Simpson's Paradox

Simpson's paradox arises in the analysis of cross-classified categorical data when contradictory conclusions about the directions of associations[Page 893] between variables are reached as a result of several groups' being aggregated into one. A well-known example of this phenomenon is the graduate school admissions data from the University of California, Berkeley. During a particular time, 8,442 men applied for graduate school, as did 4,321 women. Each department made its own admissions decisions. In the end, approximately 44% of the male applicants and 35% of the female applicants were admitted. This looks like evidence of sex bias in admissions, and the university administration wished to determine which departments exhibited the most serious bias. Here we explore only a subset of the data, namely the six largest departments. The data, as reported by Freedman and colleagues, are in Table 1.

Looking at each department individually, it is apparent that there is no systematic bias against women; on the contrary, in all six departments, the percentage of women accepted is higher than, or almost equal to, the percentage of men accepted. Indeed, for department A, 82% of female applicants were admitted, compared to only 62% of male applicants. The department that is least favorable to women, department E, shows a discrepancy of only 4%. Yet over all six departments together, 45% of the male applicants were admitted, and only 30% of the female applicants.

Closer examination of the data reveals the source of this seemingly paradoxical situation. Departments A and B were relatively liberal in their admissions policies, accepting two thirds or more of the applicants. And most men applied to those two departments. On the other hand, departments C through F were much harder to get into, accepting about a third or less of prospective graduate students. And women applied mostly to those highly competitive departments. The overall effect is that a smaller percentage of women than men were accepted into the graduate programs at Berkeley because they were aiming for more restrictive courses of study. By aggregating over departments, university officials lost vital information.

This is the source of Simpson's paradox, namely, aggregating, or collapsing, over levels in cross-tabulated data when it is inappropriate to do so because of interactions among variables. The presence of interactions means that important and relevant information is lost when the table is collapsed. In the simple example of the graduate admissions data, the problem was easy to spot because the data were subjected to only a two-way classification. As the number of cross-classifications rises, inducing more complex and more numerous types of interactions, detection might not be so immediate. Care should be taken, therefore, when aggregating over levels of classification.

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