Nonparametric estimation of the joint and conditional survival functions of the time to an event of interest and associated integrated covariate processes

Abstract

During the treatment of chronic diseases, a covariate process may exhibit a response over the treatment period and may also be associated with the event of interest. The integrated covariate process commonly known as the area under the response curve, over a specific treatment interval is a widely used measure of a cumulative treatment response in medical research. Our interest is in the survival probability that the time to the event is larger than a fixed time t and the cumulative response up to that time is larger than a fixed value y. Because the cumulative response grows with time, the latter event of the survival probability may realise at the random time when the integrated process crosses y. A common censoring mechanism may censor both these random times. In fact, our setting gives rise to a different type of censoring mechanism of the bivariate event times. In order to handle the censoring and to use all available information, we study inverse-probability of censoring weighted estimators of the survival probability. The estimators and their efficiency differ according to the use of the information in estimation of the censoring distribution. We also give a pooled estimator of the bivariate survival function in a stratified analysis. Further, we study the conditional survival function of the time to the event given the history of the covariate process and discuss its use in the medical decision making. We suggest jackknife method for estimating variances for all the proposed estimators. The proposed estimators can be easily generalised to more than one covariate processes. We illustrate our methodology using two sets of data on the age-related macular degeneration of eye disease; one from a clinical trial and the other from a real-world study.