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I have a directed acyclic graph (DAG) that describes the causal structure of my exposure, outcome and confounders. To estimate the total effect of the exposure $X$ on the outcome $Y$, I derive a (minimal) adjustment set $\mathbf{S}$.

Now, the total effect can be estimated by running a linear regression model $Y \sim X+\mathbf{S}$ and looking at the coefficient for $X$.

Does this hold true for a generalized linear model (GLM), too?

E.g., in my case I have a log-linear model (Poisson or Negative-Binomial regression) with an offset $\log(C)$, where I would regress $\log(Y/C) \sim X+\mathbf{S}$.

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    $\begingroup$ It's not in general true that you can run a regression of $Y ~ X + S$ to adjust for $S$. An adjustment set allows you to nonparametrically identify the causal effect, but in a finite sample, you typically have to make parametric assumptions. If these are wrong, conditioning on the adjustment set as you do is not guaranteed to yield a consistent estimate of the causal effect. So the answer is that it holds neither for a linear model nor a generalized linear model in general. $\endgroup$
    – Noah
    Jun 16, 2020 at 18:45

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The same will hold true for GLM. DAG are non-parametric in the sense that they don't place any restrictions on the distributions of $\mathbf{S}$ in their relation to $X$ or $Y$. When using parametric regression models, you assume that the models are flexible enough to capture the true density. You can do this with linear regression if the outcome is Gaussian, or with a GLM that has the appropriate distributions for $Y$.

Consider the following example. There is a single continuous variable $S$ which is a common cause of both a binary treatment $X$ and binary outcome $Y$ (i.e. $S$ is a confounder in this context). The causal diagram makes no assertion on the relationship between $S$ and $Y$. We assume that DAG satisfies Markov factorization, so that we can write the density for the DAG as $$f(S, X, Y) = f(Y|X, S)f(X|S)f(S)$$ Note that says nothing about the particular functional forms of $f(Y|X, S)$. In fact, the described DAG is consistent with all of the following models $$ \Pr(Y|X,S) = \beta_0 + \beta_1 X + \beta_2 S$$ $$ \Pr(Y|X,S) = \beta_0 + \beta_1 X + \beta_2 S + \beta_4 S^2$$ $$ \Pr(Y|X,S) = expit(\beta_0 + \beta_1 X + \beta_2 X S^2 + \beta_4 S^5)$$ in addition to nonparametric structural equation models (SEM). Nonparametric SEM are actually the math underlying the causal DAGs we write.

In the case of a continuous $Y$, the DAG makes no restrictions on the distribution for the errors of the outcome. When using parametric models the important part is selecting an appropriate distribution, something DAGs cannot tell you.

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  • $\begingroup$ Sounds reasonable! Nevertheless, do you have a reference for this? $\endgroup$
    – Edgar
    Jun 16, 2020 at 15:31
  • $\begingroup$ I added a little more detail in the form of an example but I don't have a great reference that is succinct for this point. The following R library has a good description under the 'Structural Causal Graphs" section: cran.r-project.org/web/packages/ggdag/vignettes/… $\endgroup$
    – pzivich
    Jun 16, 2020 at 17:23

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