  Many research questions involve time-to-event outcomes that can be prevented
from occurring due to competing events. In these settings, we must be careful
about the causal interpretation of classical statistical estimands. In
particular, estimands on the hazard scale, such as ratios of cause specific or
subdistribution hazards, are fundamentally hard to be interpret causally.
Estimands on the risk scale, such as contrasts of cumulative incidence
functions, do have a causal interpretation, but they only capture the total
effect of the treatment on the event of interest; that is, effects both through
and outside of the competing event. To disentangle causal treatment effects on
the event of interest and competing events, the separable direct and indirect
effects were recently introduced. Here we provide new results on the estimation
of direct and indirect separable effects in continuous time. In particular, we
derive the nonparametric influence function in continuous time and use it to
construct an estimator that has certain robustness properties. We also propose
a simple estimator based on semiparametric models for the two cause specific
hazard functions. We describe the asymptotic properties of these estimators,
and present results from simulation studies, suggesting that the estimators
behave satisfactorily in finite samples. Finally, we re-analyze the prostate
cancer trial from Stensrud et al (2020).
