  Structural failure time models are causal models for estimating the effect of
time-varying treatments on a survival outcome. G-estimation and artificial
censoring have been proposed to estimate the model parameters in the presence
of time-dependent confounding and administrative censoring. However, most of
existing methods require manually preprocessing data into regularly spaced
data, which may invalidate the subsequent causal analysis. Moreover, the
computation and inference are challenging due to the non-smoothness of
artificial censoring. We propose a class of continuous-time structural failure
time models, which respects the continuous time nature of the underlying data
processes. Under a martingale condition of no unmeasured confounding, we show
that the model parameters are identifiable from potentially infinite estimating
equations. Using the semiparametric efficiency theory, we derive the first
semiparametric doubly robust estimators, in the sense that the estimators are
consistent if either the treatment process model or the failure time model is
correctly specified, but not necessarily both. Moreover, we propose using
inverse probability of censoring weighting to deal with dependent censoring. In
contrast to artificial censoring, our weighting strategy does not introduce
non-smoothness in estimation and ensures that the resampling methods can be
used to make inference.
