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:
BoW_BackDoor3.xbl (opens in a new tab)
- Again, the probabilities are fictitious and irrelevant.
- We select
Menu > Analysis > Visual > Graph > Influence Paths to Targetto analyze the paths from X to Y.
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Given the presence of a noncausal path (highlighted in pink), it becomes clear that we need to control for B to block that path.
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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.
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Now, after controlling for B, only one causal path remains, highlighted in blue, which allows us to estimate the effect of X and Y.
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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.
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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.