Simpson’s paradox, a tale of causality
Journal de la société française de statistique, Volume 161 (2020) no. 1, pp. 42-66.

For the mathematically wary and unwary alike, Simpson’s paradox may well function as a permanent invitation to error. We present Simpson’s paradox and discuss its nature based on three examples. It appears that to run afoul of Simpson’s paradox it suffices to (a) conflate an invalid probabilistic reasoning with a valid instance of unassailable causal reasoning, or (b) confuse the evidential concept of learning from observation, which for rational agents proceeds by conditioning on the evidence, with the causal concept of acting, represented in causal analysis by the operation of intervening in a causal graph.

Le paradoxe de Simpson peut induire en erreur jusqu’au mathématicien prudent. Nous présentons le paradoxe de Simpson et discutons sa nature en nous appuyant sur trois exemples. Il apparaît que pour se faire prendre à son piège, il suffit (a) de combiner un raisonnement probabiliste hasardeux avec un raisonnement causal inattaquable, ou bien (b) de confondre le concept évidentiel d’apprentissage à partir de l’observation, qui pour des agents rationnels procède par conditionnement selon les données, avec le concept causal d’action tel qu’il est représenté en analyse causale par une intervention dans un graphe.

Classification: 60-01, 62A01
Keywords: causality, Simpson’s paradox
Mot clés : causalité, paradoxe de Simpson
Chambaz, Antoine 1; Drouet, Isabelle 2

1 MAP5 (UMR CNRS 8145), Université Paris Descartes, 45 rue des Saints-Pères, 75270 Paris cedex 06.
2 Sciences, Normes, Décision (SND, FRE CNRS 3593), Sorbonne Université.
     author = {Chambaz, Antoine and Drouet, Isabelle},
     title = {Simpson{\textquoteright}s paradox, a tale of causality},
     journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique},
     pages = {42--66},
     publisher = {Soci\'et\'e fran\c{c}aise de statistique},
     volume = {161},
     number = {1},
     year = {2020},
     mrnumber = {4125248},
     zbl = {1451.60002},
     language = {en},
     url = {}
AU  - Chambaz, Antoine
AU  - Drouet, Isabelle
TI  - Simpson’s paradox, a tale of causality
JO  - Journal de la société française de statistique
PY  - 2020
SP  - 42
EP  - 66
VL  - 161
IS  - 1
PB  - Société française de statistique
UR  -
LA  - en
ID  - JSFS_2020__161_1_42_0
ER  - 
%0 Journal Article
%A Chambaz, Antoine
%A Drouet, Isabelle
%T Simpson’s paradox, a tale of causality
%J Journal de la société française de statistique
%D 2020
%P 42-66
%V 161
%N 1
%I Société française de statistique
%G en
%F JSFS_2020__161_1_42_0
Chambaz, Antoine; Drouet, Isabelle. Simpson’s paradox, a tale of causality. Journal de la société française de statistique, Volume 161 (2020) no. 1, pp. 42-66.

[1] Beckett, Samuel En attendant Godot, Editions de Minuit, Paris, 1952

[2] Blyth, Colin R. On Simpson’s paradox and the sure-thing principle, J. Amer. Statist. Assoc., Volume 67 (1972), p. 364-366, 373–381 (With comments by D. V. Lindley, I. J. Good, Robert L. Winkler and John W. Pratt, and a rejoinder by Colin R. Blyth) | DOI | MR | Zbl

[3] Bandyoapdhyay, Prasanta S.; Nelson, Davin; Greenwood, Mark; Brittan, Gordon; Berwald, Jesse The logic of Simpson’s paradox, Synthese, Volume 181 (2011) no. 2, pp. 185-208 | DOI | MR | Zbl

[4] Caillois, Roger Les jeux et les hommes, Editions Gallimard, 1958 (Le masque et le vertige)

[5] Carnap, Rudolf Logical foundations of probability, University of Chicago press, Chicago, 1962 | MR

[6] Cohen, Morris R; Nagel, E. An Introduction to Logic and the Scientific Method, 1934 | JFM

[7] Chuang, John S.; Rivoire, Olivier; Leibler, Stanislas Simpson’s paradox in a synthetic microbial system, Science, Volume 323 (2009) no. 5911, pp. 272-275 | DOI

[8] Charig, Clive R; Webb, David R; Payne, Stephen Richard; Wickham, John E Comparison of treatment of renal calculi by open surgery, percutaneous nephrolithotomy, and extracorporeal shockwave lithotripsy., BMJ (Clin Res Ed), Volume 292 (1986) no. 6524, pp. 879-882 | DOI

[9] Fitelson, Branden Confirmation, causation, and Simpson’s paradox, Episteme, Volume 14 (2017) no. 3, pp. 297-309 | DOI

[10] Gibbard, Allan; Harper, William L. Counterfactuals and two kinds of expected utility, Ifs. Conditionals, Belief, Decision, Chance and Time (Harper, William L.; Stalnaker, R.; Pearce, G., eds.) (The Western Ontario Series in Philosophy of Science), Springer, 1978, pp. 153-190 | Zbl

[11] Julious, Steven A.; Mullee, Mark A. Confounding and Simpson’s paradox, BMJ, Volume 309 (1994) no. 6967, pp. 1480-1481 | DOI

[12] Joyce, James M. The foundations of causal decision theory, Cambridge Studies in Probability, Induction, and Decision Theory, Cambridge University Press, Cambridge, 1999, xii+268 pages | DOI | MR

[13] Memetea, Sonia Simpson’s paradox in epistemology and decision theory, Ph. D. Thesis , University of British Columbia (2015)

[14] Pearl, Judea Causality, Cambridge University Press, Cambridge, 2000, xvi+384 pages (Models, reasoning, and inference) | MR

[15] Pearson, Karl Mathematical contributions to the theory of evolution. VII. On the correlation of characters not quantitatively measurable, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Volume 195 (1900), pp. 1-405 | JFM

[16] Pearl, Judea Simpson’s paradox: An anatomy (2011) ( Technical report )

[17] Pearl, Judea Comment: understanding Simpson’s paradox, The American Statistician, Volume 68 (2014) no. 1, pp. 8-13 | DOI | MR

[18] Pearl, Judea The Sure-Thing Principle, Journal of Causal Inference, Volume 4 (2016) no. 1, pp. 81-86 | DOI | MR

[19] Quine, Willard V. The ways of paradox, The ways of paradox and other essays, Harvard University Press, Cambridge, Massachusetts and London, England (1976), pp. 2-18

[20] Salmon, Wesley C Confirmation and relevance, Minnesota Studies in the Philosophy of Science, Volume 6 (1975), pp. 3-36

[21] Savage, Leonard J. The foundations of statistics, Dover Publications, Inc., New York, 1972, xv+310 pages | MR

[22] Shpitser, Ilya Identification in Causal Models With Hidden Variables, Journal de la Société Française de Statistique (2019) | MR

[23] Yule, G Udny On the association of attributes in statistics: with illustrations from the material of the childhood society, &c, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Volume 194 (1900), pp. 257-319 | JFM