Moderator, Mediator, Confounding Variable in Causal Diagram
This is a quick information discussion of moderator, mediator, and confounding variables: what they are, and how they are drawn in a causal diagram.
Basic Causal Diagram of Mediator
Mediating relationship explains how a third variable C is associated with the relationship between two other variables A and B. It means that A affects B through C.
On the left, we can see that A has a direct effect on B, and another effect mediated by C.
On the right, we can see that the effect of A to B is mediated by C
Basic Mediator Testing
There have been many developments ever since the original work by Baron and Kenny on identifying mediators, but here I briefly discuss the intuition behind one of the earliest works (perhaps, as far as I know) on testing mediation outlined in "The moderator-mediator variable distinction in social psychological research: conceptual, strategic, and statistical consideration" in 1986 JPSP
Three regressions can be used to test mediation:
, first, we want to see the independent variable A affects the mediator C.
, second, we want to see that independent variable A affects the dependent variable B
, third, we want to see that mediator C affects the dependent variable B
If all three is true, then we know the effect of the independent variable A on dependent variable B must be mediated by C.
Perfect mediation (top right) holds that when regression shows that A has no significant effect and C has. (Because A must affect C, having both A and C in the regression already can introduce other issues, but this is temporarily outside of the scope of this note).
Basic Causal Diagram of Moderator
Moderating relationship also explains how a third variable C is associated with the relationship between two other variables A and B. It means that the effect of A on B is dependent on C.
However, in a causal graph, displaying a moderating relationship is a bit more chaotic. Below are four different ways you might encounter them.
Top left: The technically correct way
Surprise! Technically this is the correct version. Because in a causal diagram only describes what variables affect what variables, it does not display how. So the graph only tells us that both A and C affect B, but below are some different ways all represented by the same diagram
: In this example, we can see that A and C are independently affecting B
: In this example, we can see that A is affecting B, and also there is a moderation effect of C on the relationship between A and B.
: In this example, we can see that A, and C are both independently affecting C, and there is also a moderation effect.
As Judea Pearl said in this Tweet, In causal diagrams every variable in understood to be a moderator.
Top right: The most explicit way
Although the top left representation is correct, it is a bit hard to explicitly show the moderating relationship. The top right is one way to show the moderating relationship very explicitly, this is also featured in Baron & Kenny, 1986. This sort of causal diagram assumes that an arrow from A to B only means that A is independently affecting B. Here is a breakdown:
The arrow from A to B: A is affecting B
The arrow from A to A*C: A is affecting the interaction term A*C
The arrow from C to A*C: C is affecting the interaction term A*C
The arrow from A*C to is affecting B (this is the interaction effect)
No arrow from C to B: This means that C only affects B through the interaction term
Bottom: Common Alternative and Mathematical Equivalency
Another way to represent the moderation relationship is by pointing the arrow to another arrow instead of a node. So in the bottom left, we can see that C has an arrow pointing to the arrow from A to B. This means that the C is affecting how A affects B, or, the effect of A to B is moderated by C.
Mathematically, interaction is symmetrical. So if the effect of A to B is moderated by C, then the effect of C to B is also moderated by A. That's what the bottom right graph shows. However, in terms of theoretical interpretation, they are different.
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