Modelling the relationship between continuous covariates and clinical events using isotonic regression.

In a medical study we are often interested in graphically displaying the relationship between continuous variables and clinical events indicating disease progression. Often, it is reasonable to make the minimal assumption that the risk of progression is an arbitrary monotone function of the continuous variable. Sometimes the continuous variable is a disease marker which is recorded longitudinally, and so the goal is to provide a graphical display of the relationship between the hazard for progression and the most recent measurement of the longitudinal marker. For example, we know that for a patient with HIV infection, declining CD4 count is associated with an increased risk of opportunistic infection. The goal of this paper is to extend isotonic regression techniques to failure time data with a continuous covariate, to obtain a non-parametric estimate for the hazard of disease progression as a monotonic function of the continuous variable. We propose two methods for modelling the relationship of the hazard and covariate: the first assumes that the hazard is constant over time, and the second allows the hazard to be an arbitrary function of time. These methods will be applied to graphically display the risk for an AIDS patient of an opportunistic infection as a function of CD4 count. Copyright 2003 John Wiley & Sons, Ltd.