Bayesian inference for small-sample studies

Not necessarily: with a small sample, a non-significant result may point to a lack of power to detect an effect compatible with the clinical hypothesis, not necessarily to an absence of effect. This applies to rare diseases, pilot studies, subgroups and short series, where gathering a large sample is not an option and a frequentist analysis cannot reach a conclusion. In that setting a non-significant p-value is often mistaken for "it doesn't work", when what it usually indicates is that the data cannot distinguish precisely between no effect, a small effect and a clinically relevant one. The Bayesian approach changes the question. Instead of testing against the null hypothesis with power you do not have, it quantifies what is probable in light of the available data and prior knowledge, and reports the uncertainty rather than returning a binary verdict. It lets you state which conclusions are compatible with the data and the prior knowledge, and with how much uncertainty.

A Bayesian analysis is the right choice when the sample is necessarily small, there is defensible prior knowledge, or you want a direct probability statement about a hypothesis. It works well in rare diseases, pilot studies and subgroup analyses, where it combines the observed data with explicit prior information. A frequentist approach remains a sound option when the sample is sufficient and prior information is either not wanted or cannot be incorporated. They are tools with different assumptions, and we choose between them based on the question, the design, the available data and the quality of the prior evidence. There is a limit: with very sparse data and no defensible prior information, a Bayesian analysis does not create new information. It will make the uncertainty explicit, but it will not turn an insufficient sample into conclusive evidence. The rationale for the chosen approach is documented as part of the analysis.

The prior is not assumed in secret: it is built from explicit sources, documented and put through a sensitivity analysis that shows how the conclusion would change with different priors. This applies to any Bayesian analysis, where the usual objection is that prior information can dominate or overly condition the result. The methodological answer is traceability. Each prior is specified from published evidence, historical data or structured expert knowledge, with a record of where it comes from. When there is no basis for informing it, weakly informative priors are used by default and that choice is documented. We also check whether the prior is on the same scale as the model parameter and whether its degree of information is consistent with the available evidence. Above all, the prior sensitivity analysis tests whether the conclusion holds under more and less informative assumptions: if it depends heavily on the prior, that is reported, not hidden. A prior that is justified, documented and put through a sensitivity analysis is an explicit assumption that can be evaluated, not a hidden decision.

You need the study data and, where available, the prior information that can inform the priors: earlier studies, registries or structured expert judgement. The client collects the data and provides or validates the expert prior knowledge, as the party that knows the literature and the clinical context; the service specifies the model, builds and justifies the priors, and estimates, diagnoses and interprets. Abundant prior information is not essential. When there is none, we work with weakly informative priors, and the analysis can still be valid, but its conclusions will depend more on the observed data and the uncertainty will be greater. What does help is gathering and honestly assessing what is already known about the problem, because wasting defensible knowledge in a small-sample setting undercuts one of the main advantages of the Bayesian approach. That phase of gathering and justifying the prior evidence is part of the analysis, not a preliminary.

A 95% credible interval can be read as meaning that, given the model, the data and the specified prior, there is a 95% posterior probability that the parameter lies within that range. This applies to any Bayesian result and is one of its advantages for communication. The difference from the frequentist confidence interval is real and often misunderstood: a confidence interval does not allow you to say there is a 95% probability that the parameter lies inside it (a very widespread misreading), whereas a credible interval does allow that direct interpretation. This makes the result easier to communicate to a clinical audience, because it answers the clinical question more directly: which values of the effect are plausible once the data and prior knowledge are combined. It also lets you express the probability that the effect exceeds a clinically relevant threshold, not just that it differs from zero, which is usually the real clinical question.

A Bayesian analysis is defensible when sampling convergence, model adequacy and the sensitivity of the conclusions to the priors have been verified. We check convergence using R-hat, the effective sample size and trace plots, and use posterior predictive checks to see whether the model reproduces relevant features of the observed data. We also test how much the conclusions change under other reasonable prior choices. If the model is too complex for the data, we simplify or reformulate it rather than interpret an unstable posterior distribution. The code, model specification and diagnostics are documented so that the analysis can be reproduced and audited. And uncertainty is shown as it is: with sparse data the intervals will be wide, and that width is information about what the study allows you to conclude.

No. This service focuses on the Bayesian analysis of small-sample studies and the justified incorporation of prior knowledge; it does not cover adaptive designs (interim analyses, stopping rules or adaptive allocation), which are a separate field. The limit is worth stating because Bayesian methods are often associated with those designs, especially in clinical trials. The reason for it is scope: adaptive designs, particularly when they include interim analyses or stopping rules, call for trial planning and a different kind of methodological governance, a boundary that ADI keeps clear. What is included is the analysis of data already collected in a pilot or rare-disease study, or setting up the Bayesian model for those data. If what you need is the adaptive design of a trial, it falls outside this service; if it is drawing defensible Bayesian conclusions from a small sample, it is exactly what this service is for.

We deliver the Bayesian analysis agreed for the study question, interpreted and documented: the estimate with its credible intervals, the direct probabilities about the clinical hypotheses, the convergence diagnostics, the prior sensitivity analysis and the posterior distribution plots that make the result readable for a clinical audience. This applies if you need a defensible result for a publication or to inform a clinical or research decision with limited data. It includes drafting the Bayesian methods section in line with the applicable reporting guidelines, where they exist for the design and context of the study, so that the analysis is ready to go into the manuscript. The whole process (model, priors, diagnostics) is documented so that it is reproducible and auditable. The research team authors and leads the publication; the service provides the analysis, its interpretation and the methodological justification behind it.