Standard Article

Simpson's Paradox

  1. Jianping Dong

Published Online: 15 JUL 2005

DOI: 10.1002/0470011815.b2a10055

Encyclopedia of Biostatistics

Encyclopedia of Biostatistics

How to Cite

Dong, J. 2005. Simpson's Paradox. Encyclopedia of Biostatistics. 7.

Author Information

  1. Michigan Technological University, Houghton, MI, USA

Publication History

  1. Published Online: 15 JUL 2005


Simpson's paradox occurs when the direction of a measure of association between two variables is reversed after pooling over a covariate. For example, a treatment can be effective for both males and females, but ineffective when the data for males and females are combined. Since Simpson's original example in his 1951 paper, numerous real-life examples of Simpson's paradox have been reported in many areas and the paradox has been generalized to many forms of association reversals. Necessary and sufficient conditions for Simpson's paradox as well as generalized forms of association reversals are given so we know, for example, when a high-dimensional contingency table can be safely collapsed, avoiding the paradox. It is also of interest to know what the appropriate conclusion is when Simpson's paradox does occur.


  • odds ratio;
  • association measure;
  • contingency table;
  • marginal table;
  • pooling;
  • collapsibility;
  • association reversal;
  • amalgamation paradox