## Bayesian methods

Bayesian methods are not a form of alternative clinical trial design, but rather a different method of inference. While conventional frequentist statistical methods have been used almost universally in the analysis of scientific studies, they have major limitations when used to provide inference in clinical trials. First, they do not provide a clinically applicable answer to the question clinicians find most relevant: What is the probability that the alternate hypothesis is true? Second, they lead to erroneous binary thinking regarding an intervention. Third, they are often misused, misunderstood, and, thus, misinterpreted.26 Finally, they ignore the prior evidence about the intervention, leaving the reader on their own to integrate the results of the study with the results of prior studies.

As a community, we have evolved to reporting the results of a clinical trial as positive or negative (p<0.05 vs. p>0.05), rather than interpreting the more precise measure of inference in its continuous form (eg, p=0.07). This has become common because p values are difficult to interpret and even more difficult to apply clinically. For example, in a frequentist analysis, the alternative hypothesis might be: Patients in hemorrhagic shock who receive drug A will have at least 5% reduction in mortality at 30 days compared with those who do not receive drug A. The null hypothesis will be that there is no difference in 30-day survival. In a frequentist analysis the null hypothesis is assumed to be true and the p value can only provide information relative to this assumption. The p value is the probability of finding the observed data, or more extreme, assuming that the null hypothesis is true. The p value that is generated cannot estimate the probability of the alternative hypothesis being true because the null and the alternative cannot be true at the same time.27 Bayesian methods bridge this gap by providing the more clinically meaningful outcome, the posterior probability, which clinicians can use to perform risk-benefit assessment of an intervention. Simply put, Bayesian methods speak the clinician’s language.

Bayesian methods are similar to how surgeons approach everyday clinical scenarios. When making decisions, we use our experience, prior knowledge, and training to assess the probability that one of many treatments will yield the outcome of interest. We then perform that treatment and assess the outcomes. If the treatment is successful, we increase our probability that the treatment is appropriate in that clinical scenario. If the treatment is unsuccessful, we decrease our probability that the treatment was the best option.

Bayesian methods weigh the degree of uncertainty regarding the effect size of the treatment and combine that with the probability the treatment will be beneficial or harmful. Bayesian analyses explicitly incorporate prior knowledge into the estimates of the probability of treatment effect before the study (the prior probability). When mathematically formalized, Bayes’ theorem provides an optimized rule for updating these prior probabilities with newly observed data. The result is a posterior probability distribution of treatment effect, quantifying the plausibility of both the null and alternate hypotheses (figure 2). This posterior probability statement might take the form: ‘The chance that treatment confers benefit of some magnitude or higher is X%.’ This posterior probability is more intuitive than the analogous interpretation of a p value: ‘Assuming that the null hypothesis regarding treatment is true, the chance of observing the current data, or data more extreme is Y%.’ Note that the former approach provides a direct assessment of the alternative hypothesis (the reason the clinician is engaging in a given course of action). The latter approach makes no direct mention of the alternative hypothesis as frequentist methods result in a non-comparative value that is only in relation to the null hypothesis.

Figure 2Graphical representation of a hypothetical study using Bayesian methods. Consider a trial of two treatments in which the rate of mortality was 34% (112/331) in treatment A and 40% (134/333) in treatment B. Frequentist inference would provide a risk ratio of 0.84 (95% CI 0.69 to 1.03, p=0.09). The result would be stated that no statistically significant difference between these two interventions was observed. In contrast, a Bayesian analysis, using a vague, neutral prior, would provide a risk ratio of 0.86% and 95% credible interval of 0.70–1.04. Plotting this posterior distribution would result in an area under the curve to the left of 1 (ie, decreased mortality) of 94% of the entire distribution. The result would be stated as such: there was a 94% probability that treatment A reduced mortality compared with treatment B.

In addition to providing clinically applicable answers to the question posed, Bayesian methods have other advantages in clinical trials. First, the degree of uncertainty of the treatment effect is built into the posterior probability. Second, prior knowledge is explicitly included into the estimates of the probability of treatment effect. This inclusion allows for iterative updating of the posterior probability. Third, interim analyses under the frequentist method inflate the overall type I error rate requiring sequential methods to address (‘alpha spending’). Bayesian methods have no such penalty and allow for a monitoring schedule at any stage and with any cohort size.

The main disadvantage to using Bayesian methods is the determination of the prior probability. When no data exist, the choice of a prior can be subjective and greatly affect the posterior probability. Ideally, the prior is based on already published high-quality studies when available. In scenarios where no prior data exist, conservative analyses using neutral priors with a range limited to plausible effect sizes may be used, another advantage of Bayesian methods. Alternatively, Bayesian methods permit a sensitivity analysis whereby a range of plausible prior probabilities are provided and the clinician can consider the prior they think credible.