# DAG: no back-door paths but background information shows a need for adjusting

I am interested in the effect of town of residence on income.

Though the DAG has many arrows, it's interpretation is actually very simple:

• I have 6 covariates (Cov1-6), all causing mediation scenarios, resulting in zero back-door paths.

• I also have 2 unmeasured possibly important covariates (U1-2) that also have mediation scenarios only (no back-door paths).

However, I do know that subjects in different towns are different. According to other evidence and my study data, there are differences in the 6 covariates I have. Thus, this says to me that adjusting is neccessary.

Therefore, back-door criterion should be used in this case. I should use disjunctive cause criterion (adjust everything that you have until you do not open any new back-door paths). However, daggity package does not have such option. It only has an option to use Judea Pearl's single-door criterion.

QUESTIONS:

What is the difference between disjunctive cause criterion and Pearl's single door criterion?

And if I have the aforementioned background knowledge, is it reasonable to adjust for all available six covariates?

• I agree there are no back-door paths - as yet. (THANK YOU for including a causal diagram!) So that means there are no confounders, and if you want the true causal effect of TOWN on INCOME, you should NOT condition on anything in-between. On the other hand, if you think there is an unknown variable not on your diagram affecting both TOWN and INCOME, you might have a missing variable bias going on, here. What do you think is the cause of the variation you see in the towns? – Adrian Keister Sep 4 '20 at 15:17
• Thank you! First, lets take the age distribution, which differs between TOWNS. One town has many students (who does not have a high income) and the second town has many middle-aged people (working, higher income). If I would like to know which town has a better impact on income, wouldn't it be correct to adjust for age? Second, what should I do if there are unknown variables? Just mentioning it under limitations? And why adjusting for everything is bad. The Disjunctive Cause Criterion allows that. – st4co4 Sep 4 '20 at 19:44
• Adjusting for mediators actually curbs the true causal effect of TOWN on INCOME, so that you do not get what you expect. See Section 3.2, The Adjustment Formula in Pearl's book Causal Inference in Statistics: A Primer. So about AGE: do you think the arrow should go from TOWN to AGE, or from AGE to TOWN? Clearly there should be an arrow from AGE to INCOME. I disagree with the disjunctive cause criterion. In a mediating situation, for example, controlling for the mediator will not get you the true causal effect. See the section in Pearl I already mentioned. – Adrian Keister Sep 4 '20 at 20:12
• Thank you so much! Seems I haven't considered all the relationships possible (it's not that simple as I thought!). If we consider TOWN as a place of residence, age may change this (people at a certain age move to find a better job, education; or older people move to their children's residence as they lose their independence). Thus, the arrow may be bidirected. When should I mark a relationship as bidirected and when as uncertain (--)? And both, bidirected and uncertain relationships, can open back-doors? – st4co4 Sep 5 '20 at 17:07

What is the difference between disjunctive cause criterion and Pearl's single door criterion?

The Single Door Criterion establishes conditions under which a causal bath between two variables, say $$X \rightarrow Y$$, will be consistently estimated by the regression coefficient for $$X$$ in a multivariable regression model for the response $$Y$$. Breifly, it stipulates that, for a set of variables containing various paths between them and being acyclic (ie, it's a DAG), then a subset of these variables, $$Z$$, will be sufficient provided that

• $$Z$$ contains no descendent of $$Y$$, and
• by removing the arrow in $$X \rightarrow Y$$, $$X$$ is then independent of $$Y$$

This leads to the familiar "rules" that we should condition on confounders (ie backdoor adjustment), but not mediators.

It also leads to "front door adjustment", where we are able to estimate the causal effect of $$X$$ on $$Y$$ in $$X \rightarrow M \rightarrow Y$$ even in the presence of unmeasured confounding.

The Disjunctive Cause Criterion (VanderWeele, 2019), is actually very similar to backdoor adjustment, but tries to avoid having to explicitly identify confounders, and instead seeks to adjust for variables that are causes of either the main exposure or the outcome (or indeed both), but excluding instrumental variables. However, I say "tries to", because there is still a need to include confounders:

"controlling for each covariate that is a cause of the exposure, or of the outcome, or of both; excluding from this set any variable known to be an instrumental variable; and including as a covariate any proxy for an unmeasured variable that is a common cause of both the exposure and the outcome"

VanderWeele TJ. Principles of confounder selection. Eur J Epidemiol. 2019. https://doi.org/10.1007/s10654-019-00494-6.

The problem with this approach is two-fold. First it can lead to "over-adjustment", that is, unlike Pearl's theory, it can not, except by accident, lead to a "minimally sufficient" set of covariates, so in general it will not result in a parsimonious model and could suffer from problems due to high correlations between covariates. Second, it can lead to the inclusion of mediators, which VanderWeele acknowledges would be a problem.

And if I have the aforementioned background knowledge, is it reasonable to adjust for all available six covariates?

No, I don't think this is appropriate. All 6 observed variables appear to be confounders for the causal effect of TOWN on INCOME and should not be adjusted for. This is precisely an example of the second problem with this technique, mentioned in my last paragraph. See this answer for details and examples of what can go wrong if you do adjust for mediators:

How do DAGs help to reduce bias in causal inference?

Without further details of your research question, study, and data it is difficult to advise further, but you might want to look into a multilevel structural equation model with random effects for town, although if TOWN is your main exposure this probably wouldn't be the way to go, but some kind of SEM could be worth looking at. I would suggest asking a new question about how to proceed further.