# Book of Why, causal diagram 9.5 - how does that represent Kruskal's argument?

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 -

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?

• 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).
– Eoin
Commented Feb 7, 2022 at 15:37
• @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? Commented Feb 7, 2022 at 16:26
• That's what the linked answer says, in any case. It was news to me too.
– Eoin
Commented Feb 8, 2022 at 8:08

• 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). Commented Feb 7, 2022 at 16:16
• 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. Commented Feb 7, 2022 at 16:58
• 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. Commented Feb 7, 2022 at 17:52