Probability-scale residuals for continuous, discrete, and censored data

We describe 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 (PSR) 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 PSR is useful in settings where differences are not meaningful or where the expectation of the fitted distribution cannot be calculated. We present several desirable properties of the PSR that make it useful for diagnostics and measuring residual correlation, especially across different outcome types. We demonstrate its utility for continuous, ordered discrete, and censored outcomes, including current status data, and with various models including Cox regression, quantile regression, and ordinal cumulative probability models, for which fully specified distributions are not desirable or needed, and in some cases suitable residuals are not available. The residual is illustrated with simulated data and real data sets from HIV-infected patients on therapy in the southeastern United States and Latin America. (C) 2016 Statistical Society of Canada