Abstract
There are three distinct questions associated with Simpson’s paradox. (i) Why or in what sense is Simpson’s paradox a paradox? (ii) What is the proper analysis of the paradox? (iii) How one should proceed when confronted with a typical case of the paradox? We propose a “formal” answer to the first two questions which, among other things, includes deductive proofs for important theorems regarding Simpson’s paradox. Our account contrasts sharply with Pearl’s causal (and questionable) account of the first two questions. We argue that the “how to proceed question?” does not have a unique response, and that it depends on the context of the problem. We evaluate an objection to our account by comparing ours with Blyth’s account of the paradox. Our research on the paradox suggests that the “how to proceed question” needs to be divorced from what makes Simpson’s paradox “paradoxical.”
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References
Blyth C. (1972) On Simpson’s paradox and the sure-thing principle. Journal of the American Statistical Association 67(338): 364–366 (Theory and Method Section)
Cartwright N. (1979) Causal laws and effective strategies. Nous 13: 419–437
Cartwright N. (1999) The dappled word: A study of the boundaries of science. Cambridge University Press, UK, Cambridge
Clark M. (2002) Paradoxes from A to Z. Routledge, London
Elles E., Sober E. (1983) Probabilistic causality and the question of transitivity. Philosophy of Science 50: 35–57
Freedman D., Pisani R., Purve R. (1999) Statistics (3rd ed.). W. W. Norton & Company, New York
Good I. J., Mittal Y. (1988) The amalgamation and geometry of two-by-two contingency tables. The Annals of Statistics 15(2): 694–711
Greenland S., Robins J. M., Pearl J. (1999) Confounding and collapsibility in causal inference. Statistical Science 19: 29–46
Hausman D. (1998) Causal asymmetries. Cambridge University Press, Cambridge
Hoover K. (2001) Causality in microeconomics. Cambridge University Press, England
Kahneman, D., Slovic, P., Tversky, A. (eds) (1982) Judgment under uncertainty: Heuristics and basics. Cambridge University Press, England
Kyburg H. (1997) The rule of adjunction and reasonable inference. Journal of Philosophy XCIV(3): 109–125
Lindley D., Novick M. (1981) The role of exchangeability in inference. Annals of Statistics 9(1): 45–58
Malinas G. (2001) Simpson’s paradox: A logically benign, empirically treacherous hydra. The Monist 84(2): 265–283
Meek C., Glymour C. (1994) Conditioning and intervening. British Journal for the Philosophy of Science 45: 1001–1021
Mittal Y. (1991) Homogeneity of subpopulations and Simpson’s paradox. Journal of the American Statistical Association 86: 167–172
Morton A. (2002) If you’re so smart why are you ignorant? Epistemic causal paradoxes. Analysis 62(2): 110–116
Novick M. R. (1983) The centrality of Lord’s paradox and exchangeability for all statistical inference. In: Wainer H., Messick S. (eds) Principles of modern psychological measurement. Erlbaum, Hillsdale, NJ
Otte R. (1985) Probabilistic causality and Simpson’s paradox. Philosophy of Science 52(1): 110–125
Pearl J. (2000) Causality (1st ed). Cambridge University Press, Cambridge
Pearl J. (2009) Causality (2nd ed). Cambridge University Press, Cambridge
Rothman K., Greenland S. (1998) Modern epidemiology (2nd ed.). Lippincott Williams, Philadelphia
Savage L. (1954) Foundations of statistics. Wiley, New York
Simpson H. (1951) The interpretation of interaction in contingency tables. Journal of the Royal Statistical Society. Series. B 13(2): 238–241
Skyrms B. (1980) Causal necessity. Yale University Press, New York
Sloman S. (2005) Causal models: How people think about the world and its alternatives. Oxford University Press, New York
Sober E., Wilson D. (1998) Unto others: The evolution and psychology of unselfish behavior. Harvard University Press, Mass
Spirtes P., Glymour C., Scheines R. (2000) Causation, prediction, and search (2nd ed.). MIT Press, Cambridge
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Davin Nelson was a former student of Montana State University.
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Bandyoapdhyay, P.S., Nelson, D., Greenwood, M. et al. The logic of Simpson’s paradox. Synthese 181, 185–208 (2011). https://doi.org/10.1007/s11229-010-9797-0
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DOI: https://doi.org/10.1007/s11229-010-9797-0