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In the paper The Table 2 Fallacy: Presenting and Interpreting Confounder and Modifier Coefficients by Daniel Westreich and Sander Greenland, the authors present a simple example to illustrate how to interpret coefficients in a regression model depending on the causal structure associated with the model. The question is about the effect of HIV on the risks of having a stroke depending on the age of the patient and whether the patient smokes or not. Here is the DAG associated with the model.

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And here is the first model (Model 1) they present : $$ \text{logit(Stroke|HIV, Smoking, Age)} = \beta_0 + \beta_1 \text{HIV} + \beta_2 \text{Smoking} + \beta_3 \text{Age} $$ In the section titled "Same model, different types of effect", they mention repeatedly that, in model 1, $\beta_1$ should be interpreted differently from $\beta_2$ and $\beta_3$ because the first one represents a total effect and the two others represent direct effects. I understand the general idea of not presenting all coefficients of a model as if they represent the same types of effect and be mindful of the causal structure at play. But in that particular case, doesn't $\beta_1$ also represent the direct effect of HIV in model 1 given that we adjust for Smoking and Age ?

Thanks for your help,

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    $\begingroup$ Assuming no interactions, the $\beta_1$ coefficient is the total (and direct) effect of HIV. The $\beta_2$ coefficient is the direct effect of Smoking, not its total effect. In that sense, since both are not total effects, you could say they are different quantities. But you're right you could say both are direct effects. $\endgroup$
    – Kuku
    Commented Jun 18 at 15:29
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    $\begingroup$ Should $\beta_2 \text{Age}$ be $\beta_3 \text{Age}$? $\endgroup$
    – dimitriy
    Commented Jun 18 at 15:53
  • $\begingroup$ @dimitriy yep I corrected it thanks. $\endgroup$ Commented Jun 18 at 19:21
  • $\begingroup$ @Kuku that's what I thought. Thanks for the precisions. $\endgroup$ Commented Jun 18 at 19:22

1 Answer 1

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I am dropping the logit aspect, ignoring the error terms, and making some linearity assumptions to simplify the notation/math to build intuition.

We know that $$ \text{stroke} = \beta_0 + \beta_1 \text{HIV} + \beta_2 \text{Smoking} + \beta_3 \text{Age}$$

From the arrows, there is an analogous equation for HIV: $$\text{HIV} = \gamma_0 + \gamma_1\text{Smoking} + \gamma_2 \text{Age}$$

Plugging the second equation into the first gives:

$$ \text{stroke} = \beta_0 + (\beta_1 \cdot \gamma_0 + \beta_1 \cdot \gamma_1\text{Smoking} + \beta_1 \cdot \gamma_2 \text{Age}) + \beta_2 \cdot \text{Smoking} + \beta_3 \cdot \text{Age}$$

The first term in the parentheses is the direct effect of HIV, a product of some baseline HIV rate for a young nonsmoker and its effect on stroke. The second is the indirect effect of smoking. The third is an indirect effect of age. These are very different than $\beta_2$ and $\beta_3$ effects since those aren't operating through the HIV equation $\gamma$ coefficients. For example, someone who does not have HIV has the expected value of $\beta_0 + 0 + \beta_2 \cdot \text{Smoking} + \beta_3 \cdot \text{Age}$. If someone has HIV, they also get 2-3 terms in parentheses, depending on if they smoke. You can label the $\text{Age} \rightarrow \text{HIV}$ with $\gamma_2$ and $\text{HIV} \rightarrow \text{Stroke}$ with $\beta_1$.

This distinction is what makes it a total effect: a direct plus two indirect effects. However, we can't break them out, so we lump them together into $\beta_1$ and call that the total effect of HIV.

You can also do this kind of accounting for Age and Smoking. The first term is the direct effect and the second is the indirect:

  • The total effect of one more year of age is $\beta_3 + \beta_1 \cdot \gamma_2$
  • The total effect of taking up smoking is $\beta_2 + \beta_1 \cdot \gamma_1$
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    $\begingroup$ That is correct for the model in your question. However, you could include various interactions between HIV and Smoking and Age: $$ \text{stroke} = \tilde \beta_0 + \tilde \beta_1 \text{HIV} + \tilde \beta_2 \text{Smoking} + \tilde \beta_3 \text{Age} + \tilde \beta_5 \cdot \text{Smoking}\times \text{HIV} + \tilde \beta_6 \cdot \text{Age}\times \text{HIV} + \tilde \beta_7 \cdot \text{Smoking}\times \text{HIV} \times \text{Age}$$ $\endgroup$
    – dimitriy
    Commented Jun 18 at 20:07
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    $\begingroup$ Here the marginal effect of HIV is $$\tilde \beta_1 + \tilde \beta_5 \cdot \text{Smoking} + \tilde \beta_6 \cdot \text{Age} + \tilde \beta_7 \cdot \text{Smoking} \times \text{Age},$$ which depends on smoking and age. I am using tildes above the coefficients to emphasize that these are not the same parameters as in the simpler model. $\endgroup$
    – dimitriy
    Commented Jun 18 at 20:07
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    $\begingroup$ The simple answer is that DAGs already do so implicitly since they don't make any assumptions about the functional form of the relationship. Once you have drawn your DAG, you can assume that any variables pointing to the same outcome can modify the effect of the others pointing to the same outcome. It's up to you how to represent that mathematically. Some people will add $\text{D} \rightarrow \text{D & X} \rightarrow \text{Y}$ and $\text{X} \rightarrow \text{D & X}$, but that quickly gets untidy and is not standard practice. $\endgroup$
    – dimitriy
    Commented Jun 18 at 20:49
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    $\begingroup$ For instance, I made an (arbitrary) choice to leave out $\tilde \beta_8 \cdot \text{Age} \times \text{Smoking}$. I also could have logged Age. I don't know of any tool that handles these sorts of decisions for you. For me, it's a complicated hodgepodge of the research question(s), theory, previous literature, humoring referees when possible, and how much data/variation I have relative to parameters. $\endgroup$
    – dimitriy
    Commented Jun 18 at 20:54
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    $\begingroup$ Noted. Thanks again for your answers. I'll read some literature on that to see how people usually deal with that and make my own hodgepodge haha $\endgroup$ Commented Jun 18 at 21:08

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