Simpson's paradox

Simpson's paradox (or the Yule-Simpson effect) is a statistical paradox wherein the successes of groups seem reversed when the groups are combined. This result is often encountered in social and medical science statistics [cite journal
title = Simpson's Paradox in Real Life
author = Clifford H. Wagner
year = 1982
month = February
journal = The American Statistician
volume = 36
issue = 1
pages = 46–48
doi = 10.2307/2684093] , and occurs when frequencydata are hastily given causal interpretation Judea Pearl. "Causality: Models, Reasoning, and Inference", Cambridge University Press, 2000. ISBN 0-521-77362-8.] ; the paradox disappears when causal relations are derived systematically, through formal analysis.

Though mostly unknown to laymen, Simpson's Paradox is well known to statisticians, and is described in several introductory statistics books.David Freedman, Robert Pisani and Roger Purves. Statistics (3rd edition). W.W. Norton, 1998, p. 19. ISBN 0-393-97083-3.] [David S. Moore and D.S. George P. McCabe (February 2005). "Introduction to the Practice of Statistics" (5th edition). W.H. Freeman & Company. ISBN 071676282X.] Many statisticians believe that the mainstream public should be apprised of counterintuitive results such as Simpson's paradox, [Robert L. Wardrop (February 1995). "Simpson's Paradox and the Hot Hand in Basketball". "The American Statistician", 49 (1): pp. 24–28.] in particular to caution against the inference of causal relationships based on the association between two variables. [Alan Agresti (2002). "Categorical Data Analysis" (Second edition). John Wiley and Sons. ISBN 0-471-36093-7]

Edward H. Simpson described the phenomenon in 1951 [cite journal
title=The Interpretation of Interaction in Contingency Tables
author = Simpson, Edward H.
year = 1951
journal = Journal of the Royal Statistical Society, Ser. B
volume = 13
pages = 238–241] ,along with Karl Pearson et al., [Citation
last1 = Pearson | first1 = Karl
author1-link = Karl Pearson
last2 = Lee | first2 = A.
last3 = Bramley-Moore | first3 = L.
title = Genetic (reproductive) selection: Inheritance of fertility in man
journal = Philosophical Translations of the Royal Statistical Society, Ser. A
volume = 173
pages = 534–539
year = 1899] and Udny Yule in 1903 [Cite journal
title = Notes on the Theory of Association of Attributes in Statistics
author = G. U. Yule
year = 1903
journal = Biometrika
volume = 2
pages = 121–134
doi = 10.1093/biomet/2.2.121] . The name "Simpson's paradox" was coined by Colin R. Blyth in 1972. [cite journal
title = On Simpson's Paradox and the Sure-Thing Principle
author = Colin R. Blyth
year = 1972
month = June
journal = Journal of the American Statistical Association
volume = 67
issue = 338
pages = 364–366
doi = 10.2307/2284382] Since Simpson did not discover this statistical paradox, some authors, instead, have used the impersonal names "reversal paradox" and "amalgamation paradox" in referring to what is now called "Simpson's Paradox" and the "Yule-Simpson effect". [cite journal
title = [http://links.jstor.org/sici?sici=0090-5364%28198706%2915%3A2%3C694%3ATAAGOT%3E2.0.CO%3B2-0 The Amalgamation and Geometry of Two-by-Two Contingency Tables]
author = I. J. Good, Y. Mittal
journal = The Annals of Statistics
year = 1987
month = June
volume = 15
issue = 2
pages = 694–711]

Examples

Batting averages

A common example of the paradox involves batting averages in baseball: it is possible for one player to hit for a higher batting average than another player during a given year, and to do so again during the next year, but to have a lower batting average when the two years are combined. This phenomenon, which occurs when there are large differences in the number of at-bats between years, is well-known among sports sabermetricians such as Bill James.

A real-life example is provided by Ken Ross [Ken Ross. "A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (Paperback)"Pi Press, 2004. ISBN 0131479903. 12–13] and involves the batting average of baseball players Derek Jeter and David Justice during the years 1995 and 1996: [Statistics available from http://www.baseball-reference.com/ : [http://www.baseball-reference.com/j/jeterde01.shtml Data for Derek Jeter] , [http://www.baseball-reference.com/j/justida01.shtml Data for David Justice] .]

This seems to show treatment B is more effective. If we include data about kidney stone size, however, the same set of treatments reveals a different answer:

The explanation turned out to be that women tended to apply to competitive departments with low rates of admission even among qualified applicants (such as English), while men tended to apply to less-competitive departments with high rates of admission among qualified applicants (such as engineering). The conditions under which department-specific frequency data constitute a proper defense against charges ofdiscrimination are formulated in Pearl (2000).

2006 US school study

In July 2006, the United States Department of Education released a study [H. Braun, F. Jenkins and W. Grigg, (2006) [http://www.nytimes.com/packages/pdf/national/20060715report.pdf "Comparing Private Schools and Public Schools Using Hierarchical Linear Modeling] , U.S. Department of Education, National Center for Education Statistics, Institute of Education Sciences, Washington, DC, United States Government Printing Office.] documenting student performances in reading and math in different school settings. [Diana Jean Schemo. " [http://www.nytimes.com/2006/07/15/education/15report.html Public Schools Perform Near Private Ones in Study] . The New York Times, 15 July 2006. Retrieved on 25 July 2007.] It reported that while the math and reading levels for students at grades 4 and 8 were uniformly higher in private/parochial schools than in public schools, repeating the comparisons on demographic subgroups showed much smaller differences, which were nearly equally divided in direction.

Low birth weight paradox

The low birth weight paradox is an apparently paradoxical observation relating to the birth weights and mortality of children born to tobacco smoking mothers. Traditionally, babies weighing less than a certain amount (which varies between countries) have been classified as having "low birth weight". In a given population, low birth weight babies have a significantly higher mortality rate than others. However, it has been observed that low birth weight children born to smoking mothers have a "lower" mortality rate than the low birth weight children of non-smokers. [Wilcox, Allen (2006). " [http://aje.oxfordjournals.org/cgi/content/abstract/164/11/1121 The Perils of Birth Weight — A Lesson from Directed Acyclic Graphs] ". "American Journal of Epidemiology". 164(11):1121–1123.]

Description

Suppose two people, Lisa and Bart, each edit Wikipedia articles for two weeks. In the first week, Lisa improves 60 percent of the articles she edits out of 100 articles edited, and Bart improves 90 percent of the articles he edits out of 10 articles edited. In the second week, Lisa improves just 10 percent of the articles she edits out of 10 articles edited, while Bart improves 30 percent yet out of 100 articles edited.

Both times, Bart improved a higher percentage of the quantity of articles compared to Lisa, while Lisa improved a higher percentage of the quality of articles. When the two tests are combined using a weighted average, overall, Lisa has improved a much higher percentage than Bart because the quality modifier had a significantly higher percentage. Therefore, like other paradoxes, it only appears to be a paradox because of incorrect assumptions, incomplete or misguided information, or a lack of understanding a particular concept.

This imagined paradox is caused when the percentage is provided but not the ratio by various media outlets. In this example, if only the 90% in the first week for Bart was provided but not the ratio (9:10), it would distort the information causing the imagined paradox. Even though Bart's percentage is higher for the first and second week, when two week's of articles is combined, overall Lisa had improved a greater proportion, 55% of the 110 total articles. Lisa's proportional total of articles improved exceeds Bart's total.

Here are some notations:

* In the first week:* $S\_L(1)\; =\; 60\%$ — Lisa improved 60% of the many articles she edited.:* $S\_B(1)\; =\; 90\%$ — Bart had a 90% success rate during that time.: Success is associated with Bart.

* In the second week:* $S\_L(2)\; =\; 10\%$ — Lisa managed 10% in her busy life.:* $S\_B(2)\; =\; 30\%$ — Bart achieved a 30% success rate.: Success is associated with Bart.

On both occasions Bart's edits were more successful than Lisa's. But if we combine the two sets, we see that Lisa and Bart both edited 110 articles, and:

* — Lisa improved 61 articles.
* — Bart improved only 39.
* $S\_L\; >\; S\_B\; ,$ — Success is now associated with Lisa.

Bart is better for each set but worse overall.

The paradox stems from our healthy intuition that Bart could not possibly be a better editor on each set but worse overall. Pearl (2000) in fact proved the impossibility of such happening, where "better editor" is taken in the counterfactual sense: "Were Bart to edit all items in a set he would do better than Lisa would, on those same items." Clearly, frequency data cannot support this sense of "better editor," because it does not tell us how Bart would perform on items edited by Lisa, and vice versa. In the back of our mind, though, we assume that the articles were assigned at random to Bart and Lisa, an assumption which (for large sample) would support the counterfactual interpretation of "better editor." However, under random assignment conditions, the data given in this example is impossible, which accounts for our surprise when confronting the rate reversal.

The arithmetical basis of the paradox is uncontroversial. If $S\_B(1)\; >\; S\_L(1)$ and $S\_B(2)\; >\; S\_L(2)$ we feel that $S\_B$ "must be greater" than $S\_L$. However if "different" weights are used to form the overall score for each person then this feeling may be disappointed. Here the first test is weighted for Lisa and for Bart while the weights are reversed on the second test.

*

*

By more extreme reweighting Lisa's overall score can be pushed up towards 60% and Bart's down towards 30%.

Lisa is a better editor on average, as her overall success rate is higher. But it is possible to have told the story in a way which would make it appear obvious that Bart is more diligent.

Simpson's paradox shows us an extreme example of the importance of including data about possible confounding variables when attempting to calculate causal relations. Precise criteria for selecting a set of "confounding variables,"(i.e., variables that yield correct causal relationships if included in the analysis),is given in (Pearl, 2000) using causal graphs.

While Simpson's paradox often refers to the analysis of count tables, as shown in this example, it also occurs with continuous data: [John Fox (1997). "Applied Regression Analysis, Linear Models, and Related Methods". Sage Publications. ISBN 080394540X. 136–137] for example, if one fits separated regression lines through two sets of data, the two regression lines may show a positive trend, while a regression line fitted through all data together will show a "negative" trend, as shown on the picture above.

Vector interpretation

Simpson's paradox can also be illustrated using the 2-dimensional vector space. [Jerzy Kocik (December 2001). [http://www.jstor.org/pss/2691038 Proofs without Words: Simpson's Paradox] . "Mathematics Magazine". 74 (5), p. 399.] A success rate of $p/q$ can be represented by a vector $overrightarrow\{A\}=(q,p)$, with a slope of $p/q$. If two rates $p\_1/q\_1$ and $p\_2/q\_2$ are combined, as in the examples given above, the result can be represented by the sum of the vectors $(q\_1,\; p\_1)$ and $(q\_2,\; p\_2)$, which according to the parallelogram rule is the vector $(q\_1+q\_2,\; p\_1+p\_2)$, with slope $frac\{p\_1+p\_2\}\{q\_1+q\_2\}$.

Simpson's paradox says that even if a vector $overrightarrow\{b\_1\}$ (in blue in the figure) has a smaller slope than another vector $overrightarrow\{r\_1\}$ (in red), and $overrightarrow\{b\_2\}$ has a smaller slope than $overrightarrow\{r\_2\}$, the sum of the two vectors $overrightarrow\{b\_1\}\; +\; overrightarrow\{b\_2\}$ (indicated by "+" in the figure) can still have a larger slope than the sum of the two vectors $overrightarrow\{r\_1\}\; +\; overrightarrow\{r\_2\}$, as shown in the example.

References

External links

*Stanford Encyclopedia of Philosophy: " [http://plato.stanford.edu/entries/paradox-simpson/ Simpson's Paradox] " -- by Gary Malinas.

For a brief history of the origins of the paradox see the entries "Simpson's Paradox and Spurious Correlation" in
* [http://members.aol.com/jeff570/s.html Earliest known uses of some of the words of mathematics: S]

* Pearl, Judea, " [http://bayes.cs.ucla.edu/LECTURE/lecture_sec1.htm "The Art and Science of Cause and Effect.] " A slide show and tutorial lecture.
* Pearl, Judea, " [http://bayes.cs.ucla.edu/R264.pdf Simpson's Paradox: An Anatomy.] "
* Short articles by Alexander Bogomolny at cut-the-knot:
** " [http://www.cut-the-knot.org/blue/Mediant.shtml Mediant Fractions.] "
** " [http://www.cut-the-knot.org/Curriculum/Algebra/SimpsonParadox.shtml Simpson's Paradox.] "