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For the given graph above, the controlled direct effect will be $E[Y|\operatorname{do}(X),\operatorname{do}(M)]$. This would break all the incoming edges to node X and M, so $X\rightarrow M$ is removed.

Does this mean that the controlled direct effect is equal to the combined effect of path $M\rightarrow Y$ and $X\rightarrow Y?$

If I try to do the same thing on the graph below where we have $C$ as a confounder of $X$ on $Y$, will the controlled direct effect $E[Y|\operatorname{do}(X),\operatorname{do}(C)]$ be equal to the combined effect of paths $C\rightarrow Y$ and $X\rightarrow Y$ similarly?

Sorry, I'm still pretty new to causal inference. I've been looking online for causal effect with multiple/joint interventions for a few days, but still can't figure it out.

Also, I understand that controlled direct effect is different from natural direct effect. I found this paper VanderWeele, 2011 where it stated in Section 2 that "controlled direct effects cannot be used for effect decomposition unless there is no interaction between the effects of the treatment and the mediator on the outcome". What does "no interaction between the treatment and the mediator on the outcome" mean? Does this mean there's no edge between $M$ and $X$ in the first graph? That seems a bit weird, but maybe I'm wrong.

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The answer to your first question:

Does this mean that the controlled direct effect is equal to combined effect of path $M\to Y$ and $X\to Y?$

is no. In this scenario, the combined effect is the true or total causal effect of $X$ on $Y$, some of which is mediated through $M.$ In a linear regression setting, the correct model would be $Y\sim X.$ You would not include $M$ on the RHS, because that would "cut off" by conditioning the amount of the cause mediated through $M.$ The controlled direct effect is essentially quantifying only what $X\to Y$ is.

The answer to your second question:

... will the controlled direct effect $E[Y|\operatorname{do}(X),\operatorname{do}(C)]$ be equal to the combined effect of path $C\to Y$ and $X\to Y$ similarly?

is also no, really for the same reason. In a confounder situation, you would want to include $C$ on the RHS of your model in order to condition on it, or you will get confounder bias: $Y\sim X+C.$ Not sure why you would even be interested in the combined effect in this case. If it's a confounder, it's a nuisance variable!

The answer to your third question

What does "no interaction between the treatment and the mediator on the outcome" mean?

is, I think, that there would be no $XM$ term like $Y\sim X +M+XM$ in your mediator setting, such as is often done in linear regression. The keyword "non-linearities" just before hints that the author is thinking about a linear regression setting, and whether there are non-linearities or interaction terms to worry about.

An analogy would be from the realm of differential equations, where superposition techniques are valid in the setting of linear differential equations, but invalid in non-linear differential equations.

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