Editing Simpson's paradox (section)

===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>