Normal-Endpoint Power With Known Variance

From a Posterior Decision to a Power Calculation

For a continuous clinical endpoint, a trial may declare success when the posterior probability that the treatment effect exceeds a clinically chosen threshold is sufficiently high. To plan such a trial, it is useful to turn that posterior decision into a rule about the observed data. In the normal, known-variance setting, the rule becomes a threshold on the observed treatment-control difference. Conditional Bayesian power is then the probability, under a specified true treatment effect, that repeated trial data cross that threshold.

The word conditional matters: the true effect is held fixed for this calculation. The result is not averaged over a distribution of possible true effects. That distinction separates conditional power from assurance, which averages success probability over uncertainty about the effect before observing the trial.

Model, Prior, and Posterior Success Rule

Assume independent observations in the two randomized arms. Treatment outcomes are normally distributed with mean \(\mu_T\), control outcomes with mean \(\mu_C\), and both arms have the same known variance \(\sigma^2\). With fixed sample sizes \(n_T\) and \(n_C\), the arm means are independent and \(D\mid\delta\sim N(\delta,V)\), where \(V=\sigma^2(1/n_T+1/n_C)\). The sampling standard error of \(D\) is \(\sqrt{V}\).

For the analysis, place a proper normal prior on the treatment effect, \(\delta\sim N(m_0,\tau_0^2)\), with \(\tau_0^2>0\). This is a prior for the mean difference, expressed in squared endpoint units. Let the success rule be that the posterior probability of exceeding a prespecified clinical threshold \(\delta_c\) is at least \(\gamma\), where \(0<\gamma<1\). The direction here is favorable when the treatment-minus-control effect is larger.

The posterior variance \(v\) does not depend on the observed difference \(d\). Define \(k=v/V=\tau_0^2/(\tau_0^2+V)\), which lies strictly between zero and one. The posterior mean can be written \(m(d)=m_0+k(d-m_0)\): it is a weighted average of the prior mean and the observed difference.

Let \(\Phi\) denote the standard normal cumulative distribution function and let \(z_\gamma=\Phi^{-1}(\gamma)\). Since the posterior is normal, the posterior probability of exceeding \(\delta_c\) is at least \(\gamma\) exactly when its mean is at least \(\delta_c+z_\gamma\sqrt{v}\). Because \(k>0\), this condition is equivalent to an observed-data cutoff.

The equivalence uses the continuous normal posterior, for which \(P(\delta>\delta_c\mid D=d)=\Phi((m(d)-\delta_c)/\sqrt{v})\). Applying the increasing function \(\Phi^{-1}\) gives the mean threshold; solving the resulting linear inequality for \(d\) gives \(d_{\mathrm{crit}}\). The critical observed difference is in endpoint units.

Conditional Power at a Fixed Effect

Under the design model and a fixed true effect \(\delta^\ast\), the observed difference has sampling distribution \(D\mid\delta^\ast\sim N(\delta^\ast,V)\). Since the posterior decision is equivalent to \(D\geq d_{\mathrm{crit}}\), the conditional power is the upper-tail probability of this sampling distribution.

The subscript indicates that this probability is calculated under the repeated-sampling model with true effect \(\delta^\ast\). The posterior probability used to make a decision and the conditional power used to evaluate a design answer different questions. The former describes uncertainty about \(\delta\) after the observed data; the latter is the probability of a successful decision across hypothetical trials at a fixed true \(\delta^\ast\).

Derivation and Interpretation of the Cutoff

The cutoff follows from conjugate normal updating. The likelihood for the observed difference is proportional, as a function of \(\delta\), to \(\exp[-(d-\delta)^2/(2V)]\); the prior density is proportional to \(\exp[-(\delta-m_0)^2/(2\tau_0^2)]\). Multiplying them and collecting the terms involving \(\delta\) gives a quadratic with precision \(1/V+1/\tau_0^2\) and precision-weighted mean \(d/V+m_0/\tau_0^2\). These are the posterior precision and posterior mean shown above.

The critical value also gives useful qualitative checks. When \(\gamma\) increases, \(z_\gamma\) increases, so the posterior mean requirement becomes stricter and the data cutoff rises. Raising \(\delta_c\) also raises the cutoff. A prior mean \(m_0\) above the required posterior mean can lower the observed cutoff, while a prior mean below it can raise the cutoff. These effects are conditional on the other inputs; they do not imply that one prior is appropriate for a particular trial.

For a numerical design calculation, calculate the posterior quantities and \(d_{\mathrm{crit}}\) once for the chosen sample sizes, then evaluate the power formula at each fixed effect of interest. The same cutoff is used across those fixed-effect scenarios, while the sampling distribution shifts with \(\delta^\ast\).

Assumptions, Diagnostics, and Alternatives

The calculation depends on the specified model and decision rule. In particular, outcomes must be independent across arms under the assumed sampling model; the arm means must have the stated normal distributions; the variance used in \(V\) must be known and common; and the sample sizes and allocation must be fixed as specified. The analysis prior must be the stated normal prior, and the success rule must be the one-sided posterior threshold above. If larger outcomes are unfavorable, or success is defined in the opposite direction, the inequality must be changed rather than applying this cutoff unchanged.

A practical diagnostic is to check the calculation in two ways: verify that \(v\) is smaller than both \(V\) and \(\tau_0^2\), and substitute the computed \(d_{\mathrm{crit}}\) into the posterior rule to confirm that the posterior probability is \(\gamma\), up to rounding. Also check that standard deviations and differences use endpoint units, variances use squared units, and the same variance model is used in both the posterior update and the sampling distribution. Large discrepancies often signal a swapped sign, a missing square root, or use of \(v\) where \(\sqrt v\) is required.

This analytic result is an alternative to estimating power by simulating trial datasets and applying the posterior rule to each one. Under the assumptions here, the cutoff and normal sampling distribution give the power directly, without simulation error. Simulation can be useful when the analysis or decision rule is more complicated than this model, but it must reproduce the intended data-generating assumptions and the exact success rule. The calculation is also distinct from conventional frequentist power, which evaluates a specified frequentist rejection rule, and from assurance, which averages success probability over a distribution of possible true effects.

Common Mistakes and Limitations

• Confusing the posterior probability with power. A posterior probability is computed after observing a particular dataset. Power is a repeated-trial probability at a fixed true effect. They are not interchangeable.
• Using the clinical threshold as the data cutoff. The rule is about the posterior probability of exceeding \(\delta_c\); the required observed difference is \(d_{\mathrm{crit}}\), which also depends on the prior, sample sizes, variance, and \(\gamma\).
• Mixing up the two variances. The posterior uses \(v\), while the repeated-sampling standardization for power uses \(V\). Their square roots are standard deviations on the endpoint scale.
• Reversing the effect direction. This derivation assumes larger treatment-minus-control differences are favorable. A favorable decrease requires reformulating the effect and success rule consistently.

The method is exact only for the stated normal model with known variance and the specified fixed-effect scenario. If the variance is not known, the sampling and posterior calculations change; likewise, non-normal outcomes or additional model features may prevent this simple cutoff form from applying directly. A calculated power value is therefore a property of the model, prior, decision rule, design, and chosen \(\delta^\ast\), not a guarantee about a future trial.