Signature
v_1 = 1 / (1 / v_0 + 1 / s^2); m_1 = v_1 * (m_0 / v_0 + y / s^2); z_1 = (m_1 - a) / sqrt(v_1)
| Inputs | Definition | Unit |
|---|---|---|
v_0 | Variance of the normal prior, the square of the prior standard deviation, above zero | squared effect units |
s | Standard error of y, treated as known | effect units |
m_0 | Mean of the normal prior for the effect, set before the trial | effect units |
y | Estimate of the effect from the data observed at the interim | effect units |
a | Minimum effect regarded as beneficial, zero for superiority | effect units |
v_1 | Variance of the normal posterior distribution of the effect | squared effect units, such as pounds squared |
|---|---|---|
m_1 | Mean of the normal posterior distribution of the effect | effect units, such as pounds per patient |
z_1 | Posterior mean minus a, divided by the posterior standard deviation; Phi(z_1) is the posterior probability that the effect exceeds a | none |
Function
Bayesian interim decision function for an adaptive trial
Maps a prior distribution for the treatment effect, the data observed so far and pre-specified thresholds to the quantities that interim rules consult: the posterior probability that the effect exceeds a minimum value, which summarises the evidence now, and the predictive probability that the final analysis will succeed, which averages over the results still to come. Response-adaptive rules use the same posterior to set the chance of allocation to each arm. The notation follows the Bayesian Adaptive Design article.
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Implementations
Excel
Normal posterior and posterior probability from named cells
With PriorMean, PriorVar, InterimEst, InterimSE and MinEffect named, the formulas return the posterior variance (PostVar), the posterior mean (PostMean) and the posterior probability that the effect exceeds MinEffect (PostProb).
=1/(1/PriorVar+1/InterimSE^2); =PostVar*(PriorMean/PriorVar+InterimEst/InterimSE^2); =NORM.S.DIST((PostMean-MinEffect)/SQRT(PostVar),TRUE)
Assumptions
Normal likelihood with known standard error and a normal prior
The interim estimate is approximately normal around the true effect with standard error s, and the prior is normal, so the posterior is normal with the stated mean and variance.
Prior fixed before the interim data
The prior is specified in the protocol and not chosen after seeing the data. A posterior summary is conditional on that prior, which may not be the prior an appraisal would choose.
Worked examples
Article's interim look on incremental net monetary benefit
With a prior mean of 0 pounds and standard deviation of 2,000 pounds, and an interim estimate of 1,200 pounds with standard error 800, the estimate is 6.25 times as precise as the prior. The posterior mean is about 1,034.48 pounds with standard deviation 742.8, z_1 is about 1.3927 and the posterior probability of positive net benefit about 0.918, as in the article.
m_0 = 0; v_0 = 4000000; y = 1200; s = 800; a = 0; v_1 = 551724.1; m_1 = 1034.48; z_1 = 1.3927
Informative prior centred on 500 pounds
With a prior mean of 500 pounds and standard deviation 1,000, the same interim data give a posterior mean of about 926.83 pounds, z_1 of about 1.4837 and a posterior probability of about 0.931 (computed here for illustration).
m_0 = 500; v_0 = 1000000; y = 1200; s = 800; a = 0; v_1 = 390243.9; m_1 = 926.83; z_1 = 1.4837
Minimum effect of 500 pounds in the success criterion
Requiring net benefit above 500 pounds instead of above zero, the article's prior and data give z_1 of about 0.7196 and a posterior probability of about 0.764 (computed here for illustration).
m_0 = 0; v_0 = 4000000; y = 1200; s = 800; a = 500; v_1 = 551724.1; m_1 = 1034.48; z_1 = 0.7196
Common errors
Weighting by standard deviations instead of precisions
Weighting the estimate by the ratio of standard deviations, 2.5, instead of the ratio of variances, 6.25, gives a posterior mean of about 857 pounds instead of 1,034 in the article's example (computed here for illustration).
Reading the probability of positive net benefit as the probability that the treatment works
The posterior probability that incremental net monetary benefit is positive is the probability of cost-effectiveness at the stated threshold. It depends on the threshold, costs and time horizon and differs from the probability that the clinical effect exceeds zero.
Sources
Normal prior and normal likelihood with known variance
Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. 3rd ed. Boca Raton: CRC Press; 2013 (electronic edition for non-commercial use). Section 2.5, equation 2.10: the posterior mean is the precision-weighted average of the prior mean and the observation, and the posterior precision equals the prior precision plus the data precision.
Posterior probability success criterion in FDA draft guidance
US Food and Drug Administration. Use of Bayesian Methodology in Clinical Trials of Drug and Biological Products: Draft Guidance for Industry. Silver Spring, MD: FDA; January 2026. Section IV.A: a success criterion based on the posterior probability that the treatment effect exceeds a threshold, Pr(d > a) > c, where a is the minimum effect considered beneficial and c the minimum probability that would support a conclusion of effectiveness.
Canonical Identity
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