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Is there a term for a variable that is both a mediator and a confounder at the same time? By this a mean a variable that both influences the exposure and the outcome but there is also an influence from the exposure to the variable as indicated by the diagram below:

enter image description here

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  • $\begingroup$ Is it really both a confounder and mediator, or is the pre-exposure instance of the variable a confounder and the post-exposure instance of the variable a mediator? $\endgroup$
    – Noah
    Mar 25, 2021 at 17:36
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    $\begingroup$ A concrete example be the following: The exposure is physical activity, the outcome is the risk of having some disease and the variable is for example BMI. Physicians tell us that BMI has a direct influence on the incidence of the disease and physical activity effects the BMI. However, we are worried that obesity itself has an influence on physical activity as well. $\endgroup$
    – Alex
    Mar 25, 2021 at 22:35
  • $\begingroup$ Where is this diagram from? $\endgroup$
    – Firebug
    Mar 21, 2023 at 9:13

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Your example is not a Directed Acyclic Graph (DAG) because it has a cycle in its bi-directed edge. So in terms of causal graphical models, I do not think there is a name for such a structure because it contradicts the underlying causal DAG assumption. That said, it might make sense to consider common causes of "Exposure" and "Variable", or find other ways of expressing your set-up in a graph that does not rely on a bi-directed edge.

If you really do believe there is a bidirectional effect that cannot be resolved either through a different model, or by arguing things like "the effect in one direction will take very long to show so it does not matter", then it seems the DAG modeling assumptions are simply violated. I should say that this would not be unusual - they are very strong assumptions. There is also some theory on cyclical causal graphs. I am not sure how applicable such approaches would be to your case, but it could be an idea to look into it.

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