Question

Discussion: Simpson's Paradox

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A family member can go to one of two local hospitals for heart surgery.
Checking the history for the past year, you find that each of the two hospitals has performed cardiac surgery on 1000 patients. In hospital A 710 patients survived (71%). In hospital B 540 (54%) survived.
Based on the numbers presented, which hospital do you think is superior in cardiac surgery?

Surely hospital A is better, right?

Now, let’s look at more data. The below chart summarizes three categories of patients (those entering in fair, serious and critical condition) and the survival rate from surgery (in percent) for the two local hospitals.

Patient Entering Condition	Hospital A	Hospital B	Survivors from A (# and percent)	Survivors from B (# and percent)
Fair	700	100	600 or 86%	90 or 90%
Serious	200	200	100 or 50%	150 or 75%
Critical	100	700	10 or 10%	300 or 43%
Total	1000	1000	710 or 71%	540 or 54%

Looking at the data broken down in this way, we see that Hospital B has a higher success rate in all three categories of patients but when averaged all together, Hospital A has the higher overall survival rate. Based on the numbers presented, which hospital do you think is superior in cardiac surgery?

Answer 1

solution :

Given that

A family member can go to one of two local hospitals for heat surgey.

In the example, 2 parameter were used to decide which Hospital is superior in cardiac surgery, -

1. Success rate in all 3 categories of patients

2. Overall survival rate

• For the 1st parameter, Hospital B has a higher success rate in all 3 categories of patients. So, Hospital B is superior in Cardiac surgery as per the 1^st parameter.
• For the 2^nd parameter, Hospital A has a higher overall success rate. So, Hospital A is superior in Cardiac surgery as per the 2^nd parameter.
• . But looking at the percentage of critical patients entering in Hospital   Hospital A = 100/1000 = 10%. is lower than in Hospital B is 700/1000 = 70%.
• We assume that the chance of sucess in critical cardiac surgery is very low. (as can be seen 10 % for hospital A and 43% for hospital B), so the overall success rate of B is low. ​​​​​​​
• If we normalize the data and same percentage of patients with different conditions enter the hospital A and B, and calculate number of survivors based on their past performance, we get the overall success rate as   

  Patient entering condition	Hospital A	Hospital B	Survivors from A (# and percent)	Survivors from B (# and percent)
  Fair	100	100	86(or 86%)	90 (or 90%)
  Serious	100	100	50 (or 50%)	75(or 75%)
  Critical	100	100	10 (or 10%)	43(or 43%)
  Total	300	300	146 (or 48.67%)	208 (or 69.33%)

     1) So, Hospital B has higher overall survival rate of (69.33%).

2) we determined that Hospital B has a higher success rate in all 3 categories of patients and also has higher overall survival rate compared with Hospital A So, Hospital B is superior in Cardiac surgery.