Non-Parametric Mediation Analysis of Non-Markov Illness-Death Model

The illness-death model is widely used to characterize disease progression over time. Previous work focuses either on estimation without causal interpretation or on causal interpretation under a strong Markov assumption where the terminal event depends on the status but not the timing of the intermediate event. To bridge the research gap, we propose a new definition of counterfactual hazard that relaxes the Markov assumption by considering the entire history of the intermediate event. We derive an identification formula that involves an integral with respect to the probability density function of the intermediate event time. Direct and indirect effects refer to the influence of an exposure on the terminal event not mediated by, and mediated through, the intermediate event, respectively. We propose non-parametric kernel estimators for the two effects and study their asymptotic properties. We conduct numerical simulations to examine the proposed estimators' finite-sample performance. Applying the method to a hepatitis study where the Markov assumption is violated, we show that the effect of hepatitis C on mortality is not mediated through septicemia during the first 15 years of follow-up.