Fixed effects in DAGs

Let's imagine I'm interested in studying the causal effect of beliefs in some ideas and behavior related to these ideas (say, if I believe sunscreen is good for my health, I use more sunscreen etc.). Then I run a survey asking several questions about beliefs and behavior for several items. Suppose I fit the following model:

y_ij = person_fixed_effect_i + item_fixed_effect_j + belief_ij + e_ij, e_ij ~ N(mu, sigma^2)


If I believe that person ideology and item ideology can be confounding, and assuming both person ideology and item ideology are fixed, how should I draw my DAG?

I think that both fixed effects capture every effect person and item ideology could have, and thus my DAG should be like this:

In this case, the model is identified and everything is good. This makes sense, since the fixed effect is supposed to capture everything non-observed and that does not change for individuals and items. However, I was wondering if this is correct, because the fixed effects capture many unobserved and constant variables. Because of this, perhaps it's possible for some of the effect of ideology to "escape" the fixed effect? So, my question is: Is my DAG correct? Or is it possible for person and items ideology to have a direct effect on belief, not mediated by the fixed effects?

1 Answer

A DAG encapsulates a researcher's theory about the causal structure in their data. There is no "correct" and "incorrect" DAG (apart from obvious mis-specifications which often have to do with time - a cause must occur before the effect). The best that you can do is to make your DAG consistent with your causal theory. Sometimes their could be competing DAGs representing different theories.

Whether or not it is possible for person and items ideology to have a direct effect on belief, not mediated by the fixed effects, is a matter for you to decide. Many things are possible in theory.

If you think that the 2 fixed effect variables capture more than just the ideologies, then you should add variables to your DAG as needed - whether or not you have actually observed those data, and if you are planning a study then this can inform what questions you may need to ask.

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