How do you interpret estimates when the model is the same for two exposure variables?

Question

I want to investigate the direct effect of two environmental variables $X_1$ and $X_2$ on a quantity $Y$. $X_1$ and $X_2$ are connected together and with two other environmental variables $X_3$ and $X_4$ (which have been measured as well). I built a DAG including all the relevant and plausible causal links I could think of between the $X_i$ to choose the adjustment set needed to estimate the effect of $X_1$ and $X_2$ (see figure below).

My issue is that the adjustment sets suggested by ggdag to estimate the direct effects of X1 and X2 result in the same model. For X1, I need to adjust for {X2, X3 and X4} and for X2, I need to adjust for {X1, X3 and X4}. In both cases, my model looks like this : $ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \beta_4 X_4$. What does it mean regarding the interpretability of $\beta_1$ and $\beta_2$ ?

Answer 1

First, we outline the graphical and modeling assumptions, followed by the causal queries to be examined:

Assumptions:

1. The graph (DAG on the left) accurately represents all relevant direct causal links within the system, with no latent confounding structures.
2. The parametric regression model $\mathcal{M}$: $\mathbb{E}[Y \mid X_{1:4}] = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \beta_4 X_4$ contains the true model for the conditional expectation. This is, we assume linearity and no interactions.
3. To simplify analysis, let variables $X_1$ and $X_2$ be binary.

Next, we examine the average total effect (ATE) and controlled direct effect (CDE) on $Y$ of $(X_1, X_2)$ as a multivariate exposure/treatment, and the ATE and CDE of $X_1$ and $X_2$ on $Y$ individually.

Causal Queries:

• ATE from multivariate exposure: Defined as $\mathbb{E}[Y \mid \operatorname{do}(X_1=1, X_2=1)] - \mathbb{E}[Y \mid \operatorname{do}(X_1=0, X_2=0)]$. Given that $X_4$ is a mediator for the effect of $X_1$ on $Y$, it cannot be included in the adjustment set to identify this query. Therefore, the model $\mathcal{M}$ is inappropriate, and $\beta_1$ and $\beta_2$ do not carry a causal interpretation in terms of the ATE. However, by adjusting only for $X_3$ using $\mathbb{E}[Y \mid X_{1:3}] = \alpha_0 + \alpha_1 X_1 + \alpha_2 X_2 + \alpha_3 X_3$, the ATE is given by $\alpha_1 + \alpha_2$.

• CDE from multivariate exposure: Defined as $\mathbb{E}[Y \mid \operatorname{do}(X_1=1, X_2=1, X_4=x)]-\mathbb{E}[Y \mid \operatorname{do}(X_1=0, X_2=0, X_4=x)]$. In this scenario, the model $\mathcal{M}$ is valid, as it adjusts for the confounder $X_3$ and the mediator $X_4$, thus helping identify the CDE, which equals $\beta_1 + \beta_2$ for all $x$ within the support of $X_4$.

• ATE from $X_1$ exposure: Defined as $\mathbb{E}[Y \mid \operatorname{do}(X_1=1)]-\mathbb{E}[Y \mid \operatorname{do}(X_1=0)]$. Since $X_2$ and $X_4$ are mediators, they cannot be part of the adjustment set. Thus, $\beta_1$ does not interpret the ATE, which can in fact be identified by adjusting only for $X_3$ using $\mathbb{E}[Y \mid X_{1,3}] = \gamma_0 + \gamma_1 X_1 + \gamma_3 X_3$. Here, the ATE is given by $\gamma_1$.

• CDE from $X_1$ exposure: Defined as $\mathbb{E}[Y \mid \operatorname{do}(X_1=1, X_2=z, X_4=x)]-\mathbb{E}[Y \mid \operatorname{do}(X_1=0, X_2=z, X_4=x)]$. Here, the model $\mathcal{M}$ is valid, as it adjusts for the confounder $X_3$ and the mediators $X_2$ and $X_4$, thus helping identify the CDE, which equals $\beta_1$ for all $(z, x)$ within the support of $(X_2, X_4)$.

• ATE from $X_2$ exposure: Defined as $\mathbb{E}[Y \mid \operatorname{do}(X_2=1)]-\mathbb{E}[Y \mid \operatorname{do}(X_2=0)]$. As $X_4$ is a mediator, $\beta_2$ does not interpret the ATE. The ATE can be identified by adjusting for $X_1$ and $X_3$ using $\mathbb{E}[Y \mid X_{1:3}] = \delta_0 + \delta_1 X_1 + \delta_2 X_2 + \delta_3 X_3$, and the ATE equals $\delta_2$.

• CDE from $X_2$ exposure: Defined as $\mathbb{E}[Y \mid \operatorname{do}(X_2=1, X_4=x)]-\mathbb{E}[Y \mid \operatorname{do}(X_2=0, X_4=x)]$. Here, the model $\mathcal{M}$ is valid, as it adjusts for the confounders $X_1$ and $X_3$ and the mediator $X_4$, thus identifying the CDE, which equals $\beta_2$ for all $x$ within the support of $X_4$.

In summary, given the graphical and functional assumptions, within the model $\mathcal{M}$, $\beta_1$ represents the CDE of $X_1$ on $Y$, $\beta_2$ represents the CDE of $X_2$ on $Y$, and $\beta_1 + \beta_2$ represents the CDE of the multivariate exposure $(X_1, X_2)$ on $Y$.