Simpson’s paradox is a classic puzzle discussed in introductory statistics courses worldwide. That is, when confronted with a Simpson’s paradox, where two different choices seem to compete for being the best choice depending on how the data is partitioned, which choice should one choose?
To make the problem concrete, let’s consider the first example given in the relevant Wikipedia article. It is based on a real study about a treatment for kidney stones.
Suppose you are a doctor and a test reveals that a patient has kidney stones. Using only the information provided in the table, you would like to determine whether I should adopt treatment A or treatment B.
So: A patient walks into your office. A test reveals they have kidney stones but gives you no information about their size. Which treatment should you recommend? Is there any accepted resolution to this problem?
Solution
It seems that if you know the size of the stone, then we should prefer treatment A. But if we do not, then we should prefer treatment B.
But consider another plausible way to arrive at an answer. If the stone is large, we should choose A, and if it is small, we should again choose A. So even if we do not know the size of the stone, by the method of cases, we see that we should prefer A. This contradicts our earlier reasoning.
Simpson’s paradox occurs because of confounding. In this example, the treatment is confounded with the kind of kidney stones each patient had. We know from the full table of results presented that treatment A is always better. Thus, a doctor should choose treatment A. The only reason treatment B looks better in the aggregate is that it was given more often to patients with the less severe condition, whereas treatment A was given to patients with the more severe condition. Nonetheless, treatment A performed better with both conditions. As a doctor, you don’t care about the fact that in the past the worse treatment was given to patients who had the lesser condition, you only care about the patient before you, and if you want that patient to improve, you will provide them with the best treatment available.
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