View source for Simpson's paradox

{{Short description|Error in statistical reasoning with groups}}
[[Image:Simpson's paradox continuous.svg|thumb|Simpson's paradox for quantitative data: a positive trend (<span style="display:inline-block; vertical-align:middle; width:2em; height:0px; border-style:none; border-top:2px solid blue;">&nbsp;</span>, <span style="display:inline-block; vertical-align:middle; width:2em; height:0px; border-style:none; border-top:2px solid red;">&nbsp;</span>) appears for two separate groups, whereas a negative trend (<span style="display:inline-block; vertical-align:middle; width:2em; height:0px; border-style:none; border-top:2px dashed #000;">&nbsp;</span>) appears when the groups are combined.]]
[[File:Simpsons paradox - animation.gif|thumb|Visualization of Simpson's paradox on data resembling real-world variability indicates that risk of misjudgment of true causal relationship can be hard to spot.]]

'''Simpson's paradox''' is a phenomenon in [[Probability|probability]] and [[Statistics|statistics]] in which a trend appears in several groups of data but disappears or reverses when the groups are combined. This result is often encountered in social-science and medical-science statistics,<ref>
{{cite journal
 | title = Simpson's Paradox in Real Life
 | author = Clifford H. Wagner
 |date=February 1982
 | journal = [[The American Statistician]]
 | volume = 36
 | issue = 1
 | pages = 46–48
 | doi = 10.2307/2684093
 | jstor = 2684093
}}</ref><ref>Holt, G. B. (2016).  [http://jco.ascopubs.org/content/34/9/1016.1.full Potential Simpson's paradox in multicenter study of intraperitoneal chemotherapy for ovarian cancer.] Journal of Clinical Oncology, 34(9), 1016–1016.</ref><ref name="VogelFranks2017">{{cite journal|last1=Franks|first1=Alexander|last2=Airoldi|first2=Edoardo|last3=Slavov|first3=Nikolai|title=Post-transcriptional regulation across human tissues|journal=PLOS Computational Biology|volume=13|issue=5|year=2017|pages=e1005535|issn=1553-7358|doi=10.1371/journal.pcbi.1005535|pmid=28481885|pmc=5440056|arxiv=1506.00219|bibcode=2017PLSCB..13E5535F |doi-access=free }}</ref> and is particularly problematic when frequency data are unduly given causal interpretations.<ref name="pearl">[[Biography:Judea Pearl|Judea Pearl]]. ''Causality: Models, Reasoning, and Inference'', Cambridge University Press (2000, 2nd edition 2009). {{isbn|0-521-77362-8}}.</ref> The paradox can be resolved when confounding variables and causal relations are appropriately addressed in the statistical modeling<ref name="pearl" /><ref>Kock, N., & Gaskins, L. (2016).  [http://cits.tamiu.edu/kock/pubs/journals/2016JournalIJANS_ModJCveNetCorrp/Kock_Gaskins_2016_IJANS_SimpPdox.pdf Simpson's paradox, moderation and the emergence of quadratic relationships in path models: An information systems illustration.] International Journal of Applied Nonlinear Science, 2(3), 200–234.</ref> (e.g., through [[Cluster analysis|cluster analysis]]<ref>Rogier A. Kievit, Willem E. Frankenhuis, Lourens J. Waldorp and Denny Borsboom, Simpson's paradox in psychological science: a practical guide https://doi.org/10.3389/fpsyg.2013.00513</ref>).

Simpson's paradox has been used to illustrate the kind of misleading results that the misuse of statistics can generate.<ref>Robert L. Wardrop (February 1995). "Simpson's Paradox and the Hot Hand in Basketball". ''The American Statistician'', ''' 49 (1)''': pp. 24–28.</ref><ref>[[Biography:Alan Agresti|Alan Agresti]] (2002). "Categorical Data Analysis" (Second edition). John Wiley and Sons {{isbn|0-471-36093-7}}</ref>

Edward H. Simpson first described this phenomenon in a technical paper in 1951,<ref>
{{cite journal
 | title=The Interpretation of Interaction in Contingency Tables
 | author = Simpson, Edward H.
 | year = 1951
 | journal = Journal of the Royal Statistical Society, Series B
 | volume = 13
 | pages = 238–241
}}</ref> but the statisticians [[Biography:Karl Pearson|Karl Pearson]] (in 1899<ref>
{{Cite journal
 | last1 = Pearson  | first1 = Karl
 | last2 = Lee | first2 = Alice
 | last3 = Bramley-Moore | first3 = Lesley
 | title = Genetic (reproductive) selection: Inheritance of fertility in man, and of fecundity in thoroughbred racehorses 
 | journal = Philosophical Transactions of the Royal Society A
 | volume = 192
 | pages = 257–330
 | year = 1899
 | doi = 10.1098/rsta.1899.0006
| doi-access = free
 }}</ref>) and Udny Yule (in 1903<ref name=yule>
{{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
 | issue = 2
| url = https://zenodo.org/record/1431599
 }}</ref>) had mentioned similar effects earlier. The name ''Simpson's paradox'' was introduced by Colin R. Blyth in 1972.<ref name="blyth-72">
{{cite journal
 | title = On Simpson's Paradox and the Sure-Thing Principle
 | author = Colin R. Blyth
 | date=June 1972
 | journal = Journal of the American Statistical Association
 | volume = 67
 | issue = 338
 | pages = 364–366
 | doi = 10.2307/2284382
 | jstor = 2284382
}}</ref> It is also referred to as '''Simpson's reversal''', the '''Yule–Simpson effect''', the '''amalgamation paradox''', or the '''reversal paradox'''.<ref>
{{cite journal|author=[[Biography:I. J. Good|I. J. Good]], Y. Mittal|date=June 1987|title=The Amalgamation and Geometry of Two-by-Two Contingency Tables|journal=The Annals of Statistics|volume=15|issue=2|pages=694–711|doi=10.1214/aos/1176350369|issn=0090-5364|jstor=2241334|doi-access=free}}</ref>

Mathematician [[Biography:Jordan Ellenberg|Jordan Ellenberg]] argues that Simpson's paradox is misnamed as "there's no contradiction involved, just two different ways to think about the same data" and suggests that its lesson "isn't really to tell us which viewpoint to take but to insist that we keep both the parts and the whole in mind at once."<ref>{{Cite book |last=Ellenberg |first=Jordan |url=https://www.worldcat.org/oclc/1226171979 |title=Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy and Everything Else |date=May 25, 2021 |publisher=Penguin Press |isbn=978-1-9848-7905-9 |location=New York |pages=228 |oclc=1226171979}}</ref>

== 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 [[Organization:University of California, Berkeley|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">[[Biography:David A. Freedman|David Freedman]], Robert Pisani, and Roger Purves (2007), ''Statistics'' (4th edition), W. W. Norton. {{isbn|0-393-92972-8}}.</ref><ref name="Bickel">{{cite journal
 | author = [[Biography: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
 | 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 stones.<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.

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:
# 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"/>

===Batting averages===
A common example of Simpson's paradox involves the batting averages 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 bats between the years. Mathematician 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"/>

==Vector interpretation==
[[File:Simpson_paradox_vectors.svg|thumb|Vector interpretation of Simpson's paradox]]
Simpson's paradox can also be illustrated using a 2-dimensional [[Vector space|vector space]].<ref>{{cite journal|author=Kocik Jerzy|year=2001|title=Proofs without Words: Simpson's Paradox|url=http://www.math.siu.edu/kocik/papers/simpson2.pdf |archive-url=https://web.archive.org/web/20100612220747/http://www.math.siu.edu/kocik/papers/simpson2.pdf |archive-date=2010-06-12 |url-status=live|journal=Mathematics Magazine|volume=74|issue=5|pages=399|doi=10.2307/2691038|jstor=2691038}}</ref> A success rate of <math display="inline">\frac{p}{q}</math> (i.e., ''successes/attempts'') can be represented by a vector <math>\vec{A} = (q, p)</math>, with a [[Slope|slope]] of <math display="inline">\frac{p}{q}</math>. A steeper vector then represents a greater success rate. If two rates <math display="inline">\frac{p_1}{q_1}</math> and <math display="inline">\frac{p_2}{q_2}</math> are combined, as in the examples given above, the result can be represented by the sum of the vectors <math>(q_1, p_1)</math> and <math>(q_2, p_2)</math>, which according to the parallelogram rule is the vector <math>(q_1 + q_2, p_1 + p_2)</math>, with slope <math display="inline">\frac{p_1 + p_2}{q_1 + q_2}</math>.

Simpson's paradox says that even if a vector <math>\vec{L}_1</math> (in orange in figure) has a smaller slope than another vector <math>\vec{B}_1</math> (in blue), and <math>\vec{L}_2</math> has a smaller slope than <math>\vec{B}_2</math>, the sum of the two vectors <math>\vec{L}_1 + \vec{L}_2</math> can potentially still have a larger slope than the sum of the two vectors <math>\vec{B}_1 + \vec{B}_2</math>, as shown in the example.  For this to occur one of the orange vectors must have a greater slope than one of the blue vectors (here <math>\vec{L}_2</math> and <math>\vec{B}_1</math>), and these will generally be longer than the alternatively subscripted vectors&nbsp;– thereby dominating the overall comparison.

==Correlation between variables==
Simpson's reversal can also arise in [[Correlation|correlation]]s,  in which two variables appear to have (say) a positive correlation towards one another, when in fact they have a negative correlation, the reversal having been brought about by a "lurking" confounder.  Berman et al.<ref>Berman, S. DalleMule, L.  Greene, M., Lucker, J. (2012), "[http://www.statslife.org.uk/the-statistics-dictionary/2012-simpson-s-paradox-a-cautionary-tale-in-advanced-analytics Simpson's Paradox: A Cautionary Tale in Advanced Analytics] {{Webarchive|url=https://web.archive.org/web/20200510171740/https://www.statslife.org.uk/the-statistics-dictionary/2012-simpson-s-paradox-a-cautionary-tale-in-advanced-analytics |date=2020-05-10 }}", ''Significance''.</ref> give an example from economics, where a dataset suggests overall demand is positively correlated with price (that is, higher prices lead to ''more'' demand), in contradiction of expectation.   Analysis reveals time to be the confounding variable: plotting both price and demand against time reveals the expected negative correlation over various periods, which then reverses to become positive if the influence of time is ignored by simply plotting demand against price.

==Psychology==
Psychological interest in Simpson's paradox seeks to explain why people deem sign reversal to be impossible at first, offended by the idea that an action preferred both under one condition and under its negation should be rejected when the condition is unknown. The question is where people get this strong [[Intuition|intuition]] from, and how it is encoded in the [[Philosophy:Mind|mind]].

Simpson's paradox demonstrates that this intuition cannot be derived from either [[Philosophy:Classical logic|classical logic]] or probability calculus alone, and thus led [[Biography:Philosopher|philosopher]]s to speculate that it is supported by an innate causal logic that guides people in reasoning about actions and their consequences.<ref name="pearl" /> Savage's [[Philosophy:Sure-thing principle|sure-thing principle]]<ref name="blyth-72"/> is an example of what such logic may entail. A qualified version of Savage's sure thing principle can indeed be derived from Pearl's ''do''-calculus<ref name="pearl"/> and reads: "An action ''A'' that increases the probability of an event ''B'' in each subpopulation ''C<sub>i</sub>'' of ''C'' must also increase the probability of ''B'' in the population as a whole, provided that the action does not change the distribution of the subpopulations." This suggests that knowledge about actions and consequences is stored in a form resembling Causal Bayesian Networks.

==Probability==
A paper by Pavlides and Perlman presents a proof, due to Hadjicostas, that in a random 2 × 2 × 2 table with uniform distribution, Simpson's paradox will occur with a [[Probability|probability]] of exactly {{frac|1|60}}.<ref>
{{cite journal
 | title = How Likely is Simpson's Paradox?
 | author1=Marios G. Pavlides
 | author2=Michael D. Perlman
 | name-list-style=amp |date=August 2009
 | journal = [[The American Statistician]]
 | volume = 63
 | issue = 3
 | pages = 226–233
 | doi = 10.1198/tast.2009.09007
| s2cid=17481510
 }}</ref> A study by Kock suggests that the probability that Simpson's paradox would occur at random in path models (i.e., models generated by [[Path analysis (statistics)|path analysis]]) with two predictors and one criterion variable is approximately 12.8 percent; slightly higher than 1 occurrence per 8 path models.<ref>Kock, N. (2015). [http://cits.tamiu.edu/kock/pubs/journals/2015JournalIJeC/Kock_2015_IJeC_SimpPdox.pdf How likely is Simpson's paradox in path models?] International Journal of e-Collaboration, 11(1), 1–7.</ref>

== Simpson's second paradox ==
A second, less well-known paradox was also discussed in Simpson's 1951 paper. It can occur when the "sensible interpretation" is not necessarily found in the separated data, like in the Kidney Stone example, but can instead reside in the combined data. Whether the partitioned or combined form of the data should be used hinges on the process giving rise to the data, meaning the correct interpretation of the data cannot always be determined by simply observing the tables.<ref>{{Cite journal|last1=Norton|first1=H. James|last2=Divine|first2=George|date=August 2015|title=Simpson's paradox ... and how to avoid it|journal=Significance|volume=12|issue=4|pages=40–43|doi=10.1111/j.1740-9713.2015.00844.x|doi-access=free}}</ref>

[[Biography:Judea Pearl|Judea Pearl]] has shown that, in order for the partitioned data to represent the correct causal relationships between any two variables, <math>X</math> and <math>Y</math>, the partitioning variables must satisfy a graphical condition called "back-door criterion":<ref>{{cite journal|last1=Pearl|first1=Judea|year=2014|title=Understanding Simpson's Paradox|journal=The American Statistician|volume=68|issue=1|pages=8–13|doi=10.2139/ssrn.2343788|s2cid=2626833}}</ref><ref>{{cite journal|last1=Pearl|first1=Judea|year=1993|title=Graphical Models, Causality, and Intervention|journal=Statistical Science|volume=8|issue=3|pages=266–269|doi=10.1214/ss/1177010894|doi-access=free}}</ref>
# They must block all spurious paths between <math>X</math> and <math>Y</math>
# No variable can be affected by <math>X</math>
This criterion provides an algorithmic solution to Simpson's second paradox, and explains why the correct interpretation cannot be determined by data alone; two different graphs, both compatible with the data, may dictate two different back-door criteria.

When the back-door criterion is satisfied by a set ''Z'' of covariates, the adjustment formula (see [[Confounding]]) gives the correct causal effect of ''X'' on ''Y''. If no such set exists, Pearl's ''do''-calculus can be invoked to discover other ways of estimating the causal effect.<ref name="pearl" /><ref name="pearl-bow">{{cite book |last1=Pearl |first1=J. |last2=Mackenzie |first2=D. |title=The Book of Why: The New Science of Cause and Effect |date=2018 |publisher=Basic Books |location=New York, NY}}</ref> The completeness of ''do''-calculus <ref>{{cite journal |last1=Shpitser |first1=I. |last2=Pearl |first2=J. |editor1-last=Dechter |editor1-first=R. |editor2-last=Richardson |editor2-first=T.S. |title=Identification of Conditional Interventional Distributions |journal=Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence |date=2006 |pages=437–444 |publisher=AUAI Press |location=Corvallis, OR}}</ref><ref name="pearl-bow" /> can be viewed as offering a complete resolution of the Simpson's paradox.

== Criticism ==
One criticism is that the paradox is not really a paradox at all, but rather a failure to properly account for confounding variables or to consider causal relationships between variables.<ref>{{Cite journal |last=Blyth |first=Colin R. |date=June 1972 |title=On Simpson's Paradox and the Sure-Thing Principle |url=http://www.tandfonline.com/doi/abs/10.1080/01621459.1972.10482387 |journal=Journal of the American Statistical Association |language=en |volume=67 |issue=338 |pages=364–366 |doi=10.1080/01621459.1972.10482387 |issn=0162-1459}}</ref>

Another criticism of the apparent Simpson's paradox is that it may be a result of the specific way that data is stratified or grouped. The phenomenon may disappear or even reverse if the data is stratified differently or if different confounding variables are considered. Simpson's example actually highlighted a phenomenon called noncollapsibility,<ref>{{Cite journal |last=Greenland |first=Sander |date=2021-11-01 |title=Noncollapsibility, confounding, and sparse-data bias. Part 2: What should researchers make of persistent controversies about the odds ratio? |url=https://www.jclinepi.com/article/S0895-4356(21)00182-7/fulltext |journal=Journal of Clinical Epidemiology |language=English |volume=139 |pages=264–268 |doi=10.1016/j.jclinepi.2021.06.004 |issn=0895-4356 |pmid=34119647|doi-access=free }}</ref> which occurs when subgroups with high proportions do not make simple averages when combined. This suggests that the paradox may not be a universal phenomenon, but rather a specific instance of a more general statistical issue.

Critics of the apparent Simpson's paradox also argue that the focus on the paradox may distract from more important statistical issues, such as the need for careful consideration of confounding variables and causal relationships when interpreting data.<ref>{{Cite journal |last1=Hernán |first1=Miguel A. |last2=Clayton |first2=David |last3=Keiding |first3=Niels |date=June 2011 |title=The Simpson's paradox unraveled |journal=International Journal of Epidemiology |volume=40 |issue=3 |pages=780–785 |doi=10.1093/ije/dyr041 |issn=1464-3685 |pmc=3147074 |pmid=21454324}}</ref>

Despite these criticisms, the apparent Simpson's paradox remains a popular and intriguing topic in statistics and data analysis. It continues to be studied and debated by researchers and practitioners in a wide range of fields, and it serves as a valuable reminder of the importance of careful statistical analysis and the potential pitfalls of simplistic interpretations of data.

==See also==
* {{annotated link|Aliasing}}
* {{annotated link|Anscombe's quartet}}
* {{annotated link|Berkson's paradox}}
* {{annotated link|Cherry picking}}
* {{annotated link|Condorcet paradox}}
* {{annotated link|Philosophy:Ecological fallacy}}
* {{annotated link|Social:Gerrymandering}}
* {{annotated link|Low birth-weight paradox}}
* {{annotated link|Modifiable areal unit problem}}
* {{annotated link|Prosecutor's fallacy}}
* Spurious correlation
* [[Omitted-variable bias]]

==References==
{{Reflist}}

== Bibliography ==
* Leila Schneps and [[Biography:Coralie Colmez|Coralie Colmez]], ''Math on trial. How numbers get used and abused in the courtroom'', Basic Books, 2013. {{isbn|978-0-465-03292-1}}. (Sixth chapter: "Math error number 6: Simpson's paradox. The Berkeley sex bias case: discrimination detection").

==External links==
*[http://plato.stanford.edu/entries/paradox-simpson/  Simpson's Paradox] at the [[Stanford Encyclopedia of Philosophy]], by Jan Sprenger and Naftali Weinberger.
*[http://ed.ted.com/lessons/how-statistics-can-be-misleading-mark-liddell How statistics can be misleading – Mark Liddell] – TED-Ed video and lesson.
*[[Biography:Judea Pearl|Pearl, Judea]], [https://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf "Understanding Simpson’s Paradox"] (PDF)
*[http://www.cut-the-knot.org/Curriculum/Algebra/SimpsonParadox.shtml Simpson's Paradox], a short article by [[Biography:Alexander Bogomolny|Alexander Bogomolny]] on the vector interpretation of Simpson's paradox
*[https://www.wsj.com/articles/SB125970744553071829 The Wall Street Journal column "The Numbers Guy"] for December 2, 2009 dealt with recent instances of Simpson's paradox in the news. Notably a Simpson's paradox in the comparison of unemployment rates of the 2009 recession with the 1983 recession.
*[http://www.stateoftheusa.org/content/at-the-plate-a-statistical-puz.php At the Plate, a Statistical Puzzler: Understanding Simpson's Paradox] by Arthur Smith, August 20, 2010
*[https://www.youtube.com/watch?v=ebEkn-BiW5k Simpson's Paradox], a video by Henry Reich of MinutePhysics


{{DEFAULTSORT:Simpson's Paradox}}
[[Category:Probability theory paradoxes]]
[[Category:Statistical paradoxes]]
[[Category:Causal inference]]

{{Sourceattribution|Simpson's paradox}}