A Comparison of Schoenfeld and Generalized Residuals in Survival Analysis

Abstract

Proportional hazards models are used extensively in the analysis of data that represent time-to-event responses, such as survival. Proportional hazards models may be formulated on the basis of either fully specified parametric distributions, or as semi-parametric forms in which the underlying survival distribution is not specified, known as Cox proportional hazards models. This creative component examines the use of several types of residuals that can be used in the assessment of proportional hazards models. One type of residuals, called Shoenfeld residuals, are most often used in conjunction with Cox models and do not require distributional information. Schoenfeld residuals are intended to assess the proportional hazards assumption without concern for other aspects of model assessment. Another type of residual, sometimes called generalized residuals, require specification of a distributional form for the observed responses. Here, we extend the usual definition of generalized residuals to allow the use of censored observations. Generalized residuals constitute an omnibus model assessment, taking into account both distributional forms and specification of the effects of covariates on expected values (through the hazard function for proportional hazards models). We compare the performance of Schoenfeld and generalized residuals using simulated examples from models that do, and do not, follow the proportional hazards assumption. Those examples are then extended to a small Monte Carlo exercise.