The Book of Why: The New Science of Cause and Effect

by Judea Pearl

Unlike Kruskal, we can draw a diagram and see exactly what the problem is. Figure 9.5 shows the causal diagram representing Kruskal’s counterexample. Does it look slightly familiar? It should! It is exactly the same diagram that Barbara Burks drew in 1926, but with different variables. One is tempted to say, “Great minds think alike,” but perhaps it would be more appropriate to say that great problems attract great minds. FIGURE 9.5. Causal diagram for Berkeley admissions paradox—Kruskal’s version. Kruskal argued that the analysis in this situation should control for both the department and the state of residence, and a look at Figure 9.5 explains why this is so. To disable all but the direct path, we need to stratify by department. This closes the indirect path Gender Department Outcome. But in so doing, we open the spurious path Gender Department State of Residence Outcome, because of the collider at Department. If we control for State of Residence as well, we close this path, and therefore any correlation remaining must be due to the (discriminatory) direct path Gender Outcome. Lacking diagrams, Kruskal had to convince Bickel with numbers, and in fact his numbers showed the same thing. If we do not adjust for any variables, then females have a lower admission rate. If we adjust for Department, then females appear to have a higher admission rate. If we adjust for both Department and State of Residence, then once again the numbers show a lower admission rate for females.