  In observational studies of survival time featuring a binary time-dependent
treatment, the hazard ratio (an instantaneous measure) is often used to
represent the treatment effect. However, investigators are often more
interested in the difference in survival functions. We propose semiparametric
methods to estimate the causal effect of treatment among the treated with
respect to survival probability. The objective is to compare post-treatment
survival with the survival function that would have been observed in the
absence of treatment. For each patient, we compute a prognostic score (based on
the pre-treatment death hazard) and a propensity score (based on the treatment
hazard). Each treated patient is then matched with an alive, uncensored and
not-yet-treated patient with similar prognostic and/or propensity scores. The
experience of each treated and matched patient is weighted using a variant of
Inverse Probability of Censoring Weighting to account for the impact of
censoring. We propose estimators of the treatment-specific survival functions
(and their difference), computed through weighted Nelson-Aalen estimators.
Closed-form variance estimators are proposed which take into consideration the
potential replication of subjects across matched sets. The proposed methods are
evaluated through simulation, then applied to estimate the effect of kidney
transplantation on survival among end-stage renal disease patients using data
from a national organ failure registry.
