Bayesian residual analysis for binary response regression models

In a binary response regression model, classical residuals are difficult to define 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 definitions of Bayesian residuals are proposed for binary regression data. Plots of the posterior distributions of the basic 'observed-fitted' residuals can be helpful in outlier detection. Alternatively, the notion of a tolerance random variable can be used to define latent data residuals that are functions of the tolerance random variables and the parameters. In the probit setting, these residuals are attractive in that a priori they are a sample from a standard normal distribution, and therefore the corresponding posterior distributions are easy to interpret. These residual definitions are illustrated in examples and contrasted with classical outlier detection methods for binary data.