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I working with observational data and defining assumptions for DAG seems to be more complex than often in examples provided in textbooks. For me, it would be much easier to just skip DAG part and condition for everything, and probably there will be no problem in publishing. However, I like the idea of being explicit with my causal assumptions under methods.

I'll give a simplified example with only two predictors, and thus it would be easier to follow.

Background Information

  • a crude analysis shows very clear multifold regional disparities in income between people from different towns
  • variables or nodes like sex and age differ between towns (p < 0.05), indicating a need for adjusted analysis
  • I have done different uni- and multi-level models with all kind of combinations of predictors and the result of regional disparities always holds.

DAG

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Blue arrows seem to be okay for me; however, relationships between towns and sex/age are quite hard to define. I'll bring few, maybe stupid, examples

  • town may be the cause of different sex distribution by offering more jobs for one sex (e.g. men and mining towns)
  • sex may be a reason to change residence (e.g. local policies discriminate women and they move to another town)
  • town may be polluted and shorten our expected living years (age)
  • age may be a reason to change residence (e.g. moving to another town to go to university)

As you see, causal assumptions can be unidirectional (red, green) or bidirectional (orange), or is it even more reasonable to show them as undirectional (no arrows) (black)?

Goal

  • As age and sex differed between towns, there will be a question about adjusted analysis. The goal is to use adjusted analysis to confirm the results of crude data analysis (to make them more bulletproof) - regional disparities between towns.

What would be the best way to achieve my goal?

For me, it seems that publishing the most conservative result would be reasonable since the result won't change with any adjusting.

What would be the most conservative adjusting?

  • adjusting for everything, age and sex, and even if they may partially act as mediators (unknown direction or bidirectional arrows)?
  • should I show a causal graph with undirected arrows (how should I name it then)?
  • should I show a causal graph with bidirectional arrows (still named DAG?)
  • am I right that undirected and bidirected arrows both make sex and age confounders due to opening a back-door path?

How would you solve and present this situation in your article?

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Fist, I think it is good that you are using a DAG because it requires careful thought about causality, and this is often at the heart of modelling.

adjusting for everything, age and sex, and even if they may partially act as mediators (unknown direction or bidirectional arrows)?

One approach to this is to estimate the net effect for each variable that could either be a confounder or a mediator, and then adjust as appropriate. How you estimate the net effect is another question of course. You could also just make an assumption (and state the assumption in the paper). Another idea is to fit several models where the variables are treated as either mediators or confounders and report the results of all. Since you only have 2 variables, Sex and Age, this seems like a reasonable approach; it would mean fitting 4 models.

should I show a causal graph with undirected arrows (how should I name it then)?

I would not do this, as it makes the diagram ambiguous.

should I show a causal graph with bidirectional arrows (still named DAG?)

I would not do this either, if you are fitting 4 models, as it would be inconsistent with the modelling. Also, you can't call it a DAG if it has bidirectional arcs (a DAG is dorected by definition)

I would include 4 DAGs.

am I right that undirected and bidirected arrows both make sex and age confounders due to opening a back-door path?

Not really, if you are following DAG theory, because the presence of an arc with no direction means that the graph is not directed and therefore is not a DAG.

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    $\begingroup$ Your last point is not quite accurate in terms of notational convention. Bi-directed arrows are conveniently used in DAGs to symbolize open back-door paths due to unobserved confounders. I have updated my answer with respect to this point. $\endgroup$ – persephone Sep 9 '20 at 10:32
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    $\begingroup$ @BobD in my opinion that's an abuse of notation, but I do take your point that some people do use that convention. But in any case there are no unobserved confounders here. $\endgroup$ – Robert Long Sep 9 '20 at 10:37
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    $\begingroup$ The only thing I can think of at the moment is a sruvey of other literature where the net effect has been estimated over time. With that number of variables which could be either mediators or confounders, plus the presence of (potentiallly) unmeasured confounding, unfortunately I am sorry to say that, to me, the situation looks quite hopeless for a regression model. Perhaps an SEM might be worth pursuing, but without a lot more detail I can't say. Please ask a new question, where you explain your study design, and give details of the data, about how to answer your research question(s) $\endgroup$ – Robert Long Sep 9 '20 at 11:25
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    $\begingroup$ Unless you can find evidence, or you are willing to make assumptions, about the direction of causality, there isn't much you can do with regression. There is no test to determine the direction of the causal effect when you only have cross-sectional observational data. Even with an SEM, you will need to create latent variables and then you will need to specify in which direction causation flows. $\endgroup$ – Robert Long Sep 9 '20 at 11:32
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    $\begingroup$ I side completely with Robert on these last points. And yes, I also agree that bi-directed arrows is not good convention! $\endgroup$ – persephone Sep 9 '20 at 11:51
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If you are not sure about the direction of the arrow, this is likely because you suspect (implicitly or explicitly) some potential confounding between the two variables. Hence, you should draw all plausible graphs and derive identifying assumption for each. For some you might reach the conclusion that your causal quantity of interest is not identifyable vis a vis available data, for others it might. With the DAGs you make clear under which causal assumptions a causal interpretation of your empirical estimand is internally consistent.

Generally speaking, the causal interpretation of an empirical estimand is based on the underlying causal model. That is, based on likely untestable assumptions. The DAGs are a tool to make this clear.

Bi-directed arrows are used in DAGs to indicate that there are unobserved back-door paths between two variables. You could also include this unobserved confounder explicitly, labelling it for instance $U$. This is just notational convention. However, assuming a bi-directed (or an unobserved confounder) changes, of course, the implications for identification.

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  • $\begingroup$ Thank you so much! I have also seen bidirectional arrows in multiple articles. Am I correct, that I have to test the direction of each arrow using regression with my study data? If so, could you please check my response to Robert Long and maybe help to elaborate how to do this (my variables are categorical or continuous and non-normal and zero-inflated). $\endgroup$ – st4co4 Sep 9 '20 at 11:15
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    $\begingroup$ You cannot "test" causality in observational settings (and it is also often harder than assumed with experimental data). To repeat myself, you have to make assumptions: if you can draw a plausible DAG where you can identify the quantity youre interested and estimate an effect you can interpret it, internally consistent, as causal. But other causal models might be plausible too. DAGs make this transparent. For regression, you need to invoke further assumtions about functional form etc. DAGs in the form above are for identifying causal effects in a non-parametric setting. $\endgroup$ – persephone Sep 9 '20 at 11:51

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