not go through any of the above pathways.\

We are now ready to define the cause–$j$–specific natural effect proportional hazard model, accounting for longitudinal mediators, as follows: $$\begin{aligned}
\lambda^j_{a, a^*}(t) = \lambda_0^j(t)e^{\alpha_{1j}a + \alpha_{2j}a^*}\end{aligned}$$ for all $a,a^*$, where $\lambda_0^j(t), \alpha_{1j}$ and $\alpha_{2j}$ are unknown. Under model $(1)$, the total, direct and indirect effect can be expressed on the hazard ratio scale as $HR^j_{TE} = e^{(\alpha_{1j}+\alpha_{2j})(a^* - a)}$; $HR^j_{IE} = e^{\alpha_{2j}(a^* - a)}$ and $HR^j_{DE} = e^{\alpha_{1j}(a^* - a)}$, respectively. To assess the possibility of mediator-exposure interaction, one can alternatively consider model: $$\label{eq:model}
\lambda^j_{a, a^*}(t) = \lambda_0^j(t)e^{\alpha_{1j}a + \alpha_{2j}a^* + \alpha_{3j}a.a^*}$$ Under model (2), the total causal effect on the hazard ratio scale is expressed as $HR^j_{TE} = e^{\left(\alpha_{1j}+\alpha_{2j} + \alpha_{3j}(a^*+a))\right)(a^* - a)}$ and is decomposed into the indirect effect $ HR^j_{IE} = e^{(\alpha_{2j} + \alpha_{3j}a)(a^* - a)}$ and the direct effect $HR^j_{DE} = e^{(\alpha_{1j} + \alpha_{3j}a^*)(a^* - a)}$. Finally, note that other models for survival outcomes, such as the Aalen model, can also be extended to the current context.

Estimation
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To estimate the parameters in model $(1)$, we will assume that the set of baseline covariates $L_0$ is sufficient to control for confounding of the relationship between $A$ and 