 
Summary: BAYESIAN RESIDUAL ANALYSIS
FOR BINARY RESPONSE
REGRESSION MODELS
Jim Albert
Department of Mathematics and Statistics
Bowling Green State University, Bowling Green, 43403 USA
Siddhartha Chib
Olin School of Business
Washington University, St. Louis 63130 USA
March, 1994
Summary
In a binary response regression model, classical residuals are difficult to
define and interpret due to the discrete nature of the response variable. In con
trast, Bayesian residuals have continuousvalued 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 dis
tributions of the basic ``observed \Gamma fitted'' residual 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 vari
ables and the parameters. In the probit setting, these residuals are attractive
