Adaption of bivariate frailty models for prediction, with application to biological markers as prognostic indicators

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Abstract
Serial biological markers are often used in medicine as prognostic indicators. For example, elevated tumour antigen tests presage tumour recurrence; rapid decrease in the CD4 cell counts of HIV + individuals indicate high risk for developing AIDS. Methods for evaluating the usefulness of markers should address how the prediction of future risk is revised, in a way that assists the making of decisions at the time such an indicator is observed. Proportional hazards modelling with time-varying covariates, a common approach, seems unnatural for this purpose because its orientation is explanatory; it addresses the question ‘Is the individual suffering a failure more likely to have had the marker?’, and allows use of all information prior to the failure, rather than the more relevant predictive question ‘Does the appearance of the marker alter the individual's subsequent risk enough to warrant intervention today?’. Frailty models do address this question, but available methods for their analysis make censoring assumptions which are inappropriate in the predictive setting. We adapt frailty models to relevant censoring processes, and propose a predictively oriented test statistic and a related estimation procedure based on summing concurrent martingales, each representing a ‘landmark’ analysis.

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