  Significance testing based on p-values has been implicated in the
reproducibility crisis in scientific research, with one of the proposals being
to eliminate them in favor of Bayesian analyses. Defenders of the p-values have
countered that it is the improper use and errors in interpretation, rather than
the p-values themselves that are to blame. Similar exchanges about the role of
p-values have occurred with some regularity every 10 to 15 years since their
formal introduction in statistical practice. The apparent contradiction between
the repeated failures in interpretation and continuous use of p-values suggest
that there is an inferential value in the computation of these values. In this
work we propose to attach a radical Bayesian interpretation to the number
computed and reported as a p-value for the Generalized Linear Model, which has
been the workhorse of applied statistical work. We introduce a decision
analytic framework for thresholding posterior tail areas (pi-values) which for
any given Bayesian analysis will have a direct correspondence to p-values in
non-Bayesian approaches. Pi-values are non-controversial, posterior probability
summaries of treatment effects. A predictive probability argument is made to
justify the exploration of the stochastic variation (replication probability)
of p and pi-values and culminates into a concrete proposal for the synthesis of
Likelihood and Bayesian approaches to data analyses that aim for
reproducibility. We illustrate these concepts using the results of recent
randomized controlled trials in cardiometabolic and kidney diseases and provide
R code for the implementation of the proposed methodology.
