How Likely Is Simpson’s Paradox?

@article{Pavlides2009HowLI,
  title={How Likely Is Simpson’s Paradox?},
  author={Marios G. Pavlides and Michael D. Perlman},
  journal={The American Statistician},
  year={2009},
  volume={63},
  pages={226 - 233}
}
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… 
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