Simpsons Paradox

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Simpsons Paradox

• Michael Kuykindall
• Faculty Advisor Engin Sungur

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Outline

• Introduction
• Simpsons Paradox
• Data
• Results
• Conclusion

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What is Simpsons Paradox?

• Simpsons Paradox occurs when an association
  between two variables is reversed upon observing
  a third variable.

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According to Colin R. Blyth Simpsons Paradox can
be defined mathematically as follows

• P(AB)
• while at the same time
• P(ABC) P(A BC)
• P(ABC) P(ABC).
• This is important because
• P(AB) An average of P(ABC) and P(ABC)
• P(AB) An average of P(ABC) and P(ABC),
• Which is easily seen to be true when all the
  conditional events have positive probabilities
• P(AB)P(CB)P(ABC) P(CB)P(ABC)
• P(AB)P(CB)P(ABC) P(CB)P(ABC)

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Colin R. Blyths example of Simpsons Paradox

• A doctor was planning to try a new treatment on
  patients mostly local (C) and a few in Chicago
  (C). A statistician advised him to use a table
  of random numbers and as each C patient became
  available, assign him to the new treatment with
  probability .91, leave him to the standard
  treatment with probability .09 and the same for
  C patient with probability .01 and .99
  respectively. When the doctor returned with the
  data the statistician told him that the new
  treatment was obviously a very bad one, and
  criticized him for having continued trying it on
  so many patients.

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• The doctor replied that he continued because the
  new treatment was obviously a very good one,
  having nearly doubled the recovery rate in both
  cities.

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• In this example A Alive, B New Treatment, C
  Local Patient P( ) refers to the probability
  for a patient chosen at random from among those
  recorded in the table, and coincides with
  proportions in that table, now being taken as a
  total available population. In that example we
  have
• P(AB) .11
• P(ABC) .10P(ABC).05,
• P(ABC) .95P(ABC).95.
• The initially surprising fact that an average of
  .10 and .95 is so much smaller than the average
  of .05 and .50 is easily explained by showing the
  numerical values in
• .11 .99(.10) .01(.95)
• .46 .10(.05) .90(.50)

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Survival rate
Treatment
Patient Location
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Smokers Example

• In England a study was conducted to examine the
  survival rates of smokers and non-smokers. The
  result implied a significant positive correlation
  between smoking survival rates because only 24
  of smokers died as compared to 31 of
  non-smokers. When the data were broken down by
  age group in a contingency table, it was found
  that there were more older people in the
  non-smoker group. Thus age played a very
  significant role in the outcome but since it was
  overlooked the researchers were left with
  deceiving results. (Appleton French, 1996).

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Survival rate
Smoker or Non Smoker
Age Group
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Death Penalty Example

• Effects of racial characteristics on whether
  individuals convicted of homicide receive the
  death penalty.
• The variables death penalty verdict, having
  categories yes, no. The race of the defendant
  and the race of the victim, each having
  categories European American or African American.
• Data 326 defendants were recorded as being
  indicted for homicide in 20 Florida counties
  during 1976-1977.

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Frequencies for Death Penalty Verdict and
Defendant's Race

• About 12 of European American defendants and
  about 10 African American defendants receive the
  death penalty. Ignoring victims race, the
  percentage of yes death penalty verdicts was
  lower for African Americans than for European
  Americans.

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Death Penalty Verdict by Defendant's Race and
Victim's Race

• When victim is European American, the death
  penalty was imposed about 5 percentage points
  more often for African American defendants than
  for European American defendants. When the
  victim is African American, the death penalty was
  imposed over 5 percentage points more often than
  for European American defendants.

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Death Penalty Verdict by Defendant's Race and
Victim's Race

• Controlling for victims race, the percentage of
  yes death penalty verdicts was higher for African
  American than for European Americans. The
  direction of the association is reversed.

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Odds Ratios for Death Penalty (P), Victim's Race
(V), and Defendant's Race (D)

• The estimated odds of the death penalty were 1.18
  times as high for European American defendants as
  for African American defendants. But, when the
  victim was European American, the estimated odds
  of the death penalty were .67 times as high for
  European American defendants as for African
  American defendant when the victim was African
  American, the estimated odds were .79 times as
  high for European American defendants as for
  African American defendants.

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• The odds of having killed a European American are
  estimated to be 25.99 times higher for European
  American defendants than for African American
  defendants.
• The odds ratios relating death penalty verdict
  and victims race indicate the death penalty was
  more likely when the victim was European American
  than when the victim was African American.
• So European Americans tend to kill other European
  Americans, and killing a European American is
  more likely to result in the death penalty.

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Percent Receiving Death Penalty

• Percent receiving death penalty by defendants
  race, controlling and ignoring victims race.
• Each observation is represented by a letter
  giving the level of the victims race.
• Surrounding each observation is a circle having
  area proportional to the number of observations
  at that combination of defendants race and
  victims race.
• The largest circles occur when European Americans
  kill other European Americans or African American
  kill other African Americans.
• These cause the marginal results whereby European
  Americans are more likely to receive the death
  penalty.

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EA
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Marginal effect
EA
x
10
x
AA
5
AA
0
European American
African American
x marginal effect of defendant's race,
ignoring victim's race.
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Death Penalty
Race of Defendant
Race of Victim
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NLSY79 Example

• DATA The National Longitudinal Survey Handbook
  2001
• The NLSY79 is a nationally representative sample
  of 12,686 young men and women who were 14 to 22
  years of age when first surveyed in 1979. During
  the years since that first interview, these young
  people typically have finished their schooling,
  moved out of their parents homes, made decisions
  on continuing education and training, entering
  the labor market, served in the military, married
  and started their own families. Data collected
  from the NLSY79 respondents chronicle these
  changes providing researchers with a unique
  opportunity to study in detail the life course
  experiences of a large group of adult
  representatives of all men and women born in the
  late 1950s and early 1960s and living in the
  United States when the survey began.

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Variables

• Dependent Highest grade completed
• Independent Race (Hispanic, Black, Non-Black
  Non-Hispanic), Highest grade Completed by mother

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Analysis Tools

• SAS ANOVA Linear Model (Calculate Parameter
  Estimates, Hypothesis test type for F test Type
  III
• SAS Graphs Box Plots

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Highest grade completed by subjects

• 
  
  
• Dependent Variable HIGHEST GRADE COMPLETED (REV)
  1998
  
  
  
  
• 
  Standard
  
• Parameter Estimate
  Error t Value
  Pr t
  
• 
  
  
• Intercept
  13.52547170 B 0.03722366 363.36
  
• Hispanic 1 -1.13028058 B
  0.07076467 -15.97
• Black 2 -0.69496322 B
  0.06083837 -11.42
• Non B-H 3 0.00000000 B
  . .
  .
  
• Source DF Type III SS
  Mean Square F Value Pr F
  
• 
  
  
• Race 2 1757.477371
  878.738686 149.57
  
  

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Highest grad completed by subjects their
mothers

• 
  
  
• Dependent Variable HIGHEST GRADE COMPLETED (REV)
  1998
• 
  
  
  
  
  
• 
  Standard
  
• Parameter Estimate
  Error t Value Pr
  t
  
• 
  
  
• Intercept 9.804807070 B
  0.10941378 89.61
• Hispanic 0.168059825 B
  0.07579780 2.22 0.0266
  
• Black -0.297132340
  B 0.05879723 -5.05
• Non B-H 0.000000000 B
  . .
  .
  
• HGC By Mom 0.316727781
  0.00870516 36.38
• Source DF Type III
  SS Mean Square F Value Pr
  F
• 
  
  
• Race 2
  216.301255 108.150627 21.84
  
• HGC By MOM 1 6556.118704
  6556.118704 1323.79

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Race
Childs Education
Mother Education
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CONCLUSION!!!!

• Simpsons paradox is a rare phenomenon! It does
  not occur often! Thus statisticians must be
  trained academically ethically well enough to
  make sure that if it has occurred they will
  detect and correct it. This is where practice,
  critical thinking skills, and repetition come
  into play!

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Sources

• Agresti, Alan. Categorical Data Analysis. John
  Wiley Son, Inc. Canada.1990 (135-138)
• Blyth, Colin R.. Journal of the American
  Statistical Association, Vol. 67, No. 338. (Jun.,
  1972), pp. 364-366.

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Acknowledgements

• Engin Sungur
• Jon Anderson
• Laura Argys
• Josephine Myers-Kuykindall