How Likely Is Simpson’s Paradox?

Marios G. Pavlides Marios G. Pavlides is Lecturer of Mathematics and Statistics, Frederick University Cyprus, Box 24729, 1303 Nicosia, Cyprus . Michael D. Perlman is Professor, Department of Statistics, University of Washington, Box 354322, Seattle, WA 98195 . We warmly thank Fred Bookstein, Peter Hoff, Thomas Richardson, Jon Wakefield, Jon Wellner, the Editor, Associate Editor, and an anonymous referee for many valuable comments. We are especially grateful to Petros Hadjicostas for generously contributing the results in Section 5.

Michael D. Perlman Marios G. Pavlides is Lecturer of Mathematics and Statistics, Frederick University Cyprus, Box 24729, 1303 Nicosia, Cyprus . Michael D. Perlman is Professor, Department of Statistics, University of Washington, Box 354322, Seattle, WA 98195 . We warmly thank Fred Bookstein, Peter Hoff, Thomas Richardson, Jon Wakefield, Jon Wellner, the Editor, Associate Editor, and an anonymous referee for many valuable comments. We are especially grateful to Petros Hadjicostas for generously contributing the results in Section 5.

Abstract

What proportion of all 2×2×2 contingency tables exhibit Simpson’s Paradox? An exact answer is obtained for large sample sizes and extended to 2×2×ℓ tables by Monte Carlo approximation. Conditional probabilities of the occurrence of Simpson’s Paradox are also derived. If the observed cell proportions satisfy a Simpson reversal, the posterior probability that the population parameters satisfy the same reversal is obtained. This Bayesian analysis is applied to the well-known Simpson reversal of the 1995–1997 batting averages of Derek Jeter and David Justice.

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