  In failure-time settings, a competing risk event is any event that makes it
impossible for the event of interest to occur. For example, cardiovascular
disease death is a competing event for prostate cancer death because an
individual cannot die of prostate cancer once he has died of cardiovascular
disease. Various statistical estimands have been defined as possible targets of
inference in the classical competing risks literature. Many reviews have
described these statistical estimands and their estimating procedures with
recommendations about their use. However, this previous work has not used a
formal framework for characterizing causal effects and their identifying
conditions, which makes it difficult to interpret effect estimates and assess
recommendations regarding analytic choices. Here we use a counterfactual
framework to explicitly define each of these classical estimands. We clarify
that, depending on whether competing events are defined as censoring events,
contrasts of risks can define a total effect of the treatment on the event of
interest, or a direct effect of the treatment on the event of interest not
mediated through the competing event. In contrast, regardless of whether
competing events are defined as censoring events, counterfactual hazard
contrasts cannot generally be interpreted as causal effects. We illustrate how
identifying assumptions for all of these counterfactual estimands can be
represented in causal diagrams in which competing events are depicted as
time-varying covariates. We present an application of these ideas to data from
a randomized trial designed to estimate the effect of estrogen therapy on
prostate cancer mortality.
