Download - Simpson’s Paradox
Simpson’s Paradox
Jeff Witmer
18 March 2010
G. Udny Yule
(Not Edward H. Simpson)
Treatment A 81/87 93%Treatment B 234/270 87%
Blue disease
Red diseaseTreatment C 192/263 73%Treatment D 55/80 69%
Black diseaseTreatment E 78%Treatment F 83%
273/350289/350
81+192
273Treatment ATreatment B
Treatment ATreatment B
€
ab
<AB
and cd
<CD
but a + cb + d
>A + CB + D
15<28 and
74<63 but
1+ 75+ 4
>2 + 68 + 3
Sizemore 29/134 .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
Applicants % admittedMen 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:
Airport Late Total % Late Total %Newark 957 3998 23.9 100 399 25.1LaGuardia 62 356 17.4 113 573 19.7Pittsburg 8 60 13.3 17 119 14.3Detroit 16 145 11.0 16 139 11.5
Continental United
Percent of Planes Delayed from City of OriginJanuary 2009
Totals 1043 4559 22.9 246 1230 20.0
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%
Continental United
Marginal effect (ignoring/combining airports)
X equals average for the two airports Continental at 23.4% and United at 21.9%
Continental United
Marginal effect (ignoring/combining airports)
15<28 and
74<63 but
1+ 75+ 4
>2 + 68 + 3
Hat tip: Roger Nelsen
Justice 104/411 .253
Jeter 12/48 .250
45/140 .321
183/582 .314
149/551 .270
195/630 .310
1995 1996 Combined
What level of aggregation is “right”?
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…)
White def 19/151 .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?
Regression data: Olympic 1500 winners
Hat tip: Phil Everson
State average SAT scores (1995)
Average SAT vsExpenditure per student
Average SAT vsAverage teacher salary
Average SAT vsPupil/teacher ratio
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.
Consider the fraction of students in a state who take the SAT
States with high SAT scores but low fractions taking the test
North DakotaIowa
Minnesota
Wisconsin
Utah
South Dakota
Average SAT vsFraction of students taking the test
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
Other examplesUS 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
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)).