Causal inference in DAGs and resulting structural equations

Question

I am trying to understand the difference between the two modelling approaches described below that stems from the causal graphs. Our goal is to causally measure the total treatment effect of our expenditure of two investment options denoted $X_1, X_2$ on the final outcome sales $Y$. To our help we also have measurements of mediators denoted $end\_exog\_1, end\_exog\_2$.

We know that the investment options causally affect sales in the following way:

Note that all variables are observed: yellow ovals denote exogenous variables and blue ovals denote endogenous variables.

To model this we set up the following system of equations(just as an example, note that the graph does not explicitly state any concrete computations, just relations between variables): \begin{align} Y &= b_0 + b_1 * end\_exog\_2 + b_2 * x_1 + b_3 * x_2 + \epsilon\\ end\_exog\_2 &= b_4 + b_5 * end\_exog\_1 + x_1 * b_6 + x_2 * b_7 + \epsilon\\ end\_exog\_1 &= b_8 + b_9 * x_1 + b_{10} * x_2 + \epsilon \end{align}

---

We could also get rid of our mediator variables $end\_exog\_1, end\_exog\_2$ resulting in a graph with solely two edges $X_1 -> Y, X_2 -> Y$ which we could consider e.g as a single multiple regression equation(just as an example, note that the graph does not explicitly state any concrete computations, just relations between variables): $Y = b_0 + b_1 * X_1 + b_2 * X_2$.

---

My understanding of why choose one modelling approach over the other one: since we have this extra information in $end\_exog\_1, end\_exog\_2$ and we know the causal structure it should be smart to utilize it.

Given the simple structure with just 2 edges as in case 2 the effect is indeed identifiable, as can be shown by the backdoor criterion as well as the frontdoor criterion. For situation number 1 it is a bit different: the backdoor criterion is violated due to the mediator variables being descendants of our exogenous variables $X_1, X_2$. However, the frontdoor criterion is still valid.

---

Which is the preferred modelling way given that we want to understand the total treatment effect of $X_1, X_2$ on $Y$ and why is that?

Answer 1

Since you talk about identification, I will focus on the causal considerations of your problem. At this level, which modelling approach to choose is not a matter of whether we have "extra information", but rather what is our question of interest. In technical terms, it is a matter of defining our estimand$^1$. You should answer whether you care about the total effect of $X_1$ and $X_2$ on $Y$, or their direct effect on $Y$, or their indirect effect that is mediated through mediators $M_1$ and $M_2$.

Once you have chosen an estimand, you will see that they limit which models you can use to estimate it. For instance, a decomposition of the total effect into direct and indirect components will force you to include mediators in the model (as well as making stronger assumptions).

However, if your estimand is the total treatment effect, you could use both models to estimate this quantity. Note however that a DAG encodes non-parametric relationships: it is not making any assumptions about the functional form of the model for $Y$ for the different covariates. Both models you have suggested assume that there is no interaction between $X_1$ and $X_2$, i.e. no $X_1X_2$ term, as well as that the effects are linear and additive. Without further justification for these assumptions, neither of these models might help you "understand the effect of $X_1, X_2$ on $Y$", which starts to involve estimation and thus statistical considerations, which go beyond mere identification.

---

Note also that you say that

the backdoor criteria [sic] is violated due to the mediator variables being descendents of our exogenous variables X1,X2

, however there is no backdoor path from $X_1$ or $X_2$ to $Y$---no arrow is entering $X_1$ or $X_2$, so perhaps you would want to re-check your definition.

---

1. On the fundamental importance of defining your estimand, you can see "What is your estimand? Defining the target quantity connects statistical evidence to theory"