Estimating the Population Survival Function Using Additional Information Recorded Over Time: a Filter Based Approach

Survival studies often collect information about covariates. If these covariates are believed to contain information about the life-times, they may be considered when estimating the underlying life-time distribution. We propose a non-parametric estimator which uses the recorded information about the covariates. Various forms of incomplete data, e.g. right-censored data, are allowed. The estimator is the conditional mean of the true empirical survival function given the observed history, and it is derived using a general filtering formula. Feng & Kurtz (1994) showed that the estimator is the Kaplan–Meier estimator in the case of right-censoring when using the observed life-times and censoring-times as the observed history. We take the same approach as Feng & Kurtz (1994) but in addition we incorporate the recorded information about the covariates in the observed history. Two models are considered and in both cases the Kaplan–Meier estimator is a special case of the estimator. In a simulation study the estimator is compared with the Kaplan–Meier estimator in small samples.