Simpson’s paradox

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Simpson’s paradox



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.

EXPLANATION:

success rate

 

STONE dia < 2

STONE dia>=2

TOTAL

Open Surgery

93% (81/87)

73% (192/263)

78% (273/350)

Newer PN Method

83% (234/270)

69% (55/80)

83% (289/350)

 

 

 

 

 

Here, Open Surgery is better for both cases. Total is higher is lesser as total samples for Stone dia>2 is more ie 263 comparing to Stone dia<2 ie 87. So, the Total will be nearer to 73 and far from 93 ie 78. In Newer PN Method since total samples for Stone dia<2 is more ie 270 than Stone dia >2 ie 80, the Total is more nearer to 83 than 69. Hence, the total depends on no of samples in each group. Since, It is not equally grouped, the total is unreliable.

Technical Explanation:

According to Multiplication rule & Total Probability Theory, The overall success rate depends on the success rates for each stone size group, but also the probability of the groups.It is the weighted average of the group success rate weighted by the group size as follows P(overall success) = P(succes rate of stone dia>=2)*P(stone dia>=2))+ P(success rate of stone dia < 2)*P(stone dia < 2 ). 

Here, Probability of sample percentage of each stone size in each method varies.

For open surgery, the dominant group (stone dia>=2) has a smaller success rate while for PN, the dominant group (stone dia < 2) has a larger success rate.


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