# Causal mediation effect decomposition when you have multiple mediators

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?

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

• 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