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How do we adjust for the confounder of a confounder in order to compute unbiased estimates of the treatment effect of $A$ on $D$? See the causal graph (DAG) below:

confounder of a confounder

What do we call the confounder $C$ (of a confounder $B$) - does we give such variables/confounders any special name?

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Thank you for including the DAG!

The answer here is pretty straight-forward: you simply condition on both $C$ and $B.$ Neither $C$ nor $B$ is part of a collider, so you're not opening up new paths by doing so, and this conditioning closes all backdoor paths from $A$ to $D,$ enabling you to get the unbiased effect of $A$ on $D.$

I'm not aware of any special name for the $C$ relative to $B$ except "parent". I'm also not sure I would call $C$ a "confounder of a confounder". A confounder is a variable that sets up a backdoor path from the cause in which you're interested to the effect in which you're interested. But you're not interested in $B$ as a cause of anything, so it strikes me as moot in this situation. I would just call $C$ the parent of $B$ and leave it at that.

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    $\begingroup$ I agree. "C" is simply another confounder, because of its residual effect even after conditioning on "B". You're right, "confounder of a confounder" is misleading because a confounder is defined by causing two things: if C caused only B and A OR if C caused only B and D, then C would not be a confounder, and adjusting for only B would satisfy the backdoor criterion. $\endgroup$
    – AdamO
    Feb 21, 2023 at 20:13

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