Resolution of Simpson's paradox via the common cause principle
This provides a theoretical resolution to a statistical paradox, which is incremental as it builds on existing formulations but clarifies decision-making in specific contexts.
The paper tackles Simpson's paradox by analyzing scenarios where a common cause influences the variables, showing that for minimal cases with binary or Gaussian variables, conditioning over the lurking variable resolves the paradox consistently.
Simpson's paradox is an obstacle to establishing a probabilistic association between two events $a_1$ and $a_2$, given the third (lurking) random variable $B$. We focus on scenarios when the random variables $A$ (which combines $a_1$, $a_2$, and their complements) and $B$ have a common cause $C$ that need not be observed. Alternatively, we can assume that $C$ screens out $A$ from $B$. For such cases, the correct association between $a_1$ and $a_2$ is to be defined via conditioning over $C$. This setup generalizes the original Simpson's paradox: now its two contradicting options refer to two particular and different causes $C$. We show that if $B$ and $C$ are binary and $A$ is quaternary (the minimal and the most widespread situation for the Simpson's paradox), the conditioning over any binary common cause $C$ establishes the same direction of association between $a_1$ and $a_2$ as the conditioning over $B$ in the original formulation of the paradox. Thus, for the minimal common cause, one should choose the option of Simpson's paradox that assumes conditioning over $B$ and not its marginalization. The same conclusion is reached when Simpson's paradox is formulated via 3 continuous Gaussian variables: within the minimal formulation of the paradox (3 scalar continuous variables $A_1$, $A_2$, and $B$), one should choose the option with the conditioning over $B$.