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In this chapter, we have described the fundamentals of the classic frequentist statistics. Over the last few years, several studies have employed the Bayesian approach in the design of randomized clinical trials. Indeed, the US.Food and Drug Administration has recently recognized that Bayesian analyses could enable smaller trials to be conducted without reduction in the validity of the results.

Conceptually, these approaches are complementary, yet provide distinct interpretations to the underlying hypothesis being tested. The most frequent of these estimates the probability of detecting a difference as large or larger than what was observed under the assumption of the null hypothesis. This is not immediately intuitive, while the Bayesian approach provides a probability estimate for the alternative hypothesis being true. Anytime a frequentist approach is used, it is necessary to take repeated experiments into consideration: in 95% of these experiments, the observed results will be in the range expressed by the confidence interval. Conversely, Bayesian statistics is strictly connected with conditional probability: Bayes’ theorem says that we can set up a prior information – based on our knowledge – and then calculate posterior probabilities of this prior information. These prior probabilities are updated through an iterative process of data collection. Moreover, they should incorporate information from all relevant research before we perform the new experiment. Of note, as aforementioned, the frequentist statistical approach uses results from completed studies to design a new study but does not formally incorporate these data in the evaluation of the new study hypotheses. Unlike the frequentist approach, Bayesian methods allow prior information to be synthesized with new data.

An example is represented by the Biotronik Prospective Randomized Multicenter Study to Assess the Safety and Effectiveness of the Orsiro Sirolimus Eluting Coronary Stent System in the Treatment of Subjects with Up to Three De Novo or Restenotic Coronary Artery Lesions V (BIOFLOW V) trial.[11] In this study, the authors took advantage of Bayesian methods to synthesize prior information with new data: data from the similarly designed BIOFLOW II [12] and IV [13] (same treatment arms, primary endpoint definition, follow‐up and event adjudication, and very similar inclusion criteria) were quantitatively combined with those of the BIOFLOW V using the Bayes’ theorem, in order to provide a more precise estimation of the clinical performance of the Orsiro and Xience drug‐eluting stents. However, it is important to note that this approach may be feasible and result in a smaller sample size only if high quality data from very similar studies are available. In the Surgical Replacement and Transcatheter Aortic Valve Implantation (SURTAVI) trial [14], the Bayesian analytic method was applied to elaborate a statistically reliable predictive model of the primary endpoint when all patients were enrolled and a sufficient number had reached two year follow‐up (Figure 6.4). Late events in patients at an earlier follow‐up phase were predicted by leveraging data from patients that had already completed a follow‐up. Similar to the BIOFLOW V study, the use of Bayesian methods in the SURTAVI trial required a homogeneous population as different features between patients enrolled in the early and late enrolment phases would have led to unreliable predictive models [15].

Figure 6.4 Graphic explanation of the results from the SURTAVI trial, reporting posterior probability distribution for the difference in the primary end point (death from any cause or disabling stroke at 24 months).

These models must recognize the uncertainty in the prior information and that differences might exist between posterior and prior evidence. Indeed, the most important limitation of this approach is that if the prior is wrong, all the following steps of the inductive process will be wrong.