Probability-Scale Residuals for Continuous, Discrete, and Censored Data

• Activity Number: 80
• Type: Contributed
• Date/Time: Sunday, August 9, 2015 : 4:00 PM to 5:50 PM
• Sponsor: Biometrics Section
• Author(s): Bryan Shepherd* and Chun Li and Qi Liu
• Companies: Vanderbilt University and Case Western Reserve University and Vanderbilt University
• Keywords: nonparametric statistics ; ordinal data ; residuals ; diagnostics ; quantile regression ; rank statistics

Abstract

We propose a new residual for general regression models, defined as pr(Y*< y)-pr(Y*>y), where y is the observed outcome and Y* is a random variable from the fitted distribution. This probability-scale residual can be written as E{sign(y,Y*)} whereas the popular observed-minus-expected residual can be thought of as E(y-Y*). Therefore, the probability-scale residual is useful in settings where differences are not meaningful or where the expectation of the fitted distribution cannot be calculated (e.g., ordered categorical or right-censored outcomes). The residual has several desirable properties that make it better for diagnostics than traditional residuals in certain situations. We demonstrate its utility for continuous, ordered discrete, and censored outcomes, and compare it with observed-minus-expected, deviance, Pearson, and martingale residuals. We also demonstrate estimation of probability-scale residuals with quantile regression and semi-parametric models, where fully specified distributions are not desirable or needed. The residual is illustrated with simulations and data from HIV-infected patients starting therapy in Latin America.