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

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 $$\beta_2$$ ?

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 $$Y$$.

• Hi @Johan. Thanks for your detailed answer it helps a lot. Commented Jun 18 at 19:53