Hypothesis test for causal mediation of time-to-event mediator and outcome

Hepatitis B has been a well-documented risk factor of liver cancer and mortality. To what extent hepatitis B affects mortality through increasing liver cancer incidence is of research interest and remains to be studied. We formulate the research question as a hypothesis testing problem of causal mediation where both the mediator and the outcome are time-to-event variables. The problem is closely related to semicompeting risks because time to the intermediate event may be censored by an occurrence of the outcome. We propose two hypothesis testing methods: a weighted log-rank test (WLR) and an intersection-union test (IUT). A test statistic of the WLR is constructed by adapting a nonparametric estimator of the mediation effect; however, the test may be conservative regarding its Type I Error rate. To address this, we further propose the IUT, the test statistic of which is constructed under the composite null hypothesis. Asymptotic properties of the two tests are studied, showing that the IUT is a size $$ test with better statistical power than the WLR. The theoretical properties are supported by extensive simulation studies under finite samples. Applying the proposed methods to the motivating hepatitis study, both WLR and IUT provided strong evidence that hepatitis B had a significant mediation effect on mortality via liver cancer incidence.