The Book of Why: The New Science of Cause and Effect

by Judea Pearl

In the last chapter, we looked at three rules that tell us how to stop the flow of information through any individual junction. I will repeat them for emphasis: (a) In a chain junction, A B C, controlling for B prevents information about A from getting to C or vice versa. (b) Likewise, in a fork or confounding junction, A B C, controlling for B prevents information about A from getting to C or vice versa. (c) Finally, in a collider, A B C, exactly the opposite rules hold. The variables A and C start out independent, so that information about A tells you nothing about C. But if you control for B, then information starts flowing through the “pipe,” due to the explain-away effect. We must also keep in mind another fundamental rule: (d) Controlling for descendants (or proxies) of a variable is like “partially” controlling for the variable itself. Controlling for a descendant of a mediator partly closes the pipe; controlling for a descendant of a collider partly opens the pipe. Now, what if we have longer pipes with more junctions, like this: A B C D E F G H I J? The answer is very simple: if a single junction is blocked, then J cannot “find out” about A through this path. So we have many options to block communication between A and J: control for B, control for C, don’t control for D (because it’s a collider), control for E, and so forth. Any one of these is sufficient. This is why the usual statistical procedure of controlling for everything that we can measure is so misguide...