Simpson's Paradox and Related Phenomena

Abstract

Simpson's paradox is viewed as one of a natural and coherent collection of association reversal phenomena that are of fundamental importance in statistical practice. Association reversal means that the direction of association between two variables X and Y is changed by collapsing (unconditioning) over a covariate Z; an example is Simpson's paradox for contingency tables. This article gives necessary and sufficient conditions for Simpson's paradox and for more general forms of association reversal. Close connections are noted with amalgamation paradoxes, defined as situations where an unconditional measure of association between X and Y lies outside the range of the conditional (on Z) measures. Emphasis throughout is on statistical interpretation and on commonalities and contrasts between the paradoxical phenomena in various settings.

KEY WORDS:

• Amalgamation
• Association
• Confounding
• Contingency table