Bayesian Residual Analysis for Binary Response Regression Models

In a binary response regression model, classical residuals are diicult to deene and interpret due to the discrete nature of the response variable. In contrast, Bayesian residuals have continuous-valued posterior distributions which can be graphed to learn about outlying observations. Two deenitions of Bayesian residuals are proposed for binary regression data. Plots of the posterior distributions of the basic (observed-tted) residual can be helpful in outlier detection. Alternatively, the notion of a tolerance random variable can be used to deene latent data residuals that are a function of the tolerance random variables and the parameters. In the probit setting, these residuals are attractive in that apriori they are a sample from a standard normal distribution, and therefore the corresponding posterior distributions are easy to interpret. These residual deenitions are illustrated in examples and contrasted with classical outlier detection methods for binary data.