Simple nested Bayesian hypothesis testing for meta-analysis, Cox, Poisson and logistic regression models

Many would probably be content to use Bayesian methodology for hypothesis testing, if it was easy, objective and with trustworthy assumptions. The Bayesian information criterion and some simple bounds on Bayes factor are closest to fit this bill, but with clear limitations. Here we develop an approximation of the so-called Bayes factor applicable in any bio-statistical settings where we have a d-dimensional parameter estimate of interest and the d x d dimensional (co-)variance of it. By design the approximation is monotone in the p value. It it thus a tool to transform p values into evidence (probabilities of the null and the alternative hypothesis, respectively). It is an improvement on the aforementioned techniques by being more flexible, intuitive and versatile but just as easy to calculate, requiring only statistics that will typically be available: e.g. a p value or test statistic and the dimension of the alternative hypothesis.