and final statistics with unit marginal variances. Under a specified treatment-effect scenario, let \(\mu_M=1.0\), \(\mu_F=2.2\), and \(\rho=0.60\). The observed intermediate statistic is \(z_M=1.5\), and the final one-sided boundary is \(c=1.96\). Assume the joint normal model is appropriate and the final success direction is upward.

The conditional mean and standard deviation are \(\mu_{F\mid M}=2.2+0.60(1.5-1.0)=2.50\) and \(\sigma_{F\mid M}=\sqrt{1-0.60^2}=\sqrt{0.64}=0.80\).

Therefore, \(CP=1-\Phi((1.96-2.50)/0.80)=1-\Phi(-0.675)=\Phi(0.675)\approx0.750\).

Under this model and assumed effect scenario, the probability that the final statistic exceeds 1.96, conditional on the observed intermediate statistic of 1.5, is about 75.0%. This is not a 75% probability that the treatment is effective; it is the model-based chance of crossing the specified final boundary.

Worked Examples: Sensitivity and Future Information

Worked Example: How Correlation Changes the Update

Consider another hypothetical calculation with unit marginal variances, \(\mu_M=0.80\), \(\mu_F=1.80\), observed \(z_M=1.40\), and boundary \(c=1.96\). First use \(\rho=0.50\). Assume these means and the correlation describe the joint distribution under the stated treatment-effect scenario.

The conditional mean is \(1.80+0.50(1.40-0.80)=2.10\), and the conditional standard deviation is \(\sqrt{1-0.50^2}=\sqrt{0.75}\approx0.8660\). Hence \(CP=1-\Phi((1.96-2.10)/0.8660)=1-\Phi(-0.1617)=\Phi(0.1617)\approx0.564\).

If instead \(\rho=0\), the intermediate statistic gives no update to the final statistic in this model. The conditional mean is then 1.80 and the conditional standard deviation is 1, so \(CP=1-\Phi(1.96-1.80)=1-\Phi(0.1600)\approx0.436\). The difference illustrates why the assumed relationship between intermediate and final evidence matters; it does not establish which correlation is correct.

Worked Example: Accounting for Planned Final Information

Suppose a hypothetical event-endpoint model expresses the assumed beneficial effect as a log-hazard reduction \(\theta=0.30\), with final standardized mean \(\mu_F=\theta\sqrt{I_F}\), where \(I_F\) is planned final event information. Use \(\mu_M=0.80\), observed \(z_M=1.40\), \(\rho=0.50\), unit marginal variances, and final boundary \(c=1.96\). Assume the same calibrated dependence model applies in both information scenarios.

With \(I_F=36\), \(\mu_F=0.30\sqrt{36}=1.80\). The calculations are those of the preceding example: conditional mean \(2.10\), conditional standard deviation \(0.8660\), and conditional power approximately \(0.564\).

With \(I_F=64\), \(\mu_F=0.30\sqrt{64}=2.40\). The conditional mean is now \(2.40+0.50(1.40-0.80)=2.70\); the conditional standard deviation remains \(0.8660\) under the stated model. Thus \(CP=1-\Phi((1.96-2.70)/\sqrt{0.75})=\Phi(0.74/\sqrt{0.75})\approx\Phi(0.8545)\approx0.804\).

The larger planned information raises the assumed final mean and, in this illustration, raises conditional power from about 56.4% to 80.4%. In an actual design, the correlation and variance model should be checked for each information scenario rather than assumed unchanged automatically.

Assumptions, Diagnostics, and Alternatives

A numerical conditional-power estimate is only as credible as its inputs. Before interpreting it, check that:

The final statistic and success direction are defined correctly, and the boundary \(c\) matches the planned final test for that arm and endpoint.
The assumed final effect and future event information are plausible. For a time-to-event endpoint, information depends on event accrual, not simply the number of participants enrolled.
The joint distribution, including \(\rho\), reasonably represents how the intermediate evidence relates to the final statistic. A correlation estimated imprecisely from limited data can make the conditional update unstable.
The interim measure is sufficiently mature and measured as assumed. If the prediction model ignores delayed availability or changes in follow-up, its conditional mean may be poorly calibrated.

A diagnostic is to examine conditional power across a reasonable range of assumed final effects and correlations, not just report one plug-in value. Also check whether the resulting conditional means and variances behave sensibly and whether simulations under the design reproduce the proposed joint distribution. If an upper-tail test has standardized conditional mean \(m\) and standard deviation \(s\), then its success probability is \(1-\Phi((c-m)/s)\). For fixed \(m<c\), increasing \(s\) increases this upper-tail probability; for fixed \(m>c\), increasing \(s\) decreases it. Thus, greater uncertainty does not always lower conditional power.

The main alternative is to calculate power under a fixed effect without conditioning on the observed interim evidence. That gives a design-level probability before the look, rather than a data-dependent probability after it. Conditional power is more responsive to the observed evidence but depends on the assumed effect scenario. A probability that averages over uncertainty in the effect is a different quantity; conditional power as defined here does not perform that averaging.

In a multi-arm design, calculate the quantity for the relevant arm using its final event statistic and the correct final boundary. The calculation alone does not authorize an arm-selection rule or change the test’s error-rate calibration. It is also limited by the model: misspecified intermediate-to-final relationships, uncertain nuisance parameters, non-proportional hazards, or departures from the assumed information pattern can make the reported probability misleading.

Common Mistakes

Changing the assumed effect without saying so. Conditional power is conditional on observed evidence and on a specified model, including an effect scenario. Report that scenario explicitly.
Using the intermediate statistic as if it were the final event statistic. The intermediate evidence updates the distribution of the final statistic through the model; it does not replace the final endpoint analysis.
Using the wrong tail or boundary. If benefit corresponds to a lower statistic, reverse the direction or use the appropriate lower-tail probability. A boundary from a different test is not interchangeable.
Calling conditional power the probability the null is true or the treatment will work. It is a conditional sampling probability under stated assumptions, not a probability assigned to a hypothesis.
Assuming that more conditional variance always reduces upper-tail probability. Its effect depends on whether the conditional mean is below or above the success boundary, as shown by the normal-tail formula.

Key takeaway: Conditional power is the upper-tail probability for the final event statistic under its distribution conditional on observed intermediate evidence. State the assumed treatment effect, dependence model, future information, and final boundary alongside the result.

Check Your Understanding

Use the stated model and assumptions in each question. Solutions are included below the questions.

With unit variances, \(\mu_M=0.5\), \(\mu_F=1.7\), \(\rho=0.4\), and \(z_M=1.0\), find the conditional mean and standard deviation of \(Z_F\mid Z_M=z_M\).
If the conditional mean is 2.10, the conditional standard deviation is 0.80, and the upper boundary is 1.96, calculate conditional power to three significant figures.
In the preceding setting, explain the effect on upper-tail probability of increasing the conditional standard deviation while holding the conditional mean fixed below the boundary.
Why is a conditional-power value not a probability that the null hypothesis is true or false?

Solutions. (1) The conditional mean is \(1.7+0.4(1.0-0.5)=1.90\), and the standard deviation is \(\sqrt{1-0.4^2}=\sqrt{0.84}\approx0.9165\). (2) \(CP=1-\Phi((1.96-2.10)/0.80)=1-\Phi(-0.175)=\Phi(0.175)\approx0.569\). (3) With mean \(m<c\), \((c-m)/s\) decreases toward zero as \(s\) increases, so \(1-\Phi((c-m)/s)\) increases. (4) The calculation gives a sampling probability under a specified model and effect scenario, conditional on observed intermediate evidence; it does not assign probability to either hypothesis.