Problem 131E
An article in the British Medical Journal [“Comparison of treatment of renal calculi by operative surgery, percutaneous nephrolithotomy, and extracorporeal shock wave lithotripsy” (1986, Vol. 82, pp. 879–892)] provided the following discussion of success rates in kidney stone removals. Open surgery had a success rate of 78% (273/350) and a newer method, percutaneous nephrolithotomy (PN), had a success rate of 83% (289/350). This newer method looked better, but the results changed when stone diameter was considered. For stones with diameters less than 2 centimeters, 93% (81/87) of cases of open surgery were successful compared with only 83% (234/270) of cases of PN. For stones greater than or equal to 2 centimeters, the success rates were 73% (192/263) and 69% (55/80) for open surgery and PN, respectively. Open surgery is better for both stone sizes, but less successful in total. In 1951, E. H. Simpson pointed out this apparent contradiction (known as Simpson’s paradox), and the hazard still persists today. Explain how open surgery can be better for both stone sizes but worse in total.
Solution :
Step 1 of 1:
From the given information the success rate in kidney stone removals table is given below.
The the total open surgery had success rate is
Type 
Percent 
Fraction 
Open surgery 
78% 

PN 
83% 
Then stone diameter is less than 2 centimeters.
The table is given below.
Type 
Percent 
Fraction 
Open surgery 
93% 

PN 
83% 
Our goal is:
We need to explain how open surgery can be better for both stone sizes but worse in total.
Now the stone diameter is greater or equals to 2 centimeters.
Then the table is given below.
Type 
Percent 
Fraction 
Open surgery 
73% 

PN 
69% 
Now we have to explain how open surgery can be better for both stone sizes but worse in total.
The open surgery table is given below.
Success 
Failure 
Sample size 
Sample percentage 
Conditional success rate 

Large stone 
192 
71 
263 
75% 
73% 
Small stone 
81 
6 
87 
25% 
93% 
Total 
273 
77 
350 
100% 
78% 
The PN surgery table is given below.
Success 
Failure 
Sample size 
Sample percentage 
Conditional success rate 

Large stone 
55 
25 
80 
23% 
69% 
Small stone 
234 
36 
270 
77% 
83% 
Total 
289 
61 
350 
100% 
83% 
From the above table there are a lot more open surgeries the stone have diameter is less than 2 centimeters, so when the stones have a diameter of less than 2 centimeters then there are a lot of PN’s.
Here the total success rate is depends on the success rate for each stone size, but same as the probability group.
So this the weighted average of the group success rate weighted by the group size is
P(overall success) = P.
So open surgery, the large stone has a smaller success rate while for PN, the small stone has a larger rate.