- The probability of male patients recovering following treatment is greater than the probability of their recovering following no treatment.
- The probability of female patients recovering following treatment is greater than the probability of their recovering following no treatment.
- Therefore, the probability of (male and female) patients recovering following treatment is greater than the probability of their recovering following no treatment.
The arithmetical structures that underlie facts like the one above invalidate a cluster of arguments that many people, at least initially, take to be intuitively valid. E.g., despite intuitions to the contrary, the following argument is invalid.
In probability and statistics, Simpson's paradox (or the Yule-Simpson effect) is a paradox in which a correlation (trend) present in different groups is reversed when the groups are combined. This result is often encountered in social-science and medical-science statistics, and it occurs when frequency data are hastily given causal interpretations. Simpson's Paradox disappears when causal relations are brought into consideration
Practical significance
The practical significance of Simpson's paradox surfaces in decision making situations where it poses the following dilemma: Which data should we consult in choosing an action, the aggregated or the partitioned?
Example of Simpson's paradox
An examination of racial differences in the management of localized prostate cancer in Pennsylvania simultaneously revealed that whites were more likely to receive prostate surgery than blacks, that whites and blacks were equally likely to get surgery, and that blacks were more likely to get surgery than whites. This example statistical analysis used hypothetical data. All of the above conclusions were correct, but they reflected answers to subtly different questions that relied on different parsings of the same aggregate data.
