causal effect in linear causal model

Question

I have the following linear causal model:

$B = \epsilon_B$

$C = \epsilon_C$

$A = \beta_1 B + \epsilon_A$

$Z = \beta_2 B + \beta_3 C + \epsilon_Z$

$D = \beta_4 C + \epsilon_D$

$X = \beta_5 A + \beta_6 Z + \epsilon_X$

$W = \beta_7 X + \epsilon_W$

$Y = \beta_8 W + \beta_9 Z + \beta_{10} D + e_Y$

all structural errors are independent each others. In terms of DAG the causal model is:

My questions are, how amount the following quantities?

$E[Y|do(X),B]$ (b-specific effect of $X$ on $Y$

$E[Y|do(X),C]$ (c-specific effect of $X$ on $Y$

$E[Y|do(X),Z]$ (z-specific effect of $X$ on $Y$

$E[Y|do(X),do(B)]$ (combined effect of $X$ and $B$ on $Y$)

$E[Y|do(X),do(C)]$ (combined effect of $X$ and $C$ on $Y$)

$E[Y|do(X),do(Z)]$ (combined effect of $X$ and $Z$ on $Y$)

Them should be expressed with structural coefficients, if any. Moreover should be explicitated the linear regressions needed for identified those effects/parameters, then the relations among structural and regression parameters involved.

Answer 1

$\newcommand{\eps}{\varepsilon}\newcommand{\doop}{\operatorname{do}}$My other answer has some value, I think, but rather than edit it, I think it might be better to simply add another answer. Let's take the Structural Equation Model (SEM): \begin{align*} B&=\eps_B\\ C&=\eps_C\\ A&=\beta_1B+\eps_A\\ Z&=\beta_2B+\beta_3C+\eps_Z\\ D&=\beta_4C+\eps_D\\ X&=\beta_5A+\beta_6Z+\eps_Z\\ W&=\beta_7X+\eps_W\\ Y&=\beta_8W+\beta_9Z+\beta_{10}D+\eps_Y. \end{align*} We take the first computation: $E[Y|\doop(X=x),B=b].$ As mentioned in my other answer, a sufficient set that satisfies the backdoor criterion in this case is $\{Z\},$ so that we must compute $E[Y|\doop(X=x),B=b,Z=z].$ The $\doop$ operator means that we replace the structural equation for $X$ with $X=x$ and substitute in everywhere: \begin{align*} B&=\eps_B\\ C&=\eps_C\\ A&=\beta_1B+\eps_A\\ Z&=\beta_2B+\beta_3C+\eps_Z\\ D&=\beta_4C+\eps_D\\ X&=x\\ W&=\beta_7x+\eps_W\\ Y&=\beta_8W+\beta_9Z+\beta_{10}D+\eps_Y. \end{align*} With a conditional like $B=b$ or $Z=z,$ while we replace the variable, we don't delete the equation. So, to compute: \begin{align*} E[Y|\doop(X=x),B=b,Z=z] &=\beta_8E[W|\doop(X=x),B=b,Z=z]\\ &\quad+\beta_9E[Z|\doop(X=x),B=b,Z=z]\\ &\quad+\beta_{10}E[D|\doop(X=x),B=b,Z=z]+0\\ &=\beta_8\beta_7x+\beta_9z+\beta_{10}E[D|\doop(X=x),B=b,Z=z]. \end{align*} Now this last expression is what we must work on. For $Z$ to equal $z,$ (and since this is not the $\doop,$ we do not delete the equation for $Z$) we must have \begin{align*} z&=\beta_2b+\beta_3C\\ C&=\frac{z-\beta_2b}{\beta_3}, \end{align*} which forces $$E[D]=\beta_4\cdot \frac{z-\beta_2b}{\beta_3}.$$ Hence $$E[Y|\doop(X=x),B=b,Z=z]=\beta_8\beta_7x+\beta_9z+\beta_{10}\beta_4\, \frac{z-\beta_2b}{\beta_3}.$$ For the $c$-specific effect, the computations will be similar. For the $z$-specific effect, one "gotcha" is that you will need to condition on either $A,B,C,$ or $D$ to block the backdoor path $X\leftarrow A\leftarrow B\to Z\leftarrow C\to D\to Y,$ which is opened up because of conditioning on $Z.$

For a combined effect computation, the work is actually easier in some ways. Let's take $E[Y|\doop(X=x),\doop(B=b)].$ The difference between this and the conditioned version $E[Y|\doop(X=x),B=b]$ is that in the $\doop$ version, we replace $B=\eps_B$ with $B=b$ and substitute $b$ for $B$ everywhere in the SEM, thus: \begin{align*} B&=b\\ C&=\eps_C\\ A&=\beta_1b+\eps_A\\ Z&=\beta_2b+\beta_3C+\eps_Z\\ D&=\beta_4C+\eps_D\\ X&=x\\ W&=\beta_7x+\eps_W\\ Y&=\beta_8W+\beta_9Z+\beta_{10}D+\eps_Y. \end{align*} The result should be $$E[Y|\doop(X=x),\doop(B=b)]=\beta_8\beta_7x+\beta_9\beta_2b.$$

Answer 2

Partial Solution:

According to Rule 2 on page 70 of Causal Inference in Statistics: A Primer, by Pearl, Glymour, and Jewell:

The $z$-specific effect $P(Y=y|\operatorname{do}(X=x), Z=z)$ is identified whenever we can measure a set $S$ of variables such that $S\cup Z$ satisfies the backdoor criterion. Moreover, the $z$-specific effect is given by the following adjustment formula \begin{align*} &\quad P(Y=y|\operatorname{do}(X=x),Z=z)\\ &=\sum_s P(Y=y|X=x,S=s,Z=z)\,P(S=s|Z=z). \end{align*} This modified adjustment formula is similar to Eq. (3.5) with two exceptions. First, the adjustment set is $S\cup Z,$ not just $S,$ and, second, the summation goes only over $S,$ not including $Z.$

Once you have the probability distribution computed, you can calculate the expected value with the usual formula: $$E[Y=y|\operatorname{do}(X=x),Z=z]=\sum_y y\cdot P(Y=y|\operatorname{do}(X=x),Z=z).$$

So here is a worked-out example: your first quantity. We must first find $$P(Y=y|\operatorname{do}(X=x),B=b).$$ The adjustment set $S$ must be one of the following: \begin{align*} S&=\{Z\}\\ S&=\{Z,C\}\\ S&=\{Z,D\}\\ S&=\{Z,C,D\}. \end{align*} We do not need to worry about the backdoor path starting with $A,$ because conditioning on $B$ already blocks it. That leaves the backdoor path beginning with $Z.$ You must condition on $Z,$ else the backdoor path $X\leftarrow Z\to Y$ is open. The reason just $\{Z\}$ can work is that $B$ is already conditioned on, blocking the collider that opens up at $Z$ when conditioning on it. Hence the probability distribution you can write as $$P(Y=y|\operatorname{do}(X=x),B=b)= \sum_z P(Y=y|X=x,B=b,Z=z)\,P(Z=z|B=b).$$ The expectation you would write in a $\operatorname{do}$-free fashion as $$E[Y=y|\operatorname{do}(X=x),B=b]= \sum_y y\cdot \sum_z P(Y=y|X=x,B=b,Z=z)\,P(Z=z|B=b).$$ The other cases can be worked out similarly. The reason I am calling this a partial solution is that I do not know how to express these probabilities in terms of the edge weights $\beta_i.$ But perhaps this partial solution will help. I will say this: if $Y\in\{0,1\},$ then you can dispense with the $y$-summation in the final expression.