Computing Binary Endpoint Power With Beta-Binomial Updates

From a Posterior Decision to Conditional Power

For a binary endpoint, the analysis after a trial can be simple: count the successes in each arm, update beta priors, and assess whether the posterior probability of clinically meaningful benefit is high enough. Planning requires a different calculation. Conditional Bayesian power is the probability, under a specified pair of true event rates, that the planned trial will meet that posterior decision rule. It averages over possible trial data, not over uncertainty about the true rates.

This calculation uses the conditional-power idea from the earlier material on conditional Bayesian power under a fixed treatment effect, but now makes the binary sampling model and beta updates explicit. It also differs from assurance: assurance averages success probability over a design distribution for the true rates, whereas conditional power fixes those rates at a particular scenario.

Model, Update, and Decision Rule

Assume participants have a fully observed binary endpoint, with success defined in advance so that a higher probability is better. Conditional on the fixed rates \(p_T\) and \(p_C\), outcomes are independent within and between arms, and each participant in an arm has the same success probability. Thus

For the analysis, assign independent beta priors \(p_T\sim\operatorname{Beta}(a_T,b_T)\) and \(p_C\sim\operatorname{Beta}(a_C,b_C)\), with positive shape parameters. Given observed counts \(x_T,x_C\), beta-binomial conjugacy gives

Choose a clinical benefit margin \(\delta\geq 0\) and a posterior probability threshold \(\gamma\), usually near 1. For a particular pair of counts, declare treatment beneficial when the posterior probability that the treatment success rate exceeds the control rate by more than \(\delta\) exceeds \(\gamma\). The distinction between “exceeds” and “equals” should be specified in the protocol; here the rule is strictly greater than \(\gamma\).

With independent beta posteriors, this probability can be calculated by integrating over the treatment rate:

where \(f_T\) is the treatment posterior density and \(F_C\) is the control posterior cumulative distribution function. The lower limit is \(\delta\) because \(u-\delta\) must be nonnegative; the control CDF is zero below zero. This integral can be evaluated numerically, or by suitable beta-distribution calculations. It is a posterior probability for a given dataset, not the power itself.

Why the Power Calculation Is a Finite Sum

For fixed sample sizes, there are only \(n_T+1\) possible treatment success counts and \(n_C+1\) possible control success counts. Each pair either satisfies the posterior decision rule or does not. The sampling probability of a pair is the product of the two binomial probabilities under the fixed scenario. Summing those probabilities over all pairs that pass the rule gives the exact conditional power:

Here \(I\{\cdot\}\) is 1 when the condition holds and 0 otherwise. The expression follows by partitioning trial outcomes into their mutually exclusive count pairs: the probability of decision success is the sum of the probabilities of precisely those pairs that trigger the decision. The product form requires independent arm outcomes under the specified model. If the design induces dependence, or the analysis rule includes additional features such as a shared nuisance parameter, the joint sampling probabilities or decision calculation must reflect them.

Implementation, Diagnostics, and Alternatives

For a design with moderate sample sizes, a direct calculation is usually straightforward: enumerate all possible count pairs, compute the posterior probability for each, mark those that pass the rule, then sum their scenario sampling probabilities. For larger designs, the decision region can still be enumerated; numerical integration may be needed for posterior probabilities. A simulation can approximate the sum by repeatedly generating trial counts under the scenario and applying the same posterior rule, but it introduces Monte Carlo error. Exact enumeration is preferable when it is computationally feasible.

Check the calculation at several points. Posterior parameters must include the observed successes and failures in the correct arm. Each arm’s binomial probabilities should sum to one, and the total probability over all count pairs should also equal one. Confirm that the decision region uses the intended inequality, benefit direction, margin, and threshold. A useful unit test is a scenario where treatment and control are identical: the resulting decision probability should be consistent with the rule, priors, and sample sizes, not automatically assumed to equal a particular nominal level.

The main alternative is simulation-based operating-characteristic estimation. Simulation is flexible when the design includes features that make enumeration cumbersome, but its estimate has sampling variability and can be unstable when decision success is rare. Analytic or numerical enumeration computes the finite sum directly and avoids that Monte Carlo error. Neither approach changes the estimand: for conditional power, the true rates are fixed at the scenario values. Averaging the result over a distribution of rates instead answers an assurance question.

The calculation depends on the model and decision rule being appropriate for the planned trial. Unequal follow-up, missing outcomes, clustering, noncompliance, or varying participant risks may invalidate the simple independent-binomial sampling model unless explicitly incorporated. The posterior decision can also be sensitive to the analysis priors, especially for small samples. Recompute power under reasonable prior choices and clinically relevant rate scenarios rather than treating one result as universal. Finally, conditional power at a single scenario does not summarize performance across all plausible scenarios; scenario-based operating-characteristic assessment is needed for that purpose.

Common Pitfalls

• Confusing posterior probability with power. A posterior probability such as 0.95 is evaluated for one realized dataset. Power is the probability, across repeated datasets at fixed true rates, that the posterior rule is met.
• Using the analysis prior as the distribution of true rates. The beta analysis priors update the observed data. They do not define the fixed scenario in a conditional-power calculation.
• Counting only a single “ideal” outcome. Every count pair that passes the decision rule contributes its sampling probability.
• Ignoring the clinical margin. A rule about \(p_T-p_C>\delta\) is not the same as a rule about \(p_T>p_C\) when \(\delta>0\).
• Reporting a scenario result as overall assurance. A conditional result applies to the stated \(p_T,p_C\); it is not averaged across uncertainty in those rates.