Bayesian statistical methods

In this week's BMJ, Lilford and Braunholtz (p 603) explain the basis of Bayesian statistical theory.1 They explore its use in evaluating evidence from medical research and incorporating such evidence into policy decisions about public health. When drawing inferences from statistical data, Bayesian theory is an alternative to the frequentist theory that has predominated in medical research over the past half century. As explained by Lilford and Braunholtz, the main difference between the two theories is the way they deal with probability. Consider a clinical trial comparing treatments A and B. Frequentist analysis may conclude that treatment A is superior because there is a low probability that such an extreme difference would have been observed when the treatments were in fact equivalent. Bayesian analysis begins with the observed difference and then asks how likely is it that treatment A is in fact superior to B. In other words, frequentists deduce the probability of observing an outcome given the true underlying state (in this case no difference between treatments), while Bayesians induce the probability of the existence of the true but as yet unknown underlying state (in this case, A is superior to B) given the data. The difference is quite profound, and, although the conclusions reached by applying the two methods may be qualitatively the same, the mode of expressing those conclusions will always be different. For example, a frequentist may conclude that the difference between treatments A and B is highly significant (P = 0.002), meaning that the chance of observing such an extreme difference when A and B are in fact …