Pearl's Causal Inference In Statistics, equation 3.11 - Calculation of group specific causal effects

Question

In the book Causal Inference In Statistics by Pearl, page 63, while referring to the below DAG, it says -

Thus to compute the $w$-specific causal effect, written $P(y|do(x),w)$, we adjust for $T$, and obtain

$P(Y=y|do(X=x),W=w)$ $=$ $\sum_t {P(Y=y|X=x,W=w,T=t)P(T=t|X=x,W=w)}$ (3.11)

I have the following queries -

1. Why does it say - "to compute the $w$-specific causal effect, written $P(y|do(x),w)$"? Given the definition of $do(x)$ presented here, it cannot be guaranteed that $P(y|do(x),w)$ calculates the respective causal effect, when conditioning on $w$ opens up a non-causal path (highlighted in pink in the figure). Am I understanding the definition of $do(x)$ incorrectly here?
2. In the equation if the summation on the right-hand side is performed, $\sum_t {P(Y=y|X=x,W=w,T=t)P(T=t|X=x,W=w)}$
   $=\sum_t {P(Y=y,T=t|X=x,W=w)}$
   $=P(Y=y|X=x,W=w)$
   which should not be the causal-effect as it seems to be including the association rising from the non-causal path. What am I missing here?

Answer 1

The $w$-specific causal effect of $X$ on $Y$ is quite distinct from the causal effect of $X$ on $Y.$ The causal effect of $X$ on $Y$ is just $P(y|do(x)).$ The $w$-specific causal effect of $X$ on $Y$ is as you've written: $P(y|do(x),w).$ Essentially, you are stratifying the causal effect of $X$ on $Y$ by values of $w.$ Now, in the particular case in question, there are backdoor or other non-causal paths opened up by conditioning on $w,$ so it is necessary to stop those up to get the true $w$-specific causal effect (just condition on $Z$ or $T$ as well to stop up the undesired path - hence the first formula in your question). I don't think you're necessarily misunderstanding the $do$ operator.

In your second question, you need to understand what's meant by "including the association rising from the non-causal path". You could include that association in more than one way. In this example, we're "including" that association by adjusting for it so that it does not bias our results.