Viewing Simpson's Paradox

TL;DR

This paper explains Simpson's paradox using simple algebra and geometry, clarifying when and how the reversal of associations occurs, and discusses its implications in predictive and causal contexts.

Contribution

It provides an accessible geometric and algebraic explanation of Simpson's paradox, explicitly states the conditions for its occurrence, and discusses its relevance in practical and causal reasoning contexts.

Findings

The paradox can occur under non-extreme dependence between variables.

It is always possible to define a third variable algebraically to produce the paradox.

The occurrence of the paradox depends on the context and interpretation by domain experts.

Abstract

Well known Simpson's paradox is puzzling and surprising for many, especially for the empirical researchers and users of statistics. However there is no surprise as far as mathematical details are concerned. A lot more is written about the paradox but most of them are beyond the grasp of such users. This short article is about explaining the phenomenon in an easy way to grasp using simple algebra and geometry. The mathematical conditions under which the paradox can occur are made explicit and a simple geometrical illustrations is used to describe it. We consider the reversal of the association between two binary variables, say, $X$ and $Y$ by a third binary variable, say, $Z$. We show that it is always possible to define $Z$ algebraically for non-extreme dependence between $X$ and $Y$, therefore occurrence of the paradox depends on identifying it with a practical meaning for it in a given context of interest, that is up to the subject domain expert. And finally we discuss the paradox in predictive contexts since in literature it is argued that the paradox is resolved using causal reasoning.