Simpson’s Paradox

1. Simpson’s Paradox Jeff Witmer 18 March 2010 G. Udny Yule (Not Edward H. Simpson)

2. Blue disease 81 +192 Red disease 273 Black disease

4. Left-handed pitchers Right-handed pitchers All pitchers

5. UC Berkeley 1973 grad school admissions Applicants % admitted Men 8442 44% Women 4321 35% 6 large departments: Department Men Women Applicants % admitted Applicants % admitted A 825 62% 108 82% B 560 63% 25 68% C 325 37% 593 34% D 417 33% 375 35% E 191 28% 393 24% F 272 6% 341 7%

6. January 2009 Percent of Planes Delayed from City of Origin Continental United

7. Showing two of the four airports graphically: Circles correspond to Continental (Newark much larger than LGA) Squares correspond to United Airlines (LGA larger than Newark) X equals average for the two airports Continental at 23.4% and United at 21.9% Marginal effect (ignoring/combining airports) Continental United

8. X equals average for the two airports Continental at 23.4% and United at 21.9% Marginal effect (ignoring/combining airports) Continental United

9. Hat tip: Roger Nelsen

10. What level of aggregation is “right”? Combined 1995 1996

11. Unemployment higher now than in 1980s recession for each education level, but lower overall WSJ article, 2 December 20009 (graph only shows two education levels…)

12. 1977 data from 20 counties in Florida Sentence = death penalty? White victim Black victim All victims The probability of a convicted murderer being given the death penalty (vs life in prison) depends more on the victim’s race than on the defendant’s race Hat tip: Mike Radelet (and Alan Agresti) and Jimmy Doi

13. Regression data: Olympic 1500 winners Hat tip: Phil Everson

14. State average SAT scores (1995) Average SAT vs Expenditure per student Average SAT vs Average teacher salary

15. Average SAT vs Pupil/teacher ratio

16. The 10 states with the lowest per pupil spending included four -- North Dakota, South Dakota, Tennessee, Utah -- among the 10 states with the top SAT scores. Only one of the 10 states with the highest per pupil expenditures -- Wisconsin -- was among the 10 states with the highest SAT scores. New Jersey has the highest per pupil expenditures, an astonishing $10,561, which teachers' unions elsewhere try to use as a negotiating benchmark. New Jersey's rank regarding SAT scores? Thirty-ninth... The fact that the quality of schools... [fails to correlate] with education appropriations will have no effect on the teacher unions' insistence that money is the crucial variable. -- George F. Will, (September 12, 1993), "Meaningless Money Factor," The Washington Post, C7.

17. Consider the fraction of students in a state who take the SAT North Dakota Iowa Minnesota Utah Wisconsin South Dakota States with high SAT scores but low fractions taking the test Average SAT vs Fraction of students taking the test

18. Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1051.88710 20.82477 50.51 < 2e-16 *** expend 7.91371 3.49820 2.26 0.028 * frac -6.38069 0.70362 -9.07 8.3e-12 *** I(frac^2) 0.04741 0.00916 5.18 4.9e-06 *** Added variable plot for “expend” Hat tip: Stacey Hancock and Albyn Jones

19. Other examples US Senate: replace moderate Dems with moderate Reps and the Senate could become more more conservative while each party caucus becomes more liberal The 20-yr death rate was higher for non-smokers than for smokers in a UK city – but lower in (almost) each age group. Hat tip: Jo Hardin

20. G. U. Yule (1903). "Notes on the Theory of Association of Attributes in Statistics". Biometrika 2: 121–134. Simpson, Edward H. (1951). "The Interpretation of Interaction in Contingency Tables". Journal of the Royal Statistical Society, Ser. B 13: 238–241. Stigler, S. M. (1980). Stigler's law of eponymy. Transactions of the New York Academy of Sciences, 39: 147-58 (Merton Frestschrift Volume, F. Gieryn (ed)).