Separable pathway effects of semi-competing risks using multi-state sodels

Semi-competing risks refer to the phenomenon where a primary event (such as mortality) can ``censor'' an intermediate event (such as relapse of a disease), but not vice versa. Under the multi-state model, the primary event consists of two specific types: the direct outcome event and an indirect outcome event developed from intermediate events. Within this framework, we show that the total treatment effect on the cumulative incidence of the primary event can be decomposed into three separable pathway effects, capturing treatment effects on population-level transition rates between states. We next propose two estimators for the counterfactual cumulative incidences of the primary event under hypothetical treatment components. One estimator is given by the generalized Nelson--Aalen estimator with inverse probability weighting under covariates isolation, and the other is given based on the efficient influence function. The asymptotic normality of these estimators is established. The first estimator only involves a propensity score model and avoid modeling the cause-specific hazards. The second estimator has robustness against the misspecification of submodels. As an illustration of its potential usefulness, the proposed method is applied to compare effects of different allogeneic stem cell transplantation types on overall survival after transplantation.