# Bayesian inference of a clinical trial for clinicians

I am a clinician who is more adept than average at interpreting clinical trials in a frequentist manner. At this point, interpreting a trial as a frequentist has kind of become a procedure: check internal validity, check null and alternative hypotheses, check power assumptions, look at effect size and confidence intervals, look at p value, etc.

The Bayesian philosophy appeals more to me intuitively, though. I understand what the philosophy of Bayesian inference is because I've taken clinical epidemiology (pre-test probability of disease is updated with test results and likelihood ratio to produce a post-test probability). What I don't know yet is whether there's a similar "procedure" for interpreting a clinical trial the way there is for a frequentist.

What numbers/assumptions/figures do I need from the authors to interpret a study as a Bayesian (e.g. for a frequentist that would be p-value, confidence intervals, etc)? Is there a good "procedure" for interpreting a study as a Bayesian, much like there is for a frequentist interpretation? Are there good publications you know of that explain the process I'm asking for, with clinicians as the intended audience?

I appreciate your help!

## 1 Answer

This is quite a large topic. The short technical answer to your question is that to compute an exact posterior distribution one needs the raw data from the clinical trial. But the normal approximation (Gaussian prior/Gaussian data model) may get you close enough some of the time. For that see for example http://rgm3.lab.nig.ac.jp/RGM/R_rdfile?f=Hmisc/man/gbayes.Rd&d=R_CC .

• Thanks for answering, Dr. Harrell. A senior physician explained his rough approach to interpreting a trial from a Bayesian perspective, can you tell me if it's valid (at least in concept, if not mathematically)? If you consider a trial a diagnostic "test", then power would be your sensitivity for a "True" result (usually 0.8), and your false positive rate would be your trial's p-value (let's say .11). Therefore the likelihood ratio of the trial is 7.3. If before the trial your prior odds that the drug was effective was 2:1, your posterior odds of the drug being efficacious are now 15:1. – JJM Sep 5 '14 at 15:54
• That is technically true but "prior odds the drug was effective" is not a well-defined concept because it neglects consideration of "how effective". It is more sensible to describe the state of prior knowledge with a continuous prior distribution that may have more mass at zero effect than in other places but that allows for all kinds of possibilities. The analogy is that anyone considering a diagnostic test as + or - is oversimplifying a truly underlying continuous test outcome. – Frank Harrell Sep 5 '14 at 19:08
• The Lehmann paper about a web-based tool for readers of medical literature to calculate posterior probabilities from trial data is a fantastic idea. To your knowledge, has anyone implemented such a tool for a Bayesian trial? – JJM Sep 7 '14 at 15:52
• That same tool can be used for a Bayesian trial; it's just that for a Bayesian trial you wouldn't need to approximate the posterior distribution - you could calculate it exactly. More general software could handle the exactly calculations - I seem to remember a method somewhere for re-weighting posterior draws when you change the prior so that simulations would not have to be re-run. – Frank Harrell Sep 8 '14 at 1:13
• No more questions for me, but I just wanted to share this Annals article with you: ncbi.nlm.nih.gov/pubmed/?term=10383350 . While I can't speak to the mathematics of it, I think the concept of the minimum Bayes Factor is a good bridge for clinicians from frequentism, because many more clinicians are familiar with the use of likelihood ratios for diagnostic tests. Just thought you might be interested in sharing it with some of your clinical colleagues. – JJM Sep 9 '14 at 3:06