Conditional Power and Predictive Probability Compared

Two Ways to Describe the Chance of Success

After an interim analysis, a trial team may ask how likely the study is to meet its final success criterion. One answer fixes a treatment effect and calculates the probability of success if that effect were the true effect. Another averages over uncertainty about the effect using its posterior distribution. These answers can differ even when they use the same interim data and the same final boundary, because they answer different questions.

This comparison uses a canonical normal group sequential model with one remaining analysis. It isolates the distinction between a fixed-effect calculation and a posterior average; it does not change the stopping rule or the interim evidence.

Definition: Conditional power at an assumed effect $\theta$ is the probability, conditional on the observed interim data and the trial continuing, that the remaining data will lead to success, calculated as if $\theta$ were the true effect. Predictive probability is the same future-success probability averaged over the posterior distribution of $\theta$ given the interim data.

Model, Interim Data, and Success Criterion

Let $Z_1$ be the interim standardized statistic at information fraction $t$, and let $Z_2$ be the final statistic at full information. Use the canonical normal model: under a fixed standardized treatment effect $\theta$, $Z_1\sim N(\theta\sqrt{t},1)$, and the final statistic has the conditional distribution

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

Here $\theta$ is the standardized effect on the full-information scale. The model assumes the canonical information-based normal approximation, a prespecified information fraction, and independent future increments conditional on the effect. For illustration, suppose the trial has reached an interim analysis at $t=0.50$, observed $z_1=1.20$, and continues to one final look. Success means $Z_2\geq c$, where $c=1.96$. We assume no other remaining stopping boundary alters this final success event.

The observed interim statistic is below the final boundary, so the trial has not already met this success criterion. Given the interim result, the fixed-effect conditional power is

\operatorname{CP}(\theta\mid z_1) = P(Z_2\geq c\mid Z_1=z_1,\theta) = 1-\Phi\!\left( \frac{c-\sqrt{t}\,z_1-\theta(1-t)}{\sqrt{1-t}} \right).

This expression is conditional on the data observed so far but treats $\theta$ as fixed. Predictive probability instead averages this expression over the posterior distribution of $\theta$:

\operatorname{PP}\mid z_1 = \int \operatorname{CP}(\theta\mid z_1)\, \pi(\theta\mid z_1)\,d\theta,

where $\pi(\theta\mid z_1)$ is the posterior density. Thus, conditional power is a probability under a specified effect, whereas predictive probability incorporates both future sampling variability and posterior uncertainty about the effect.

Worked Example: Conditional Power at Two Assumed Effects

Use the interim result $t=0.50$, $z_1=1.20$, and final boundary $c=1.96$. Calculate conditional power first at $\theta=0.20$, then at $\theta=0.38$.

The shared interim contribution to the conditional mean is $\sqrt{0.50}(1.20)=0.848528$, and the remaining standard deviation is $\sqrt{0.50}=0.707107$. At $\theta=0.20$, the conditional mean is $0.848528+0.20(0.50)=0.948528$. Therefore,

\operatorname{CP}(0.20\mid z_1=1.20) = 1-\Phi\!\left(\frac{1.96-0.948528}{0.707107}\right) = 1-\Phi(1.4304) \approx 0.0763.

At $\theta=0.38$, the conditional mean is $0.848528+0.38(0.50)=1.038528$. The corresponding calculation is

\operatorname{CP}(0.38\mid z_1=1.20) = 1-\Phi\!\left(\frac{1.96-1.038528}{0.707107}\right) = 1-\Phi(1.30316) \approx 0.09626.

Under these assumptions, success is more likely at the larger assumed effect, but the calculated probabilities remain below 0.10 at both effects. Neither result is the probability that the effect equals the assumed value.

A Posterior Average Is Not a Plug-In Calculation

For a direct comparison, suppose a prespecified normal prior and the interim likelihood give the posterior $\theta\mid z_1\sim N(0.30,0.20^2)$. This posterior is an illustrative input to the calculation. Under the model above, the interim likelihood is $Z_1\mid\theta\sim N(\theta\sqrt{0.50},1)$; a normal prior $N(0.271489,1/24.5)$, for example, gives posterior variance $1/(24.5+0.50)=0.04$ and posterior mean 0.30 after observing $z_1=1.20$.

Conditional power evaluated at the posterior mean is still a fixed-effect calculation. Predictive probability averages over the full posterior. The difference can be derived by representing the final statistic as its conditional mean plus a future normal increment, then integrating over the normal posterior for $\theta$.

Result: If $\theta\mid z_1\sim N(m,s^2)$, then under the canonical model the posterior predictive distribution of the final statistic is $$ Z_2\mid z_1\sim N\!\left(\sqrt{t}\,z_1+(1-t)m,\ (1-t)+(1-t)^2s^2\right). $$ Consequently, predictive probability is the upper-tail probability of this normal distribution at the success boundary.

To justify the result, write the conditional final statistic as $Z_2=\sqrt{t}\,z_1+(1-t)\theta+\varepsilon$, where, conditional on $\theta$ and the interim data, $\varepsilon\sim N(0,1-t)$. The posterior uncertainty in $\theta$ is independent of this future increment under the model. The expectation of $Z_2$ over the posterior and future increment is $\sqrt{t}\,z_1+(1-t)m$. Its variance is the sum of the future-increment variance and the variance contributed by $(1-t)\theta$:

\operatorname{Var}(Z_2\mid z_1) = (1-t)+(1-t)^2s^2.

The sum is normal because it is a linear combination of independent normal variables. This gives the predictive distribution and, by taking its probability above $c$, the predictive probability. The extra variance term is the mathematical source of the distinction: predictive probability includes effect uncertainty, while conditional power at one effect does not.

Worked Example: Comparing Predictive Probability With Power at the Posterior Mean

Continue with $t=0.50$, $z_1=1.20$, $c=1.96$, and posterior $\theta\mid z_1\sim N(0.30,0.20^2)$. First calculate conditional power at the posterior mean. Its conditional mean is $0.848528+0.30(0.50)=0.998528$, so

\operatorname{CP}(0.30\mid z_1=1.20) = 1-\Phi\!\left(\frac{1.96-0.998528}{0.707107}\right) = 1-\Phi(1.35973) \approx 0.08696.

For predictive probability, the posterior predictive mean is also $0.998528$, but the variance includes posterior uncertainty:

\operatorname{Var}(Z_2\mid z_1) = 0.50+(0.50)^2(0.20)^2 = 0.51, \qquad SE=\sqrt{0.51}=0.714143.

Therefore,

\operatorname{PP}\mid z_1 = 1-\Phi\!\left(\frac{1.96-0.998528}{0.714143}\right) = 1-\Phi(1.34633) \approx 0.08910.

The predictive probability is about 0.0891, compared with conditional power of about 0.0870 at $\theta=0.30$. Both use the same interim result and success boundary. Their difference arises because predictive probability also averages over the posterior spread in $\theta$, not because the trial has a different final criterion.

Posterior Mixtures and the Meaning of Averaging

A posterior need not be normal. For a discrete posterior, predictive probability is the weighted sum of conditional powers at its support points. More generally, it is the posterior expectation of conditional power. This identity clarifies that predictive probability does not require a single “representative” effect; it weights possible effects according to their posterior probabilities.

Worked Example: Averaging Over a Two-Point Posterior

Keep the same interim data and final boundary. Suppose instead that the posterior assigns probability 0.40 to $\theta=0.20$ and probability 0.60 to $\theta=0.38$. The conditional powers calculated for these effects are approximately 0.0763 and 0.09626. The predictive probability is

\operatorname{PP}\mid z_1 = 0.40(0.0763)+0.60(0.09626) = 0.088276 \approx 0.08828.

This is the posterior-weighted probability of crossing the final boundary. It is not the conditional power at a selected effect unless the posterior is concentrated at that effect or the numerical values happen to coincide. The posterior mean here is $0.40(0.20)+0.60(0.38)=0.308$, but calculating conditional power at 0.308 would discard the posterior spread and would not, in general, produce the same answer.

Interpretation, Diagnostics, and Limitations

The main alternative to predictive probability is conditional power at a prespecified design effect or another fixed assumed effect. It is transparent about the effect being assumed and useful when the question is explicitly “what if this effect were true?” Predictive probability addresses a different question: given the interim data and the chosen prior model, how likely is final success after averaging over remaining effect uncertainty? Neither quantity should be described as the probability that the null hypothesis is true or as a guarantee of trial success.

Several checks help detect a calculation that is inconsistent with the stated method:

Check the conditioning. Both calculations must use the same interim result, information fraction, future analysis plan, and success boundary. Changing any of these while comparing values confounds the comparison.
Check the remaining information. In this model the future standard deviation is $\sqrt{1-t}$, not $1-t$. The effect contributes $\theta(1-t)$ to the conditional mean.
Check the posterior average. Predictive probability must weight conditional success probabilities by the posterior distribution. Substituting the posterior mean into the conditional-power formula is not that average.
Check the event being predicted. A final-statistic boundary is not interchangeable with a rule involving multiple future looks, futility stopping, or a different two-sided success criterion. The probability must match the actual event.

The normal derivation relies on the canonical model, a correctly specified posterior, and a future sampling distribution consistent with the design. If the variance is estimated, outcomes are discrete, the information schedule changes, or the stopping rule includes additional future decisions, the simple normal expression may not apply without modification. Predictive probability also depends on the prior: different defensible priors can yield different posterior averages. Conditional power avoids averaging over a prior but depends directly on the fixed effect selected. These are limitations to state, not reasons to treat either quantity as universally preferable.

Key takeaway: Conditional power fixes the effect and calculates a future success probability. Predictive probability averages that same future success probability over posterior effect uncertainty. Use the one that answers the question being asked, and make the fixed-effect assumption or posterior model explicit.

Check Your Understanding

Answer each question using the definitions and formulas above. Brief solutions follow the questions.

With $t=0.50$, $z_1=1.20$, and $c=1.96$, what does conditional power at $\theta=0.20$ condition on, and what does it treat as fixed?
Why is predictive probability generally not equal to conditional power at the posterior mean?
If the posterior for $\theta$ has mean $m$ and variance $s^2$, state the predictive variance of $Z_2$ in this one-remaining-look model.
For a two-point posterior with weights 0.25 and 0.75 at effects $\theta_a$ and $\theta_b$, how is predictive probability calculated?
Name two features that must be held fixed when comparing conditional power with predictive probability for the same interim scenario.

Solutions. (1) It conditions on the interim result and continuation to the final analysis, while treating $\theta=0.20$ as the true fixed effect; the resulting probability is approximately 0.0763. (2) Predictive probability averages over the full posterior distribution, including its spread, whereas conditional power at the posterior mean uses only one effect value. (3) The variance is $(1-t)+(1-t)^2s^2$. (4) It is $0.25\operatorname{CP}(\theta_a\mid z_1)+0.75\operatorname{CP}(\theta_b\mid z_1)$. (5) Examples include the observed interim statistic, information fraction, final success boundary, and remaining analysis plan; the same choices must be used for both calculations.