Narrative · Simpson's paradox
Kidney stones
Open surgery is more successful for small and for large stones, but less successful overall. The theory gives no verdict here, and it says why.
Chapter 1
Two operations
In 1986 Charig and colleagues compared two procedures for removing kidney stones in 700 patients: open surgery and percutaneous nephrolithotomy, an intervention through a small opening. Overall, the percutaneous treatment was more successful: 82.6% versus 78.0% (Vol. III, §8.2).
T open surgery (one) or percutaneous intervention (zero) · SEV large stone (one) or small stone (zero).
Separated by the size of the stones, the picture reverses. For small stones open surgery was more successful: 93.1% versus 86.7% (Vol. III, §8.2).
For large stones as well: 73.0% versus 68.8%. Open surgery is better in both groups and worse overall. That is Simpson's paradox (Vol. III, §8.2).
Chapter 2
Who received which treatment
The explanation lies in the assignment. Of the 350 patients who had open surgery, 263 had large stones; of the 350 treated percutaneously, only 80, just under a third as many. Large stones are harder to treat under both procedures (Vol. III, §8.2).
The study was not randomised. The severity of the case travelled with the assignment of treatment: whoever had a large stone was more likely to receive open surgery. Gate B is thereby violated. A circumstance that co-determines success occurs much more often in one arm than in the other, and the comparison of the overall rates measures it too.
Chapter 3
The suspended verdict
Gate A is violated as well. In each of the four groups there are patients with and without success, with the same values of both recorded factors: four contradictory configurations. To remove all contradictions, at least 138 of the 700 cases, just under 20%, would have to be removed or separated by further factors (Vol. III, §8.2).
The theory therefore pronounces no causal verdict on the overall data. That is its result, not an omission: it names the reason, and it names what would have to be done, namely to separate the fields and to extend the factor set.
An obvious interpretation is not justified either: that stone size is a necessary partner of open surgery in a cluster. Such a partner would show itself in the advantage of surgery appearing in only one of the two groups. Here it is present in both groups at a similar size, 6.4 and 4.3 percentage points. The pattern is that of a bias due to the assignment, not that of a conjunction (Vol. III, §8.2).