Simpson’s Paradox

Simpson’s paradox, first defined by Edward H. Simpson in 1951, is a statistical phenomenon in which the association between two variables reverses or disappears when examining aggregate versus disaggregate data of a population via a third variable. Alternative known names of Simpson’s paradox are Yule effect, reversal paradox, or amalgamation paradox.

The practical implication to decision making that Simpson’s paradox raises is the question of which level of data aggregation presents the results of interest. This question further raises the challenge of identifying potential variables and then establishing a criterion for deciding if and which of the potential variables should influence the decision making.

Figure 1 Simpson’s paradox illustration for categorical cause and outcome variables

Simpson’s paradox is commonly defined for a categorical cause variable (C) and a categorical outcome variable (E) as the phenomenon whereby an event C increases the probability of E in a given population p, at the same time, decreases the probability of E in every subpopulation of p (see Figure 1).

Mathematically, the paradox is defined for two events and their complements: Y = {E, Ec} and X = {C, Cc}, and a population Z with subpopulations {p1, p2, ..., pn}, for which the following relationship holds:

$$\begin{array}{c}P(E|C)>P(E|C^c),\text{and}\\ P(E|C{p}_i)<P(E|C^c{p}_i),\text{for each}{p}_i\subseteq Z.\end{array}$$

These inequalities can also be encountered in the form where the symbols < and > are reversed.

For continuous cause (C) and effect (E) variables, the association between Y and X are defined by two functions: the monotonic function f: Y = f(X) for the overall data, and a monotonic function g for each of the data subpopulations: Y = gi(X|pi), which has the opposite sign (see Figure 2).

Simpson’s paradox commonly arises when the underlying causal structure of the data, which is unidentifiable only from the data, is not considered by the researcher. Therefore, combining observational data with causal theory can resolve the paradox and determine the correct level of data aggregation.

We use the notation X to denote the cause variable, Y is the outcome variable, and Z is the potential reversal variable (or a vector of multiple potential reversal variables).

Figure 2 Simpson’s paradox illustration for continuous cause and outcome variables