Editing Simpson's paradox (section)

== Examples ==

===UC Berkeley gender bias===
One of the best-known examples of Simpson's paradox comes from a study of gender bias among graduate school admissions to [[University of California, Berkeley]]. The admission figures for the fall of 1973 showed that men applying were more likely than women to be admitted, and the difference was so large that it was unlikely to be due to chance.<ref name="freedman">[[David A. Freedman|David Freedman]], Robert Pisani, and Roger Purves (2007), ''Statistics'' (4th edition), [[W. W. Norton & Company|W. W. Norton]]. {{isbn|0-393-92972-8}}.</ref><ref name="Bickel">{{cite journal
 | author = [[Peter J. Bickel|P.J. Bickel]], E.A. Hammel and J.W. O'Connell
 | year = 1975
 | title = Sex Bias in Graduate Admissions: Data From Berkeley
 | journal = [[Science (journal)|Science]]
 | volume = 187
 | pages = 398–404
 | doi = 10.1126/science.187.4175.398
 | pmid = 17835295
 | issue = 4175
 | bibcode = 1975Sci...187..398B
 | s2cid = 15278703
 | url=http://homepage.stat.uiowa.edu/~mbognar/1030/Bickel-Berkeley.pdf |archive-url=https://web.archive.org/web/20160604220121/http://homepage.stat.uiowa.edu/~mbognar/1030/Bickel-Berkeley.pdf |archive-date=2016-06-04 |url-status=live
}}</ref>

{| class="wikitable" style="margin-left:auto; margin-right:auto; border:none; text-align:right;"
|-
! rowspan="2" |
! colspan="2" | All
! colspan="2" | Men
! colspan="2" | Women
|-
! Applicants
! Admitted
! Applicants
! Admitted
! Applicants
! Admitted
|-
! Total
| 12,763
| 41%
| 8,442
| style="background: #9EFF9E;" | 44%
| 4,321
| 35%
|}

However, when taking into account the information about departments being applied to, the different rejection percentages reveal the different difficulty of getting into the department, and at the same time it showed that women tended to apply to more competitive departments with lower rates of admission, even among qualified applicants (such as in the English department), whereas men tended to apply to less competitive departments with higher rates of admission (such as in the engineering department). The pooled and corrected data showed a "small but statistically significant bias in favor of women".<ref name="Bickel" />

The data from the six largest departments are listed below:

{| class="wikitable" style="margin-left:auto; margin-right:auto; border:none; text-align:right;"
|-
! rowspan="2" | Department
! colspan="2" | All
! colspan="2" | Men
! colspan="2" | Women
|-
! Applicants
! Admitted
! Applicants
! Admitted
! Applicants
! Admitted
|-
! A
| 933
| 64%
| style="background: #FE9;" | '''825'''
| 62%
| 108
| style="background: #9EFF9E;" | 82%
|-
! B
| 585
| 63%
| style="background: #FE9;" | '''560'''
| 63%
| 25
| style="background: #9EFF9E;" | 68%
|-
! C
| 918
| 35%
| 325
| style="background: #9EFF9E;" | 37%
| style="background: #FE9;" | '''593'''
| 34%
|-
! D
| 792
| 34%
| style="background: #FE9;" | 417
| 33%
| 375
| style="background: #9EFF9E;" | 35%
|-
! E
| 584
| 25%
| 191
| style="background: #9EFF9E;" | 28%
| style="background: #FE9;" | '''393'''
| 24%
|-
! F
| 714
| 6%
| style="background: #FE9;" | 373
| 6%
| 341
| style="background: #9EFF9E;" | 7%
|-
! Total
! 4526
! 39%
! 2691
! 45%
! 1835
! 30%
|-
| colspan="7" style="text-align:left;" |
Legend:<br>
{{legend|#9EFF9E|greater percentage of successful applicants than the other gender}}
{{legend|#FE9|greater number of applicants than the other gender}}
'''bold''' - the two 'most applied for' departments for each gender
|}

The entire data showed total of 4 out of 85 departments to be significantly biased against women, while 6 to be significantly biased against men (not all present in the 'six largest departments' table above). Notably, the numbers of biased departments were not the basis for the conclusion, but rather it was the gender admissions pooled across all departments, while weighing by each department's rejection rate across all of its applicants.<ref name="Bickel" />

===Kidney stone treatment===
Another example comes from a real-life medical study<ref>{{Cite journal
 | author1=C. R. Charig
 | author2=D. R. Webb
 | author3=S. R. Payne
 | author4=J. E. Wickham
 | title = Comparison of treatment of renal calculi by open surgery, percutaneous nephrolithotomy, and extracorporeal shockwave lithotripsy
 | journal = [[Br Med J (Clin Res Ed)]]
 | volume = 292
 | issue = 6524
 | pages = 879–882
 | pmid = 3083922
 | date = 29 March 1986
 | doi = 10.1136/bmj.292.6524.879
 | pmc = 1339981
}}</ref> comparing the success rates of two treatments for [[kidney stone]]s.<ref name="KidneyParadox">{{Cite journal
 | author1 = Steven A. Julious
 | author2 = Mark A. Mullee
 | title = Confounding and Simpson's paradox
 | journal = [[BMJ]]
 | pages = 1480–1481
 | url = http://bmj.bmjjournals.com/cgi/content/full/309/6967/1480
 | pmid = 7804052
 | date = 3 December 1994 | volume = 309
 | issue = 6967
 | pmc = 2541623
 | doi=10.1136/bmj.309.6967.1480
}}</ref> The table below shows the success rates (the term ''success rate'' here actually means the success proportion) and numbers of treatments for treatments involving both small and large kidney stones, where Treatment A includes open surgical procedures and Treatment B includes closed surgical procedures. The numbers in parentheses indicate the number of success cases over the total size of the group.

{| class="wikitable" summary="results accounting for stone size" style="margin-left:auto; margin-right:auto; border:none;"
! {{diagonal split header|Stone size &nbsp; &nbsp;|Treatment}}
! Treatment A
! Treatment B
|- align="center"
! Small stones
| ''Group 1''<br>'''93% (81/87)''' || ''Group 2''<br>87% (234/270)
|- align="center"
! Large stones
| ''Group 3''<br>'''73% (192/263)''' || ''Group 4''<br>69% (55/80)
|- align="center"
! Both
| 78% (273/350) || '''83% (289/350)'''
|}
The paradoxical conclusion is that treatment A is more effective when used on small stones, and also when used on large stones, yet treatment B appears to be more effective when considering both sizes at the same time. In this example, the "lurking" variable (or [[confounding|confounding variable]]) causing the paradox is the size of the stones, which was not previously known to researchers to be important until its effects were included.{{citation needed|date=April 2024}}

Which treatment is considered better is determined by which success ratio (successes/total) is larger. The reversal of the inequality between the two ratios when considering the combined data, which creates Simpson's paradox, happens because two effects occur together:{{citation needed|date=April 2024}}
# The sizes of the groups, which are combined when the lurking variable is ignored, are very different. Doctors tend to give cases with large stones the better treatment A, and the cases with small stones the inferior treatment B. Therefore, the totals are dominated by groups 3 and 2, and not by the two much smaller groups 1 and 4.
# The lurking variable, stone size, has a large effect on the ratios; i.e., the success rate is more strongly influenced by the severity of the case than by the choice of treatment. Therefore, the group of patients with large stones using treatment A (group 3) does worse than the group with small stones, even if the latter used the inferior treatment B (group 2).
Based on these effects, the paradoxical result is seen to arise because the effect of the size of the stones overwhelms the benefits of the better treatment (A). In short, the less effective treatment B appeared to be more effective because it was applied more frequently to the small stones cases, which were easier to treat.<ref name="KidneyParadox"/>

[[Edwin Thompson Jaynes|Jaynes]] argues that the correct conclusion is that though treatment A remains noticeably better than treatment B, the kidney stone size is more important.<ref>{{Cite book |last1=Jaynes |first1=E. T. |title=Probability theory: the logic of science |last2=Bretthorst |first2=G. Larry |date=2003 |publisher=Cambridge University Press |isbn=978-0-521-59271-0 |location=Cambridge, UK; New York, NY |chapter=8.10 Pooling the data}}</ref>

===Batting averages===
A common example of Simpson's paradox involves the [[batting average (baseball)|batting average]]s of players in [[professional baseball]]. It is possible for one player to have a higher batting average than another player each year for a number of years, but to have a lower batting average across all of those years. This phenomenon can occur when there are large differences in the number of [[at bat]]s between the years. Mathematician [[Kenneth A. Ross|Ken Ross]] demonstrated this using the batting average of two baseball players, [[Derek Jeter]] and [[David Justice]], during the years 1995 and 1996:<ref name="RossBaseball">Ken Ross. "''A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (Paperback)''" Pi Press, 2004. {{isbn|0-13-147990-3}}. 12–13</ref><ref>Statistics available from [[Baseball-Reference.com]]: [https://www.baseball-reference.com/j/jeterde01.shtml Data for Derek Jeter]; [https://www.baseball-reference.com/j/justida01.shtml Data for David Justice].</ref>

{| class="wikitable" style="margin-left:auto; margin-right:auto; border:none;"
|-
! {{diagonal split header|Batter &nbsp;|Year}}
! colspan="2" | 1995
! colspan="2" | 1996
! colspan="2" | Combined
|-
| Derek Jeter
| 12/48
| .250
| 183/582
| .314
| 195/630
| '''.310'''
|-
| David Justice
| 104/411
| '''.253'''
| 45/140
| '''.321'''
| 149/551
| .270
|}

In both 1995 and 1996, Justice had a higher batting average (in bold type) than Jeter did. However, when the two baseball seasons are combined, Jeter shows a higher batting average than Justice. According to Ross, this phenomenon would be observed about once per year among the possible pairs of players.<ref name="RossBaseball"/>