Causal mediation analysis for time-to-event mediator and outcome in the presence of left truncation.

We propose a causal mediation approach to semi-competing risks under left truncation sampling by considering an intermediate event as a mediator and a terminal event as an outcome. We focus on the causal relationship from exposure to the terminal outcome in relation to the intermediate event. In particular, we study the direct effect, the effect of exposure on the terminal event that is not through the intermediate event, and the indirect effect-the effect of exposure on the terminal event that is mediated through the intermediate event. We propose nonparametric and semiparametric methods, both accounting for left truncation. The nonparametric estimator can be viewed as a model-free time-varying Nelson-Aalen estimator that is robust to model misspecification. The semiparametric estimator calculated with the Cox proportional hazards model enjoys flexibility in adjusting for potential confounders as covariates. The asymptotic properties for both estimators, including uniform consistency and weak convergence, were established using the martingale theorem and functional delta method. The finite sample performance of the proposed estimators was evaluated through extensive numerical studies that investigated the influences of left truncation, confounding, and sample size. The utility of the proposed methods was illustrated using a hepatitis study.