# Multiple mediators and interaction effects- causal mediation effect

Assume we have a structural equation system with treatment $$T,$$ mediator $$1$$ (discrete) and mediator $$2$$ (continuous). We write this as: $$Y=\beta_{0}+\beta_{1}M_{1}+\beta_{2}M_{2}+\beta_{3}TM_{2}+\beta_{3}T+\epsilon$$ $$M_{1}=\alpha+\alpha_{1}T+v$$ $$M_{2}=\gamma+\gamma_{1}T+e$$

For $$M_{1},$$ the causal mediation effect is easy to obtain- $$\alpha_{1}\beta_{1}.$$ However, given that we have an interaction effect with $$M_{2}$$, would the causal mediation effect be: $$\gamma_{1}\beta_{2}+\gamma_{1}\beta_{3}M_{2}$$

If this is the case, would a reasonable'' estimate of the causal mediation effect be: $$\gamma_{1}\beta_{2}+\gamma_{1}\beta_{3}\mathbb{E}\left[M_{2}\right]$$ where we take the average value of $$M_{2}?$$

• Are you assuming each error ($\epsilon, v, e$) is independent?
– Ben
May 8, 2022 at 21:28
• Yes @Ben, these are indeed independent. May 8, 2022 at 22:09

An important first step of an analysis is define the estimand, or estimation target. When you write, "causal mediation effect", the estimand is left ambiguous. There are several possible terms you could be describing. I'll take you to mean the indirect natural effect.

For convenience, this answer will use counterfactual notation, with an asterisk denoting a counterfactual. With a single mediator, this is defined as $$\mathbb{E}[Y_{t, M_{1}^*}^* - Y_{t, M_{0}^*}^*],$$ for a possible treatment value $$t$$. It is interpreted as giving the causal effect of the treatment that flows through a mediator $$M$$ when the treatment is held at $$t$$ (to block the direct effect).

Your problem has multiple mediators, but their analysis is simplified because neither is a descendant of the other. Thus we can study the effect of each mediator on its own.

For the first mediator, the indirect natural effect is \begin{align*} & \mathbb{E}[Y_{t, M_{1,1}^*}^* - Y_{t, M_{1,0}^*}^*] \\ =\, & \mathbb{E} \left[ \{\beta_0 + \beta_1(\alpha + \alpha_1 + v) + \beta_2M_2 + \beta_3tM_2 + \beta_4t + \epsilon\} - \\ \{\beta_0 + \beta_1(\alpha + v) + \beta_2M_2 + \beta_3tM_2 + \beta_4t + \epsilon\} \right] \\ =\, & \beta_1\alpha_1, \end{align*} for any $$t$$. This is the value you described.

For the second mediator, the indirect natural effect is \begin{align*} & \mathbb{E}[Y_{t, M_{2,1}^*}^* - Y_{t, M_{2,0}^*}^*] \\ =\, & \mathbb{E} \left[ \{\beta_0 + \beta_1 M_1 + \beta_2(\gamma + \gamma_1 + e) + \beta_3t(\gamma + \gamma_1 + e)+ \beta_4t + \epsilon\} - \\ \{\beta_0 + \beta_1M_1 + \beta_2(\gamma + e) + \beta_3t(\gamma + e) + \beta_4t + \epsilon\} \right] \\ =\, & \gamma_1 ( \beta_2 + \beta_3 t) \end{align*} for any $$t$$. Thus the natural indirect effect depends on the treatment value $$t$$. Holding the treatment at $$t=0$$ or at $$t=1$$ will lead to a different effect of the second mediator $$M_2$$.

Each indirect effect can be consistently estimated by plugging in a consistent estimator of the regression coefficients.

• thank you. For the second mediator, as the effect depends on T, the reason I have the expectation is because I wanted a measure of an "average" effect. In that case, could the average effect in fact be written as what I have written? May 9, 2022 at 16:02
• @ChinG You can't -- you've written the term as depending on $m_2$, but the answer shows it actually depends on $t$. You certainly could average the natural indirect effect over the marginal distribution of the treatment $T$, although I think that would complicate the interpretation.
– Ben
May 9, 2022 at 16:36
• Ah yes. Got it! The treatment is continuous. This sounds like what I need. I have also asked other mediator-related questions which are bountied. Please feel free to answer them as well!!! May 9, 2022 at 17:05
• one more clarification. You mentioned that the indirect effect is "It is interpreted as giving the causal effect of the treatment that flows through a mediator M, when the treatment is held at t (to block the direct effect)". I am not sure how this would happen- if T is held at t, then the mediator would not change, correct? How can you hold T constant, while still varying the mediator? May 11, 2022 at 18:21
• @ChinG You could ask the same question about the direct effect, which is the part of the causal effect that does not flow through the mediators. The point is that neither the direct effect or the natural indirect effect are the total causal effect: they are a partition of the total causal effect, giving the parts explained by each variable.
– Ben
May 11, 2022 at 19:02