1 Simpsons paradoxheart disease patients Today we are going

1) Simpson‘s paradox—heart disease patients Today we are going to look at common numerical arguments often used in newspapers and in television. Often when newspapers and other media provide statistics, we do not know how data was gathered, or the different ways data was manipulated before being presented to us. This lesson will look at how, on the surface, the numbers may tell us one thing; but as we dig deeper, the numbers may tell us something entirely different. The following example is a generalized case of what happened several years ago in the United States. A large drug manufacturer presented information from their study suggesting that their drug reduced the number of heart attacks in patients with heart disease. Patients were randomly selected from several hospitals and administered the new drug. The rest of the patients continued to receive normal heart care. The table shows the results. From the table, we can calculate the proportion of patients that had heart attacks from both the normal and the new drug treatments.

Normal Treatment

New Drug Treatment

Heart attack

920

70

No Hear Attack

8880

765

Total

9800

835

a. Calculate the proportions of heart attacks for both the normal treatment

and the new drug treatment.

b. Is there a difference between the two treatments? Which treatment is

better?

Solution

a)

Getting p1^ and p2^,          
          
p1^ = x1/n1 = 920/9800 =   0.093877551   [ANSWER, NORMAL TREATMENT]  
p2 = x2/n2 = 70/835 =    0.083832335   [ANSWER, NEW DRUG]  

********************

b)

Formulating the hypotheses          
Ho: p1 - p2   =   0  
Ha: p1 - p2   =/=   0  
Here, we see that pdo =    0   , the hypothesized population proportion difference.  
          
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.010033019      
          
Thus,          
          
z = [p1 - p2 - pdo]/sd =    1.001215687      
          
As significance level =    0.05   , then the critical z is  
          
zcrit =    1.959963985      
          
Also, the p value is          
          
P =    0.316722544      
          
As z < 1.95996, and P > 0.05, we FAIL TO REJECT HO.

Thus, there is no significant difference between the treatments at 0.05 level. [CONCLUSION]