Simpson's Paradox with Any Given Number of Factors

TL;DR

This paper generalizes Simpson's Paradox to any number of factors, showing that adding more variables can repeatedly reverse the perceived effect of a factor, challenging assumptions about data and inference accuracy.

Contribution

It introduces the concept of n-factor Simpson's Paradox, providing a formal definition and geometric construction demonstrating its existence for any number of factors.

Findings

Existence of probability distributions exhibiting n-factor Simpson's Paradox for any n

Construction of explicit examples for n=3

Reveals complexity of statistical inference with multiple confounders

Abstract

Simpson's Paradox is a well-known phenomenon in statistical science, where the relationship between the response variable $X$ and a certain explanatory factor of interest $A$ reverses when an additional factor $B_1$ is considered. This paper explores the extension of Simpson's Paradox to any given number $n$ of factors, referred to as the $n$-factor Simpson's Paradox. We first provide a rigorous definition of the $n$-factor Simpson's Paradox, then demonstrate the existence of a probability distribution through a geometric construction. Specifically, we show that for any positive integer $n$, it is possible to construct a probability distribution in which the conclusion about the effect of $A$ on $X$ reverses each time an additional factor $B_i$ is introduced for $i=1,...,n$. A detailed example for $n = 3$ illustrates the construction. Our results highlight that, contrary to the intuition that more data leads to more accurate inferences, the inclusion of additional factors can repeatedly reverse conclusions, emphasizing the complexity of statistical inference in the presence of multiple confounding variables.