Lurking variables and relationships between categorical variables

The relationship between two categorical variables, X and Y, can also be strongly influenced by a third lurking variable, Z. Repeating the points made before,

The marginal relationship between X and Y can be...

compared to their conditional relationship, given Z.

When the direction of the relationship reverses, the effect is called Simpson's paradox. As with other 'paradoxes', there is no real contradiction; it just takes a bit more thought to understand why your initial intuition is wrong.

College admission rates for males and females

The table below describes admission rates of male and female students into the Business and Law Schools of a university.

A naive examination of the data suggests that females have a lower probability of being admitted — is there discrimination?

Click Slice to see the corresponding data for the two different degrees (MBA and Law). Observe that the probability of admission is higher for the females in each degree programme. The programme is a lurking variable and the conditional relationship between gender and admission is the reverse of their marginal relationship.


This reversal can be illustrated in a proportional Venn diagram. In this diagram,

The area of the rectangle for any combination of gender, programme and admission is proportional to the number of applicants in that combination.


Initially the rectangles are arranged to display the conditional proportions. Click any blue rectangle to display the proportion admitted to that programme for males and females, and observe that P(admission) is higher for females than males for each degree.

Now select Group by Gender from the pop-up menu. The rectangles are rearranged to show the overall proportions of males and females admitted. Observe that the overall proportion of females getting admission is less than the overall proportion of males.

The reason for the reversal of the relationship is that most males applied to the Business school with its high admission rate, increasing the marginal proportion of admissions for the males.

The reason for the reversal of the relationship is that most males applied to the Business school with its high admission rate, increasing the marginal proportion of admissions for the males.