The paper Good for Women, Good for Men, Bad for People: Simpsons Paradox and the Importance of

Question:

The paper €œGood for Women, Good for Men, Bad for People: Simpson€™s Paradox and the Importance of Sex Specific Analysis in Observational Studies€ (Journal of Women€™s Health and Gender Based Medicine [ 2001]: 867€“ 872) described the results of a medical study in which one treatment was shown to be better for men and better for women than a competing treatment. However, if the data for men and women are combined, it appears as though the competing treatment is better. To see how this can happen, consider the accompanying data tables constructed from information in the paper. Subjects in the study were given either Treatment A or Treatment B, and their survival was noted. Let S be the event that a patient selected at random survives, A be the event that a patient selected at random received Treatment A, and B be the event that a patient selected at random received Treatment B.
a. The following table summarizes data for men and women combined:

i. Find P(S).
ii. Find P(S| A).
iii. Find P(S| B).
iv. Which treatment appears to be better?
b. Now consider the summary data for the men who participated in the study:

i. Find P(S).
ii. Find P(S| A).
iii. Find P(S| B).
iv. Which treatment appears to be better?
c. Now consider the summary data for the women who participated in the study:

i. Find P(S).
ii. Find P(S| A).
iii. Find P(S| B).
iv. Which treatment appears to be better?
d. You should have noticed from Parts (b) and (c) that for both men and women, Treatment A appears to be better. But in Part (a), when the data for men and women are combined, it looks like Treatment B is better. This is an example of what is called Simpson€™s paradox. Write a brief explanation of why this apparent inconsistency occurs for this data set. (Hint: Do men and women respond similarly to the two treatments?)