A framework for consistent prediction rules based on markers

Abstract

SUMMARY Recently interest has developed regarding the statistical properties and uses of marker processes in the context of the analysis of failure time data or survival analysis. A marker process is a stochastic process that acts as a time dependent covariate that is internal to the unit under study in the language of Kalbfleisch & Prentice (1980). As such the sample path of the process up to a certain point in time may carry information about the subsequent hazard for failure. Uses of marker processes in the analysis of survival data are manifold. Here we consider the specific area of prediction of future failure times at a point in time based on various forms of information about the history of the marker process. We provide a stochastic framework for the consideration of prediction functions, demonstrate a simple consistency condition that such functions should satisfy, and discuss construction of prediction functions in a general sense. Several examples are used to illustrate the ideas and we show that certain recently suggested imputation schemes fail to meet the consistency condition. The consistency condition also elucidates the model assumed by Cox (1983) in his work on surrogate responses. We also briefly consider a closely related backward prediction problem.