Placing Trials in Context Using Bayesian Analysis. JAMA, March 15, 1995, pp. 871-875 James M. Brophy and Lawrence Joseph
The authors point out that the standard use of p-values and confidence intervals to judge the outcome of a medical trial often lead a doctor to use a drug that does not take into account information from previous studies. They argue that a Bayesian analysis can be used to take into account previous information and can lead to a different decision about the effectiveness of a drug. The authors illustrate this in terms of a recent study called the GUSTO study that compared two different treatments, tissue-type plasminogen activator (t-PA) and streptokinase (SK) for heart attacks. The study involved 41,021 patients and t-PA had a statistically significant lower mortality rate (6.3% vs 7.3%, respectively; p = .001). Since t-PA costs about $2000 and SK only $200, and two similar previous studies indicated no significant difference, some doctors have been hesitant to use the more expensive treatment despite the latest significant result. A Bayesian analysis is carried out by choosing a prior distribution for difference in mortality rates for the two drugs. The authors suggest that people might differ in how they weight the previous studies. They give three prior distributions for the mortality difference corresponding to weighting the previous studies by 10%, 50%, or 100%. For each of these three prior distrib- utions they obtain the posterior distribution for the difference in the two drugs given the results of the GUSTO trial. From the posterior distribution for the mortality difference, it is possible to calculate the probability that t-PA has a lower mortality rate than SK. In addition, it is possible to calculate the probability that one drug is "clinically superior". In this example the authors say that a drug is clinically superior if the mortality rate is at least 1% lower. The 50% weighting of the previous studies leads to a 44% probability that t-PA has a lower mortality rate and a negligible probability that the difference is clinically significant. Ignoring previous results altogether, t-PA has a very high probability (99.95%) of having a lower mortality rate but still only a 48% chance of being clinically significant. Thus, a wide range of weightings of the previous results all provide some justification for not immediately switching to the more expensive drug. DISCUSSION QUESTION. You have a coin that you initially assume is a fair coin. Jones tosses the coin ten times and gets nine heads. You then toss the coin ten times and get five heads. You want to find the posterior probability that the coin is a fair coin weighting Jones' tosses fifty percent. How would you do this?