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The treatment that is better for everyone yet worse overall

statisticssimpsons-paradoxconfoundingbase-rates

Two treatments for kidney stones were compared in a real study. For small stones, treatment A succeeded in 81 of 87 cases (93 per cent) and treatment B in 234 of 270 (87 per cent). For large stones, A succeeded in 192 of 263 (73 per cent) and B in 55 of 80 (69 per cent). So A is better for small stones and better for large stones.

Yet overall, A succeeded in 273 of 350 cases (78 per cent) and B in 289 of 350 (83 per cent). B looks better overall.

Is this arithmetic actually possible, and if so, what is going on? Which treatment should a patient prefer?

Show a hint

Look at which treatment was given to which kind of stone. The two groups are not the same mix of easy and hard cases.

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