Bayesian Adaptive Design

Bayesian Adaptive Design with Posterior Rules, Borrowing and Value-Based Stopping

Bayesian adaptive trials matter to health economists because their estimates feed health technology assessment, and because value-based variants set the stopping point from the cost of the extra evidence rather than from a fixed probability threshold. Planned adaptation in general and the frequentist control of repeated looks are covered under adaptive trial design and alpha spending, and the Bayesian layer adds posterior and predictive stopping rules, response-adaptive randomisation, borrowing of external evidence and FDA expectations on operating characteristics. One illustrative interim look, worked three ways, shows how such evidence is used in appraisal.

What the Bayesian layer adds to an adaptive trial

The label covers a family of designs rather than one method. The FDA guidance on adaptive designs (2019) lists typical features, namely predictive modelling to govern the timing and rules of interim analyses, assumed dose-response relationships to guide dose escalation, explicit borrowing from previous trials or registries through informative priors, and posterior probabilities as success criteria. All rest on Bayes theorem, shown below.

$$p(\theta \mid y) \propto p(y \mid \theta)\, p(\theta)$$

where $\theta$ is the unknown treatment effect, $y$ is the trial data so far, $p(\theta)$ is the prior distribution, $p(y \mid \theta)$ is the likelihood, and $p(\theta \mid y)$ is the posterior distribution that each interim rule consults.

The same guidance states that the principles for adaptive designs apply equally when Bayesian features are present, and that such a design may still rely on frequentist inference for its conclusions. A trial can therefore use posterior calculations to steer recruitment while its confirmatory claim is judged against a type I error rate.

Posterior and predictive probability as stopping rules

Two quantities do most of the work at an interim analysis. The first summarises the evidence now; the second projects forward to the final analysis. The FDA draft guidance on Bayesian methodology (2026) describes a success criterion of the first kind, shown below.

$$\Pr(d > a \mid y) > c$$

where $d$ is a population-level summary of the treatment effect, $a$ is the minimum effect regarded as beneficial (zero for superiority), $c$ is the minimum posterior probability that would support a conclusion of effectiveness, and $y$ is the data. The draft guidance adds that a trial with interim decision points needs a criterion at each one.

The second quantity is the predictive probability of success. In the design of Lee and Liu for single-arm phase II trials, the interim data are used to average over every possible result from the patients still to be enrolled, as follows.

$$PP = \sum_{i=0}^{m} \Pr(Y = i \mid x)\, I(B_i > \theta_T)$$

where $x$ is the number of responses so far, $m$ is the number of patients still to come, $Y$ is their number of responses with beta-binomial predictive probability $\Pr(Y = i \mid x)$, $B_i$ is the final posterior probability that the response rate exceeds $p_0$, the pre-specified response rate of the standard treatment, if $i$ further responses occur, $\theta_T$ is the final efficacy threshold, and $I(\cdot)$ equals one when its condition holds. The trial stops for futility if $PP$ falls below a bound $\theta_L$, for efficacy if it exceeds a bound $\theta_U$, and otherwise continues. The FDA adaptive design guidance treats posterior and predictive probability as alternative scales for a stopping rule and regards the choice of scale as relatively unimportant provided the operating characteristics are adequately evaluated.

Response-adaptive randomisation

In response-adaptive randomisation, the chance that a new participant joins each arm changes with the outcomes already observed. The FDA guidance (2019) notes that the statistical, ethical and pragmatic arguments for it are controversial, that it does not generally increase the type I error probability when analysed appropriately, and that it works best when outcomes are observed quickly. A widely cited Bayesian rule, due to Thall and Wathen and set out in the review by Robertson and colleagues, allocates the next patient to the experimental arm with the probability given below.

$$\pi = \frac{P^{c}}{P^{c} + (1 - P)^{c}}$$

where $P$ is the current posterior probability that the experimental arm has the higher success rate and $c$ is a tuning constant. Setting $c = 0$ gives equal randomisation and $c = 1$ gives Thompson sampling; one half, or a value rising with the number enrolled, has been suggested to reduce the variability of allocation. With an illustrative $P = 0.918$ and $c = 1/2$, $\pi = 0.958 / (0.958 + 0.286) = 0.770$.

Robertson and colleagues show that time trends, from changes in standard care or drift in the patients recruited, can inflate type I error with standard analyses and bias estimates under some rules, while rules protecting allocation to the control arm are more resistant. For an economic model, unequal arms leave fewer control patients for costs and utilities, and a biased clinical estimate biases the incremental effect carried into the model.

Borrowing external evidence through the prior

An informative prior lets a trial use evidence it did not collect. The FDA draft guidance (2026) cites a phase 3 trial of the faecal microbiota product REBYOTA whose primary analysis formally incorporated an earlier phase 2 placebo-controlled study. Borrowing is static when fixed in advance and dynamic when it shrinks as the new data diverge from the external source.

The draft guidance describes a power prior, which raises a prior based on external data to a fixed discounting factor between zero (no borrowing) and one (external patients count fully); mixture priors with prespecified weights; and hierarchical models that borrow across exchangeable groups, such as the related diseases in a basket trial. It lists effective sample size among the metrics that have been used to summarise a prior, asks for such metrics to be compared with those of a non-informative prior, and warns of prior-data conflict.

Borrowing also matters at appraisal. The NICE manual (PMG36, section 3.4.19) states that informative prior distributions for relative effectiveness are not recommended in evidence synthesis except under very specific circumstances, such as very sparse adverse event data. A trial result that rests on a borrowed prior may therefore need presenting without the borrowing too, and external trials already in the prior should not be counted again in a network meta-analysis.

Operating characteristics and calibration

A Bayesian design is judged by how it behaves across many hypothetical trials, not by its posterior alone. The FDA draft guidance (2026) distinguishes two routes. In the first, the threshold $c$ is calibrated, usually by simulation for complex designs, so that the familywise type I error stays at a set level, typically 0.025 one-sided; the guidance considers this suitable where Bayesian methods run a complex adaptive design rather than combine information sources. In the second, used mainly when external information enters the prior, the posterior is interpreted directly and the design is described by Bayesian power, the probability of meeting the success criterion averaged over a design prior.

Simulation of the whole design is therefore central, and the 2019 guidance notes that reliance on Markov chain Monte Carlo makes Monte Carlo simulation of Bayesian designs particularly resource intensive.

Worked example of one interim look viewed three ways

The figures are illustrative. A publicly funded two-arm trial monitors the incremental net monetary benefit $\delta$, the difference in mean net monetary benefit per patient between arms, with health valued at an illustrative £25,000 per QALY. The prior for $\delta$ is normal with mean £0 and standard deviation £2,000. At the interim, with 100 patients per arm, the estimate is £1,200 with standard error £800; a second stage would add 100 patients per arm at £4,000 each. The protocol stops for conclusive net benefit if the posterior probability that $\delta > 0$ exceeds 0.99, and for futility if the predictive probability of final success (a final posterior probability above 0.975) falls below 0.10.

1. Update the prior. The interim estimate is $(2000/800)^2$, or 6.25 times, as precise as the prior, so with a normal prior and likelihood the posterior is as follows.

$$m_1 = \frac{6.25 \times 1200}{1 + 6.25} = \frac{7500}{7.25} = 1034.5, \qquad v_1 = \frac{2000^2}{7.25} = 551724$$

where $m_1$ is the posterior mean of $\delta$ in pounds and $v_1$ is its posterior variance, so the posterior standard deviation is 742.8.

2. Posterior probability. The probability of positive net benefit is as follows.

$$\Pr(\delta > 0 \mid y) = \Phi\left(\frac{1034.5}{742.8}\right) = \Phi(1.3927) = 0.918$$

where $\Phi$ is the standard normal distribution function. This is below 0.99, so the conclusive net benefit rule does not stop the trial. At £25,000 per QALY it is also the probability that the new treatment is cost-effective.

3. Predictive probability. After a second stage the precision would be 13.5 times the prior's (1 + 6.25 + 6.25), giving a final standard deviation of 544.3. Success needs a final posterior mean above 1.96 times 544.3, or 1,066.8; as that mean is 6.25 times (1,200 plus the second-stage estimate) over 13.5, the second-stage estimate must exceed £1,104.3. Given the interim posterior, that estimate is predicted to be normal with mean 1,034.5 and variance 551,724 plus 800 squared, a standard deviation of 1,091.7, so the predictive probability is as follows.

$$PP = 1 - \Phi\left(\frac{1104.3 - 1034.5}{1091.7}\right) = 1 - \Phi(0.0639) = 1 - 0.5255 = 0.4745$$

where $PP$ is the predictive probability of final success. This exceeds 0.10, so the protocol says continue.

4. A value-based check. For two options with normally distributed net benefit, the expected value of perfect information per patient is the posterior standard deviation times the unit normal loss, calculated below.

$$L(z) = \phi(z) - z\,[1 - \Phi(z)] = 0.15126 - 1.3927 \times 0.08185 = 0.03727$$

where $z$ is the posterior mean divided by its standard deviation, 1.3927, and $\phi$ is the standard normal density. Per patient this is 742.8 times 0.03727, or £27.68; for an illustrative 20,000 future patients, about £553,600. The second stage would cost 200 patients at £4,000, or £800,000. Since no study is worth more than perfect information, continuing cannot pay for itself, and a value-based rule would stop and adopt the new treatment on its positive posterior mean.

The rules disagree because they answer different questions, namely whether the evidence is conclusive, and whether more evidence is worth its cost to the health system. With 40,000 future patients the value of information bound rises to about £1.1 million, but the expected value of sample information of the actual second stage is about £151,000, so its expected net benefit of sampling is still negative.

Value-based and decision-theoretic designs

Decision-theoretic designs replace a fixed probability threshold with an explicit trade-off between better decisions and the cost of obtaining them. In the model of Chick, Forster and Pertile, applied by Forster and colleagues to the ProFHER trial of surgery against sling for proximal humeral fracture, the prior is placed on expected incremental net monetary benefit. The stopping rule maximises the expected net benefit of running the trial and then recommending one technology for the patients expected to benefit, net of fixed and variable research costs and switching costs, allowing for the delay before outcomes are observed.

Applied retrospectively, the rule could have stopped ProFHER early, cutting the sample size by about 14% and the research budget by about 5% with the same recommendation. The FDA draft guidance (2026) also describes success criteria that minimise an expected loss, weighing the consequences of approving an ineffective drug against not approving an effective one.

Evidence from Bayesian adaptive trials in technology appraisal

The NICE manual (PMG36, 2022, last updated 13 October 2026) does not mention Bayesian or adaptive trial designs, so its general requirements on evidence and uncertainty apply. The NICE Decision Support Unit's evidence synthesis checklist (TSD 7) states that trials stopped early under prespecified stopping rules should be included in an evidence synthesis without adjustment for early stopping.

Two features need attention in a health technology assessment. A posterior summary is conditional on the trial's prior, which may not be the prior an appraisal would choose. Where evidence is too uncertain, PMG36 (section 6.4.14) allows a committee to recommend use while more research is done, after weighing the value to the NHS of more evidence, the trade-off that value-based designs formalise.

Where Bayesian adaptive designs are misread

A Bayesian design is sometimes presented as immune to repeated looks because the posterior does not depend on the stopping rule, yet regulators still assess behaviour across trials, and calibrated designs may need simulations showing overall error control across all decision points. The probability that a treatment works also differs from the probability that it is cost-effective, which depends on the threshold, costs and time horizon. A prior that is too informative can overwhelm the trial data, and a value-based rule rests on uncertain inputs, such as the population that will benefit, that need their own sensitivity analysis.

Sources

• Ades AE, Caldwell DM, Reken S, Welton NJ, Sutton AJ, Dias S. NICE DSU Technical Support Document 7: Evidence synthesis of treatment efficacy in decision making: a reviewer's checklist. Sheffield: Decision Support Unit, ScHARR; January 2012. Section A3.2.
• Forster M, Brealey S, Chick S, Keding A, Corbacho B, Alban A, Pertile P, Rangan A. Cost-effective clinical trial design: application of a Bayesian sequential model to the ProFHER pragmatic trial. Clinical Trials. 2021;18(6):647-656.
• Lee JJ, Liu DD. A predictive probability design for phase II cancer clinical trials. Clinical Trials. 2008;5(2):93-106.
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, last updated 13 October 2026. Sections 3.4.19 and 6.4.14.
• Robertson DS, Lee KM, López-Kolkovska BC, Villar SS. Response-adaptive randomization in clinical trials: from myths to practical considerations. Statistical Science. 2023;38(2):185-208.
• US Food and Drug Administration. Adaptive Designs for Clinical Trials of Drugs and Biologics: Guidance for Industry. Silver Spring, MD: FDA; November 2019. Sections V, V.E and VI.B.
• 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. Sections III, IV and V.