Simpson's Paradox

Illustration

Understanding Simpson's paradox is easiest in the context of a simple example. The inspiration for the one presented here is the classic article published by Peter Bickel and his colleagues.

Suppose that a university is concerned about gender bias during the admission process to graduate school. To study this, applicants to the university's graduate programs are classified based on gender and admissions outcome. These data, which are summarized in the cross-classification table depicted in Table 1, would seem to be consistent with the existence of a gender bias because men (40%) were more likely to be admitted to graduate school than women (25%).

To identify the source of the difference in admission rates for men and women, the university subdivides applicants based on whether they applied to a department in the natural sciences or to one in the social sciences and then conducts the analysis again (see Table 2). Surprisingly, the

Table 1 Graduate Admissions by Gender

Table 2 Graduate Admissions by Gender by Department

university finds that the direction of the relationship between gender and outcome has reversed! In natural science departments, women (80%) were more likely to be admitted to graduate school than men (46%); similarly, in social science departments, women (20%) were more likely to be admitted to graduate school than men (4%).

Although the reversal in association that is observed in Simpson's paradox might seem bewildering, it is actually straightforward. In this example, it occurred because both gender and admissions were related to a third variable, namely, the department. First, women were more likely to apply to social science departments, whereas men were more likely to apply to natural science departments. Second, the acceptance rate in social science departments was much less than that in natural science departments. Because women were more likely than men to apply to programs with low acceptance rates, when department was ignored (i.e., when the data were collapsed over department), it seemed that women were less likely than men to be admitted to graduate school, whereas the reverse was actually true. Although hypothetical examples such as this one are simple to construct, numerous real-life examples can be found easily in the social science and statistics literatures.

Definition

Consider three random variables X, Y, and Z. Define a 2 × 2 × K cross-classification table by assuming that X and Y can be coded either 0 or 1, and Z can be assigned values from 1 to k.

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