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Suppose we have outcome variable $Y,$ one treatment $T,$ and two mediator variables, $M_{1}$ and $M_{2}.$ We write the structural model as: $$ Y=\beta_{0}+\beta_{1}T+\beta_{2}M_{1}+\beta_{3}M_{2}+\epsilon $$ $$ M_{1}=\alpha+\alpha_{1}T+v $$ $$ M_{2}=\gamma+\gamma_{1}T+e $$ Here, the direct effect of $T$ is straightforward, which is $\beta_{1}$. Is the total effect $\beta_{1}+\beta_{2}\alpha_{1}+\gamma_{1}\beta_{3}?$ Also, is the causal medion effect is for $M_{1}$ given $\beta_{2}\alpha_{1}$ while that for $M_{2}$ given as $\gamma_{1}\beta_{3}?$ Does it make sense to have two different mediation effects here?

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  • $\begingroup$ Can you please define: i) "total effect"; ii) "causal medion effect"? $\endgroup$ Commented May 7, 2022 at 19:27

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Based on the model that you have specified you are correct

  • Direct effect: $\beta_1$
  • Indirect effect through $M_1$: $\alpha_1\cdot \beta_2$
  • Indirect effect through $M_2$: $\gamma_1\cdot \beta_3$

It makes sense to decompose the indirect effect into a component that is due to each of the 2 mediators. The total indirect effect would be $\alpha_1\cdot \beta_2+\gamma_1\cdot \beta_3$

A detailed introduction to this decomposition can be found in Vanderweele 2014, which also describes how to incorporate a potential $M_1\cdot M_2$ interaction

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