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

€

a

b<

A

B and

c

d<

C

D but

a + c

b + d>

A + C

B + D

1

5<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)

1

5<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 Dakota

Iowa

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)).