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

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Treatment A 81/8793% Treatment B234/27087% Blue disease Red disease Treatment C192/26373% Treatment D 55/8069% Black disease Treatment E78% Treatment F 83% 273/350 289/350 81 +192 273 Treatment A Treatment B Treatment A Treatment B

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Sizemore29/134.216 Valbuena 8/39.205 79/302.262 84/329.255 108/416.248 92/368.250 Left- handed pitchers Right- handed pitchers All pitchers

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Applicants % admitted Men 8442 44% Women 4321 35% 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% UC Berkeley 1973 grad school admissions 6 large departments:

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AirportLateTotal %LateTotal % Newark 957399823.9100 39925.1 LaGuardia 62 35617.4113 57319.7 Pittsburg 8 6013.3 17 11914.3 Detroit 16 14511.0 16 13911.5 ContinentalUnited Percent of Planes Delayed from City of Origin January 2009 Totals1043455922.9246123020.0

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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% ContinentalUnited Marginal effect (ignoring/combining airports)

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

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Hat tip: Roger Nelsen

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Justice104/411.253 Jeter12/48.250 45/140.321 183/582.314 149/551.270 195/630.310 19951996 Combined What level of aggregation is “right”?

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

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White def19/151.126 Black def 11/63.175 0/9.000 6/103.058 19/160.119 17/166.102 White victim Black victim All victims 1977 data from 20 counties in Florida White def Black def 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 Sentence = death penalty?

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Regression data: Olympic 1500 winners Hat tip: Phil Everson

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

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Average SAT vs Pupil/teacher ratio

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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.

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

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Added variable plot for “expend” 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 *** Hat tip: Stacey Hancock and Albyn Jones

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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

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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)).

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