Names for some canonical directed causal graphs/illustrations of some canonical causal relationships?

Question

Certain names are used for structures or node relationships that appear in acyclic, directed graphs (DAGs). Often these DAGs are interpreted causally. Here's a partial list for relationships that nodes might have to each other in such DAGs:

1. Instrument (eg, $Z$ is an instrument for $X$)
2. Cause (eg, $X$ causes $Y$)
3. Confounder (eg, $Z$ is a confounder of the influence that $X$ has on $Y$)
4. Collider (eg, $Y$ is a collider between $X$ and $Z$).

Can anyone provide a canonical list of such relationships, and illustrations of the DAGs they entail?

Answer 1

After some thought, I would say that the core relationships are (1) cause; (2) mediator; (3) collider; and (4) confounder. DAGs to illustrate these relationships and the instrument since you asked about it are below.

1. Cause

   $X \rightarrow Y$

   X is a cause of Y. X and Y are marginally associated in this DAG, by which I mean that $E[Y|X] \neq E[Y]$, and vice versa.

2. Mediator

   $X \rightarrow Z \rightarrow Y$

   Z is a mediator (intermediate) for the effect of X on Y. X and Y are marginally associated in this DAG. Conditional on Z, X and Y are independent in this DAG. So, $E[Y|X] \neq E[Y]$, but $E[Y|X,Z] = E[Y|Z]$. Note, that there can be other paths from X to Y, not mediated by Z, so independence conditional on the Z is not required for Z to be a mediator.

3. Collider, aka common effect

   $X \rightarrow Z \leftarrow Y$

   Z is a collider on the path from X to Y. Z can also be described as a common effect of X and Y. Marginally, X and Y are independent in this DAG. Conditional on Z, X and Y are associated in this DAG.

4. Confounder, aka common cause

   $X \leftarrow Z \rightarrow Y$

   Z is a confounder for the effect of X on Y in this DAG. Z can also be described as a common cause of X and Y. Note that confounding is both path specific and related to what other variables are available and conditioned on. Marginally, X and Y are associated in this DAG. Conditional on Z, X and Y are independent.

5. Instrument

   This DAG combines a cause DAG, a mediator DAG, and a confounding DAG.

   Cause/Mediator portion: $Z \rightarrow X \rightarrow Y$

   Z is a cause of X. Z is associated with Y only through X (i.e. X mediates the relationship between Z and Y). There are no common causes of Z and Y, and no paths from Z to Y not mediated through X.

   Confounding portion: $X \leftarrow U \rightarrow Y$

   Although not strictly necessary for Z to be an instrument, the purpose of using an instrument is generally because there is some unmeasured or unknown common cause(s) of X and Y which create bias (from confounding) when estimating the effect of X on Y without an instrument.

   Simple Instrument DAG (under the null hypothesis of no effect of X on Y for simplicity):

   $Z \rightarrow X \leftarrow U \rightarrow Y$

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Other DAGS can be created by combining these relationships. Selection bias occurs in a DAG when conditioning on a collider or the descendent of a collider causes the independent and dependent variables of interest (putative cause and effect, or exposure and outcome) to be associated. Confounding paths can be much longer than the simple one displayed, with mediating or common cause variables between the confounder and other variables. Instrument DAGs can get more complicated when the confounding structure is more complex or when the instrument available is a proxy for the true instrument. Identification of mediators can be complicated by confounders for the mediator-outcome or exposure-mediator relationships.

Answer 2

@AzulaR. has provided a great answer. Let me add one more, less common, possible situation that is sometimes discussed.

6. Suppression

   Here is the scenario: You are interested in estimating the relationship between X and Y, but you do not have direct access to X. You do have data on P, which can be considered a proxy for X because it is partly a function of X. However, it is also partly a function of another variable S, for which you also have data. By stipulation, S is not connected to either X or Y. You can nonetheless include S in a multiple regression model, where S will act as a suppressor. That is, including S will allow the relationship between P and Y to better approximate the underlying relationship between X and Y.

   Here is a simple DAG. Circles represent latent variables, and squares represent manifest variables.