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.