# Alternative Aggregation Method to Avoid Simpson's Paradox

The table below shows an example of Simpson's Paradox arising from some fictitious data for the success of operations performed by two doctors from the Simpsons TV show.

Dr Hilbert Dr Nick
Heart Surgery 70/90 (77%) 2/10 (20%)
Band Aid Removal 10/10 (100%) 81/90 (90%)
Total 80/100 (80%) 83/100 (83%)

Thinking about the data in terms of 4 regions:

A | B
-----
C | D


, I'm assuming, although I'm not quite sure how to prove it, that

(A > B) & (C > D) iff (A + C) > (B + D).

It seems the paradox arises due to the aggregation step - A + C are added as if when adding fractions, a/b + c/d were equal to (a + b) / (c + d).

What would be the correct way to aggregate the columns to get a better sense of the composition of the data? For example:

Dr Hilbert Dr Nick
Heart Surgery 70/90 (77%) 2/10 (20%)
Band Aid Removal 10/10 (100%) 81/90 (90%)
Total 160/90 (177.78%) 99/90 (110%)

Even though mathematically impossible, does this provide a better insight into the competence of Dr Hilbert than the original table (ignoring issues like selection bias due to expertise for now)?

Is this approach of taking the sum of the percentages a useful way to get a clearer picture of the aggregate results in cases where Simpson's Paradox is present, or can it be as misleading as the results provided by the usual method of aggregation?