R-squared for Bayesian regression models∗

The usual definition of R (variance of the predicted values divided by the variance of the data) has a problem for Bayesian fits, as the numerator can be larger than the denominator. We propose an alternative definition similar to one that has appeared in the survival analysis literature: the variance of the predicted values divided by the variance of predicted values plus the variance of the errors. This summary is computed automatically for linear and generalized linear regression models fit using rstanarm, our R package for fitting Bayesian applied regression models with Stan.