Bayesian probability of predictive agreement for comparing the outcome of two separate regressions

The concept of a Bayesian probability of agreement was recently introduced to give the posterior probabilities that the response surfaces for two different groups are within δ of one another. For example, a difference of less than δ in the mean response at fixed levels of the predictor variables might be thought to be practically unimportant. In such a case, we would say that the mean responses are in agreement. The posterior probability of this is called the Bayesian probability of agreement. In this article, we quantify the probability that new response observations from two groups will be within δ for a continuous response, and the probability that the two responses agree completely for categorical cases such as logistic regression and Poisson regression. We call these Bayesian comparative predictive probabilities, with the former being the predictive probability of agreement. We use Markov chain Monte Carlo simulation to estimate the posterior distribution of the model parameters and then the predictive probability of agreement. We illustrate the use of this methodology with three examples and provide a freely available R Shiny app that automates the computation and estimation associated with the methodology.