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I have a somewhat complicated DAG that looks like this:

enter image description here

where Y is the outcome variable. To give some context, X and Y could be two diseases with X being the precursor to disease Y. A is age, and B is treatment. Age affects both (1) how likely a person is to take the treatment, (2) how likely a person is to suffer from disease X, and (3) how likely disease X will progress to disease Y (an effect modifier). Treatment B affects both (1) how likely the person being treated will suffer from disease X and also (2) how likely disease X will progress to disease Y (an effect modifier).

The question I am interested in answering is (1) what is the effect of treatment B on the risk of getting disease X, and (2) how does treatment B affect the risk of disease X progressing into disease Y (the two red arrows).

What is the correct strategy in estimating these two effects?

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I think you could equivalently just have the effect modifier arrows going straight into $Y.$ In any case, the age variable is a confounder for both of your questions, because it sets up a backdoor path from your cause to your effect. So you need to condition on age for both questions. You can do that in one of three ways:

  1. Stratify your analysis based on age.

  2. If you are in a linear regression setting, include age on the RHS.

  3. Use the backdoor formula.

  4. and 3. should be valid in most settings, 2. obviously only in a linear regression setting.

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  • $\begingroup$ Thanks for the answer. While I agree with most of what you said and that I should control for age, I'm wondering is it really always valid to draw an arrow going straight into Y? Even though intuitively, there's no direct causal link between them? $\endgroup$ Aug 30, 2022 at 23:43
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    $\begingroup$ @DavidYoung If you really need to, why not insert another variable in-between $X$ and $Y?$ DAGs just don't have arrows that go into arrows: I've certainly never seen that. $\endgroup$ Aug 31, 2022 at 1:39
  • $\begingroup$ In general, DAGs are good for identification but are bad for specifying functional form (which is what effect modifiers are). There are graphical proposals for conveying effect modification, but they are no longer valid DAGs. See: journals.lww.com/epidem/Fulltext/2007/09000/… $\endgroup$
    – ehudk
    Oct 18, 2022 at 20:57

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