  We propose a novel methodology to quantify the effect of stochastic
interventions on non-terminal time-to-events that lie on the pathway between an
exposure and a terminal time-to-event outcome. Investigating these effects is
particularly important in health disparities research when we seek to quantify
inequities in timely delivery of treatment and its impact on patients survival
time. Current approaches fail to account for semi-competing risks arising in
this setting. Under the potential outcome framework, we define and provide
identifiability conditions for causal estimands for stochastic direct and
indirect effects. Causal contrasts are estimated in continuous time within a
multistate modeling framework and analytic formulae for the estimators of the
causal contrasts are developed. We show via simulations that ignoring censoring
in mediator and or outcome time-to-event processes, or ignoring competing risks
may give misleading results. This work demonstrates that rigorous definition of
the direct and indirect effects and joint estimation of the outcome and
mediator time-to-event distributions in the presence of semi-competing risks
are crucial for valid investigation of mechanisms in continuous time. We employ
this novel methodology to investigate the role of delaying treatment uptake in
explaining racial disparities in cancer survival in a cohort study of colon
cancer patients.
