Game 3
"In Games 1 and 2 you didn’t have to do anything, but this time you do. There is one back-door path from X to Y, X←B→Y, which can only be blocked by controlling for B. If B is unobservable, then there is no way of estimating the effect of X on Y without running a randomized controlled experiment. Some (in fact, most) statisticians in this situation would control for A, as a proxy for the unobservable variable B, but this only partially eliminates the confounding bias and introduces a new collider bias." (Pearl, p. 160)
Game 3 in BayesiaLab
As with the earlier games, we encode Game 3 as a causal Bayesian network graph:
Again, the probabilities are fictitious and irrelevant.
We select
Main Menu > Analysis > Visual > Graph > Influence Paths to Targetto analyze the paths from X to Y.
Given the presence of a noncausal path (highlighted in pink), it becomes clear that we need to control for B to block that path.
Here, "fixing the probabilities" of B are a practical way of controlling for that variable. Note that the states and the values of the variable are irrelevant.
Now, after controlling for B, only one causal path remains, highlighted in blue, which allows us to estimate the effect of X and Y.
However, if B were unobservable ("not observable" or "hidden" in BayesiaLab terminology), some statisticians would perhaps propose to control for A as a proxy of B.
Let's try that scenario as well. We are now fixing A while leaving B "open."
The Influence Path Analysis reveals that controlling for proxy A does not achieve our objective.
Not only does it not block the noncausal path X←B→Y, controlling for A introduces an additional noncausal path X→A ←B→Y, i.e., another bias that prevents us from estimating the effect of X on Y.
This phenomenon is known as "collider bias," as it is produced by conditioning on a collider, such as A.
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