In probability and statistics, Simpson's paradox, or the Yule--Simpson effect, is a paradox in which a trend that appears in different groups of data disappears when these groups are combined, and the reverse trend appears for the aggregate data. This result is often encountered in social-science and medical-science statistics, and is particularly confounding when frequency data are unduly given causal interpretations. Simpson's Paradox disappears when causal relations are brought into consideration. Though it is mostly unknown to laypeople, Simpson's paradox is well known to statisticians, and it is described in a few introductory statistics books. Many statisticians believe that the mainstream public should be informed of the counter-intuitive results in statistics such as Simpson's paradox. Edward H. Simpson first described this phenomenon in a technical paper in 1951, but the statisticians Karl Pearson, et al., in 1899, and Udny Yule, in 1903, had mentioned similar effects earlier. The name Simpson's paradox was introduced by Colin R. Blyth in 1972. Since Edward Simpson did not actually discover this statistical paradox (an instance of Stigler's law of eponymy), some writers, 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. Examples Kidney stone treatment This is a real-life example from a medical study comparing the success rates of two treatments for kidney stones. The table below shows the success rates and numbers of treatments for treatments involving both small and large kidney stones, where Treatment A includes all open surgical procedures and Treatment B is percutaneous nephrolithotomy. The numbers in parentheses indicate the number of success cases over the total size of the group. (For example, 93% equals 81 divided by 87.) 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 is more effective when considering both sizes at the same time. In this example the "lurking" variable (or confounding variable) of the stone size was not previously known to be important until its effects were included. Which treatment is considered better is determined by an inequality between two ratios (successes/total). The reversal of the inequality between the ratios, 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 the severe cases (large stones) the better treatment (A), and the milder cases (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 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 can be rephrased more intuitively as follows: Treatment A, when applied to a patient population consisting mainly of patients with large stones, is less successful than Treatment B applied to a patient population consisting mainly of patients with small stones. Berkeley gender bias case One of the best-known real-life examples of Simpson's paradox occurred when the University of California, Berkeley was sued for bias against women who had applied for admission to graduate schools there. 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. But when examining the individual departments, it appeared that no department was significantly biased against women. In fact, most departments had a "small but statistically significant bias in favor of women." The data from the six largest departments are listed below. The research paper by Bickel et al. concluded that women tended to apply to competitive departments with low rates of admission even among qualified applicants (such as in the English Department), whereas men tended to apply to less-competitive departments with high rates of admission among the qualified applicants (such as in engineering and chemistry). The conditions under which the admissions' frequency data from specific departments constitute a proper defense against charges of discrimination are formulated in the book Causality by Pearl. Low birth weight paradox Attribution: Article text available under CC-BY-SA Public domain image source in video

• published: 04 Dec 2013
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