Credible Interval

Credible Interval in Bayesian Health Economic Evaluation

A 95% credible interval gives a range within which an unknown quantity, such as a relative treatment effect or an event probability in a decision model, lies with 95% probability, given the prior, the model and the observed data. It is the Bayesian answer to the question most readers think a confidence interval answers. In health technology assessment, credible intervals are the usual way to report uncertainty from a Bayesian network meta-analysis, and the posterior distributions behind them feed probabilistic sensitivity analysis directly. The limits depend on the prior and likelihood, and two common rules for drawing the interval can give different limits from the same posterior. The frequentist counterpart is the confidence interval.

From posterior distribution to interval

A Bayesian analysis ends with a posterior distribution for each unknown parameter, formed from a prior distribution and the likelihood of the data through Bayes' theorem. The FDA guidance on Bayesian statistics in medical device trials, issued in 2010, states that Bayesian interval estimates are based on the posterior distribution and are called credible intervals, and that an interval with posterior probability 0.95 of containing the endpoint is a 95% credible interval. The FDA draft guidance for drugs and biological products, issued in January 2026, describes them as intervals that contain the parameter value with a specified probability under the posterior distribution.

$$\Pr(L \le \theta \le U \mid y) = \int_{L}^{U} p(\theta \mid y)\, d\theta = 1 - \alpha$$

where $\theta$ is the unknown parameter, $y$ is the observed data, $p(\theta \mid y)$ is the posterior density, $L$ and $U$ are the lower and upper limits, and the credible level is one minus $\alpha$, which is 0.95 for a 95% interval.

Central intervals and highest posterior density intervals

Many ranges hold 95% of the posterior probability, so a rule is needed to pick one. The FDA device guidance names the two in common use, the highest posterior density (HPD) interval and the central posterior interval. Gelman and colleagues, in Bayesian Data Analysis, define the central interval as the range with exactly $\alpha/2$ of the posterior probability below it and $\alpha/2$ above, so a 95% central interval runs from the 2.5th to the 97.5th posterior percentile. The HPD region holds the same probability but the density inside it is never lower than outside, which makes it the shortest such range.

The two are identical when the posterior is unimodal and symmetric and can differ substantially when it is highly skewed. Gelman and colleagues report that the central interval is the more common, partly because it is a pair of posterior quantiles computed directly from simulations. For a posterior with two peaks the HPD region can be two separate intervals, and they suggest such a distribution is better not summarised by any single interval. Skewed posteriors are routine in HTA, for probabilities near zero, costs and between-study heterogeneity, so a report should say which rule was used.

How credible intervals are obtained in HTA work

Simple models have posteriors in closed form. With a binomial likelihood and a uniform prior, Gelman and colleagues show that $y$ events in $n$ trials give a $\text{Beta}(y + 1, n - y + 1)$ posterior, so the limits are beta quantiles. Most HTA models have no closed form, so the posterior is sampled by Markov chain Monte Carlo and a 95% central interval is read off the 2.5th and 97.5th percentiles of the sorted draws.

NICE Decision Support Unit Technical Support Document 2, on pairwise and network meta-analysis, notes that Bayesian MCMC generates distributions, and therefore credible intervals, automatically for derived quantities such as relative risks and numbers needed to treat. Its beta-blocker example reports a pooled log odds ratio against control of minus 0.26 with a 95% credible interval of minus 0.36 to minus 0.16, read as reduced mortality. TSD 2 also asks for particular care in checking convergence and suggests at least three chains from widely different starting values, because limits from an unconverged sampler describe the sampler and not the posterior. The limits carry Monte Carlo error as well; more draws make them more precise but do not make the posterior narrower.

Choices that move the limits

The interval inherits every assumption behind the posterior, and the prior is the one most often challenged. Spiegelhalter, reviewing Bayesian ideas in health-care evaluation, uses sceptical priors centred on no effect to ask whether trial results would convince a reasonable sceptic. In the CHART lung cancer trial, a sceptical prior equivalent to 701 events, restricting plausible hazard ratios to a 95% prior interval of 0.86 to 1.14, would just have led the 95% posterior interval for the hazard ratio to include 1. He judged that degree of scepticism too great, so the survival benefit was considered convincing.

At the other end, Briggs states that a traditional frequentist cost-effectiveness analysis is equivalent to a Bayesian analysis assuming no prior information, so with vague priors and adequate data, credible and confidence limits often nearly coincide. Central intervals follow any monotone transformation, so exponentiating the limits for a log odds ratio gives the central interval for the odds ratio, while HPD intervals do not transform this way.

Worked example with a skewed admission probability

All numbers in this example are illustrative. A new day-case procedure is to be modelled, and the only evidence on the probability $p$ of an unplanned hospital admission is a small series in which 2 of 20 patients were admitted. The analysis uses a uniform $\text{Beta}(1, 1)$ prior and a binomial likelihood.

1. Find the posterior. The posterior is $\text{Beta}(1 + 2, 1 + 18) = \text{Beta}(3, 19)$, with mean 3/22 = 0.136, median 0.125 and mode 2/20 = 0.100.

2. Central 95% interval. The 2.5th and 97.5th percentiles of Beta(3, 19) are 0.031 and 0.304, so the central interval is 0.031 to 0.304, a width of 0.273.

3. HPD 95% interval. The shortest range holding 95% of the posterior is 0.018 to 0.277, a width of 0.259. It leaves 0.6% of the probability below and 4.4% above, and the density is the same (about 0.90) at both limits.

4. Contrast with a large-sample confidence interval. The Wald interval uses the standard error $\sqrt{0.1 \times 0.9 / 20} = 0.0671$ and gives 0.10 plus or minus 1.96 × 0.0671, or minus 0.03 to 0.23, an impossible negative lower limit. The beta posterior stays inside 0 to 1.

5. Direct probability statements. The posterior probability that $p$ exceeds 0.25 is 0.075, and that it exceeds 0.20 is 0.179.

6. Change the scale. At an illustrative GBP 4,000 per admission, the expected admission cost per patient is 4,000 × $p$. Its central interval is 4,000 × 0.0305 = GBP 122 to 4,000 × 0.3038 = GBP 1,215, around a posterior mean of 4,000 × 3/22 = GBP 545. An HPD interval computed on the log odds scale and converted back gives 0.034 to 0.322, not the 0.018 to 0.277 found on the probability scale.

7. Interpretation. Given the uniform prior and these 20 patients, there is a 95% probability that the admission rate in the population behind the series lies between about 3% and 30%. How many of the next 20 patients will be admitted is a predictive question, not one the interval answers.

Credible intervals in decision models and cost-effectiveness decisions

The NICE manual (PMG36, section 4.7.10) asks that distributions for probabilistic sensitivity analysis represent the available evidence on each parameter and be justified. The ISPOR-SMDM task force report on parameter estimation calls beta distributions a natural match for binomial data. It adds that, under a fully Bayesian approach, a one-way deterministic sensitivity analysis stays consistent with the probabilistic analysis if its range is based on the posterior's 95% highest density region, which in the example is 0.018 to 0.277.

O'Hagan and Stevens set out a general Bayesian framework for cost-effectiveness analysis of trial data, linking mean efficacy and mean cost to the parameters of the trial model. Credible intervals for incremental cost, incremental QALYs and incremental net benefit then come from joint posterior draws, which keep the correlation between costs and effects. The ICER needs more care. Heitjan, Moskowitz and Whang note that treatments lowering both cost and effectiveness and treatments raising both can yield identical ICERs, so their method reports posterior probabilities for the four quadrants of the cost-effectiveness plane with separate ICER intervals.

Claxton argued that decisions should rest on mean net benefit irrespective of whether differences are statistically significant or fall outside a Bayesian range of equivalence, with the spread of net benefit relevant only to whether more information is needed. PMG36 (section 4.7.15) asks for the probability that a treatment is cost effective at maximum acceptable ICERs of GBP 25,000 to GBP 35,000 per QALY gained and lists cost-effectiveness acceptability curves among appropriate presentations. Briggs argues that reading such curves as the probability of cost-effectiveness given the data requires a Bayesian interpretation.

As an illustration, suppose the posterior for incremental net monetary benefit at GBP 25,000 per QALY is close to normal with mean GBP 1,500 and standard deviation GBP 1,800. Since 1.96 × 1,800 = 3,528, the 95% credible interval is minus GBP 2,028 to GBP 5,028. The probability of positive net benefit is $\Phi(1{,}500 / 1{,}800) = \Phi(0.833) = 0.80$. The interval includes zero, yet expected net benefit is positive, so on Claxton's argument the treatment is preferred, and the 20% chance of a loss bears on the value of information from further research.

Credible, confidence and prediction intervals

The 95% of a confidence interval describes how often the procedure would cover a fixed true value across repeated samples, while the 95% of a credible interval is a probability about the parameter given this prior and these data. A prediction interval answers a third question. NICE DSU TSD 3 explains that in a Bayesian random-effects analysis a predictive distribution is obtained by drawing further samples from the distribution of effects, and argues that the predictive distribution of the effect in a new trial may often be more relevant to decision making than that of the mean effect. Because it adds between-trial variation, the predictive interval is wider than the credible interval for the mean.

Misreadings and limits

Most errors with credible intervals come from forgetting what the posterior was conditioned on.

1. Treating the limits as assumption-free. Another defensible prior or likelihood can move them, which is why the FDA drug draft asks for sensitivity analyses of results and conclusions.
2. Not stating the rule. Central and HPD limits differ for skewed posteriors.
3. Comparing two intervals by overlap. A difference between treatments is judged from the interval for the difference, computed from joint draws.
4. Rejecting a technology because the interval includes zero. The decision rests on expected net benefit; the spread informs the case for more research.
5. Reading a parameter interval as a patient range. It describes a population quantity, not variation between patients.
6. Ignoring what sits outside the model. PMG36 (section 4.7.5) notes that structural uncertainty can be brought into a probabilistic model, for example by model averaging; otherwise it lies outside the limits.

Sources

• Briggs AH. A Bayesian approach to stochastic cost-effectiveness analysis. Health Economics. 1999;8(3):257-261. https://pubmed.ncbi.nlm.nih.gov/10348420/
• Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty analysis: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force Working Group-6. Medical Decision Making. 2012;32(5):722-732. https://doi.org/10.1177/0272989X12458348
• Claxton K. The irrelevance of inference: a decision-making approach to the stochastic evaluation of health care technologies. Journal of Health Economics. 1999;18(3):341-364. https://doi.org/10.1016/S0167-6296(98)00039-3
• Dias S, Sutton AJ, Welton NJ, Ades AE. NICE DSU Technical Support Document 3: Heterogeneity: subgroups, meta-regression, bias and bias-adjustment. Decision Support Unit; September 2011, last updated April 2012. https://www.sheffield.ac.uk/sites/default/files/2022-02/TSD3-Heterogeneity.final-report.08.05.12.pdf
• Dias S, Welton NJ, Sutton AJ, Ades AE. NICE DSU Technical Support Document 2: A generalised linear modelling framework for pairwise and network meta-analysis of randomised controlled trials. Decision Support Unit; August 2011, last updated September 2016. https://www.sheffield.ac.uk/sites/default/files/2022-02/TSD2-General-meta-analysis-corrected-2Sep2016v2.pdf
• Food and Drug Administration. Guidance for the Use of Bayesian Statistics in Medical Device Clinical Trials. February 2010. https://www.fda.gov/media/71512/download
• Food and Drug Administration. Use of Bayesian Methodology in Clinical Trials of Drug and Biological Products. Draft guidance for industry, January 2026. https://www.fda.gov/media/190505/download
• Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. 3rd ed. CRC Press; 2013. Chapter 2. http://www.stat.columbia.edu/~gelman/book/BDA3.pdf
• Heitjan DF, Moskowitz AJ, Whang W. Bayesian estimation of cost-effectiveness ratios from clinical trials. Health Economics. 1999;8(3):191-201. https://pubmed.ncbi.nlm.nih.gov/10348414/
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Sections 4.7.5, 4.7.10 and 4.7.15. https://www.nice.org.uk/process/pmg36
• O'Hagan A, Stevens JW. A framework for cost-effectiveness analysis from clinical trial data. Health Economics. 2001;10(4):303-315. https://doi.org/10.1002/hec.617
• Spiegelhalter DJ. Incorporating Bayesian ideas into health-care evaluation. Statistical Science. 2004;19(1):156-174. https://doi.org/10.1214/088342304000000080