I have sometimes heard complaints from collaborators that it will be impossible to have their work published in the mainstream literature unless a p-value is reported. This post is to report yet another counterexample that was recently published; a meta-analysis for the odds of perioperative bleeding complications in patients taking one of several anticoagulant/antiplatelet drugs. In this study¹ (published by Circulation: Arrhythmia and Electrophysiology), the statistical evidence was reported using Bayesian point estimates and credible intervals. The referees had no problem with the absence of p-values. However, they did require some additional explanation of the credible interval relative to the confidence interval, and of the Bayesian approach in general because they felt that these concepts would be less familiar to their readers. Below is the text that we used. Perhaps readers here can help refine it for future work.

Bayesian analysis formalizes the notion of prior evidence about quantities under study, that is, the evidence at hand before an experiment is carried out. In this meta-analysis, the quantities under study are log-odds of perioperative bleeding. Prior evidence may be updated in light of experimental data to yield the posterior evidence. The Bayesian method encodes evidence as a probability distribution. Hence, a prior distribution is specified to reflect prior evidence, and the posterior evidence is summarized using a posterior distribution.

Another complication that arises when we take a Bayesian approach is justification of the selected prior distribution:

The log-odds were each assigned a normal-gamma prior distribution with mean zero, and small (0.1) shape and rate parameters. The resulting prior distribution was heavy-tailed and symmetric about zero. This choice of prior distribution reflected conservative (zero mean) but weak (heavy-tailed) prior evidence regarding the effects of treatment and study on the log-odds of bleeding.

Indeed, the marginal distribution of the log-odds, with these parameter values, has infinite kurtosis. Kurtosis is a measure of the degree to which a distribution is “heavy-tailed”. But, since this manuscript was intended for medical professionals, we opted for a more relevant description:

To further illustrate, suppose that an antiplatelet/anticoagulant therapy is considered “safe” with regard to bleeding complications when the estimated log-odds is zero, and there is strong evidence that the estimate is accurate to within 5 events per 100 patients (i.e., that the log-odds is confined within the interval -0.2-0.2 with high probability). Although the prior estimate is zero, there is weak prior evidence (prior probability < 0.08) that the log-odds is accurate to within 5 events per 100 patients. Hence, there is no basis for a clinical conclusion a priori. This weak prior ensures that evidence brought by experimental data will dominate the posterior evidence.

Log-odds of -0.2 and 0.2 correspond roughly to probabilities 0.45 and 0.55 respectively. Hence, if there were strong prior evidence that the log-odds occurred within this interval, then we would be confident that the prior estimate of the log-odds (zero) was accurate to within 5 patients per 100. However, the prior probability that the log-odds occur within (-0.2, 0.2) is small. We can easily form a Monte Carlo estimate of this probability using R

> # Sample from the prior distribution on the log-odds
> plo <- rnorm(1e5, mean=0, sd=1/sqrt(rgamma(1e5, shape=0.1, rate=0.1)))
> # Estimate the prior probability that the log-odds occur in (-0.2, 0.2)
> mean(plo >= -0.2 & plo < 0.2)
[1] 0.07544

I think presenting the evidence in terms of (credible or confidence) intervals will prevail as the preferred method for reporting scientific results. I another post, I will try to enumerate some of the advantages of this approach over hypothesis and significance testing.

¹Michael L. Bernard, Matthew Shotwell, Paul J. Nietert, and Michael R. Gold. 2012. Meta-Analysis of Bleeding Complications Associated with Cardiac Rhythm Device Implantation. Circulation: Arrhythmia and Electrophysiology. Pre-print, April 24 2012, doi:10.1161/CIRCEP.111.969105.