In the Book of Why, Judea Pearl, there is the case of a Berkley admission gender discrimination paradox which was solved by Peter Bickel. The solution was done by searching for discrimination department-wise (conditioning on departments). However, Kruskal later showed a counterexample clarifying that conditioning on department was correct only if department and outcome (admission result) are unconfounded.

Kruskal hypothesized a department with the following admission criteria -

accept all in-state males and out-of-state females and reject all out-of-state males and in-state females.

In the book, the causal diagram is provided in figure 9.5 as below -

Diagram 9.5

My question is, how come the dependence of the criteria on State of Residence be depicted by drawing causal arrows from State of Residence to Department and Outcome respectively. To me it seems that, it is the function behind the arrow Department$\rightarrow$Outcome that is dependent on State of Residence and Gender. How is this causal diagram then an appropriate representation of Kruskal's argument?

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    $\begingroup$ I think part of the reason this is confusing is that interaction effects are not explicitly represented in this kind of causal diagrams (discussed excellently in this answer), and Kruskal's hypothesis corresponds to a state x gender interaction (or a department x state x gender interaction, if only some departments have this policy). $\endgroup$
    – Eoin
    Commented Feb 7, 2022 at 15:37
  • $\begingroup$ @Eoin - So what you mean to say is that Department$\rightarrow$Outcome, State$\rightarrow$Outcome and Gender$\rightarrow$Outcome together will be capable of denoting the interaction type relationship assumed in Kruskal's hypothesis. Am I correct? $\endgroup$ Commented Feb 7, 2022 at 16:26
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    $\begingroup$ That's what the linked answer says, in any case. It was news to me too. $\endgroup$
    – Eoin
    Commented Feb 8, 2022 at 8:08

1 Answer 1


Well, according to the rules of causal diagrams, you can only draw arrows from one node to another. You cannot draw an arrow from a node into an arrow. So the real question is this: are the arrows from State of Residence going into Department and Outcome both correct?

  1. State of Residence into Department. Different regions of the country have different problems, and it is not unreasonable to think that residents of New York City might be more inclined to go into finance (a hot item in NYC, or at least it was), and residents of North Dakota more inclined to go into petroleum engineering (quite a hot item there). So the idea of State of Residence influencing Department seems sound.

  2. State of Residence into Outcome. This is an accepted arrow by virtue of the discussion on page 313, where you quoted the most critical part. That is, this arrow partially models the blatant discrimination that Kruskal hypothesized. The arrow from Gender into Outcome models the other half of this discrimination.

  • $\begingroup$ referring to point 2 of your answer - can you show how the hypothesis assumed by Kruskal can be captured using the Gender$\rightarrow$Outcome and State$\rightarrow$Outcome? Maybe by assuming a binary coding for Gender and State (in-state and out-of-state). $\endgroup$ Commented Feb 7, 2022 at 16:16
  • $\begingroup$ I'm not sure I understand your question. The paragraph on page 313 beginning with "In his letter to Bickel, Kruskal pointed out..." shows that the assumption includes gender influencing outcome (acceptance or rejection) and state of residence (in-state versus out-of-state) influencing outcome. $\endgroup$ Commented Feb 7, 2022 at 16:36
  • $\begingroup$ it says that the department accepts all in-state males and out-of-state females. Hence, given an in-state applicant the acceptance or rejection depends on the gender. Thus the function that might be present behind the arrow State$\rightarrow$Outcome should be depend on the state of Gender. How can such interaction terms be represented by the diagram? You can refer to the first comment by Eoin in the comments against the question. $\endgroup$ Commented Feb 7, 2022 at 16:58
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    $\begingroup$ Well, according to Noah's answer to which Eoin linked, "...DAGs already allow for interaction without any adjustment." As Noah also said, imposing a DAG as a model implies only the lightest of assumptions about the functional forms of the model. $\endgroup$ Commented Feb 7, 2022 at 17:26
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    $\begingroup$ Addendum: if the node $A$ has arrows from nodes $B$ and $C,$ you only need model this in the Structural Equation Model framework as $A=f(B,C),$ which you can see is quite general and capable of the interaction present here. $\endgroup$ Commented Feb 7, 2022 at 17:52

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