  The generalized g-formula can be used to estimate the probability of survival
under a sustained treatment strategy. When treatment strategies are
deterministic, estimators derived from the so-called efficient influence
function (EIF) for the g-formula will be doubly robust to model
misspecification. In recent years, several practical applications have
motivated estimation of the g-formula under non-deterministic treatment
strategies where treatment assignment at each time point depends on the
observed treatment process. In this case, EIF-based estimators may or may not
be doubly robust. In this paper, we provide sufficient conditions to ensure
existence of doubly robust estimators for intervention treatment distributions
that depend on the observed treatment process for point treatment
interventions, and give a class of intervention treatment distributions
dependent on the observed treatment process that guarantee model doubly and
multiply robust estimators in longitudinal settings. Motivated by an
application to pre-exposure prophylaxis (PrEP) initiation studies, we propose a
new treatment intervention dependent on the observed treatment process. We show
there exist 1) estimators that are doubly and multiply robust to model
misspecification, and 2) estimators that when used with machine learning
algorithms can attain fast convergence rates for our proposed intervention.
Theoretical results are confirmed via simulation studies.

Methodology;Applications