Directed acyclic graphs (DAGs) are efficient visual representations of qualitative causal assumptions in statistical models, but can they be used to present a regular instrumental variable equation (or other equations)? If so, how? If not, why?
Solved – Can an instrumental variable equation be written as a directed acyclic graph (DAG)
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Related Solutions
Pearl's theory of causality is completely non-parametric. Interactions are not made explicit because of that, neither in the graph nor in the structural equations it represents. However, causal effects can vary (wildly) by assumption.
If an effect is identified and you estimate it from data non-parametrically, you obtain a complete distribution of causal effects (instead of, say, a single parameter). Accordingly, you can evaluate the causal effect of tobacco exposure conditional on asbestos exposure non-parametrically to see whether it changes, without committing to any functional form.
Let's have a look at the structural equations in your case, which correspond to your "DAG" stripped of the red arrow:
Mesothelioma = $f_{1}$(Tobacco, Asbestos, $\epsilon_{m}$)
Tobacco = $f_{2}$($\epsilon_{t}$)
Asbestos = $f_{3}$($\epsilon_{a}$)
where the $\epsilon$ are assumed to be independent because of missing dashed arrows between them.
We have left the respective functions f() and the distributions of the errors unspecified, except for saying that the latter are independent. Nonetheless, we can apply Pearl's theory and immediately state that the causal effects of both tobacco and asbestos exposure on mesothelioma are identified. This means that if we had infinitely many observations from this process, we could exactly measure the effect of setting the exposures to different levels by simply seeing the incidences of mesothelioma in individuals with different levels of exposure. So we could infer causality without doing an actual experiment. This is because there exist no back-door paths from the exposure variables to the outcome variable.
So you would get
P(mesothelioma | do(Tobacco = t)) = P(mesothelioma | Tobacco = t)
The same logic holds for the causal effect of asbestos, which allows you to simply evaluate:
P(mesothelioma | Tobacco = t, Asbestos = a) - P(mesothelioma | Tobacco = t', Asbestos = a)
in comparison to
P(mesothelioma | Tobacco = t, Asbestos = a') - P(mesothelioma | Tobacco = t', Asbestos = a')
for all relevant values of t and a in order to estimate the interaction effects.
In your concrete example, let's assume that the outcome variable is a Bernoulli variable - you can either have mesothelioma or not - and that a person has been exposed to a very high asbestos level a. Then, it is very likely that he will suffer from mesothelioma; accordingly, the effect of increasing tobacco exposure will be very low. On the other hand, if asbestos levels a' are very low, increasing tobacco exposure will have a greater effect. This would constitute an interaction between the effects of tobacco and asbestos.
Of course, non-parametric estimation can be extremely demanding and noisy with finite data and lots of different t and a values, so you might think about assuming some structure in f(). But basically you can do it without that.
Best Answer
Yes.
For example in the DAG below, the instrumental variable $Z$ causes $X$, while the effect of $X$ on $O$ is confounded by unmeasured variable $U$.
The instrumental variable model for this DAG would be to estimate the causal effect of $X$ on $O$ using $E(O|\widehat{X})$, where $\widehat{X} = E(X|Z)$.
This estimate is an unbiased causal estimate if:
$Z$ must be associated with $X$.
$Z$ must causally affect $O$ only through $X$
There must not be any prior causes of both $O$ and $Z$.
The effect of $X$ on $O$ must be homogeneous. This assumption/requirement has two forms, weak and strong:
The first three assumptions are represented in the DAG. However, the last assumption is not represented in the DAG.
Hernán, M. A. and Robins, J. M. (2020). Causal Inference. Chapter 16: Instrumental variable estimation. Chapman & Hall/CRC.