A technical question for the #DAG oracles of #EconTwitter, please.

How do we describe the bias created by conditioning on a variable that is *nearly* a necessary and sufficient condition for treatment?

It can falsely seem like a 'good control'!

I'll explain in a short thread—>

In general, conditioning on an omitted variable like Z in this DAG gives an unbiased estimate of the average treatment effect of X on Y.

Z is a "good control", in this figure from the excellent guide by @analisereal, Forney, and @YudaPearl —> https://ftp.cs.ucla.edu/pub/stat_ser/r493.pdf

But suppose that Z is almost a necessary & sufficient condition for X.

E.g., we are studying the effect of schooling on test scores (Y). X is the completion of 6th grade. Z is the completion of *5th* grade.

The above DAG applies: both completing 5th and 6th grade affect scores.

But if I condition on the "good control" Z, I alter interpretation of the treatment effect E[Y|X=1] – E[Y|X=0].

Completing 5th grade is *almost* a necessary condition for completing 6th grade. ('Almost', because a few very unusual kids might enter the system at grade 6.)

Completing 5th grade is also *almost* a sufficient condition for completing 6th grade. ('Almost', because a few very unusual kids might drop out of all schooling after grade 5.)

So if I study the effect of X on Y controlling for the 'good control' Z, in this case I might well find that there is no general "effect" of completing 6th grade on scores.

But we sense that that would be a meaningless conclusion. If we condition on completing grade 5, the "effect" of X on Y is only the effect of being one of those highly unusual kids who enter or leave the system between grades 5 and 6.

That 'effect' might be completely different than the average treatment effect of grade 6 in the whole population, precisely because those kids are so unusual.

It could even be zero. But that would be uninformative to a policymaker interested in the effect of 6th grade on scores.

So Z is not at all a 'good control' in this case because it would not help us identify the effect we're interested in. Indeed it seems to take us very far away from the effect of interest.

So notwithstanding the DAG above, Z is a 'bad control' here.

This problem is closely related to @HeckmanEquation's point in his 2000 @QJEHarvard paper: https://doi.org/10.1162/003355300554674

Our original claim that the effect of interest is the effect of X on Y implicitly assumes we can cause X to vary independently of Z. (X and Z are 'variation free'.)

It is *possible* for a few students to get a 6th grade education without getting a 5th grade education, and vice versa.

But that is a different "effect", Heckman writes, than the "effect" any policymaker would be interested in: the effect of "completing 5th and then 6th grade".

My question is: What is our terminology for this bias?

The problem is worse than just reducing the useful variation in X, and thereby reducing the *precision* of the estimated treatment effect, as in this 'neutral control' case in the @analisereal, Forney, and @yudapearl paper.

Rather, in the present case conditioning on Z means estimating the treatment effect only for a tiny and perhaps unrepresentative slice of the population.

Is this a form of "bias amplification"?

If not, then what is the terminology we use to describe this bias?

Put differently, how do we analyze the bias created by conditioning on a *pre treatment* variable that is nearly a necessary and sufficient condition for treatment?

I apologize if this is an ignorant question. (I am pretty sure it is.) But I am a newcomer to @yudapearl's monumental #BookOfWhy research program. If it is easy for you to see what I'm missing, I'd greatly appreciate a pointer. Thank you.

Update: I learned that essentially this problem is also called the "Table Two Fallacy" in this insightful piece by @EpidByDesign & @Lester_Domes —> https://doi.org/10.1093/aje/kws412

That is, it might better be described as a "fallacy" than a "bias". Thank you (!) to all who commented.