Question

A paper 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. (Round your answers to three decimal places.)

(a)

The following table summarizes data for men and women combined.

	Survived	Died	Total
Treatment A	212	88	300
Treatment B	247	53	300
Total	459	141

(i)

Find

P(S).

(ii)

Find

P(S|A).

(iii)

Find

P(S|B).

(iv)

Which treatment appears to be better?

Treatment ATreatment B    

(b)

Now consider the summary data for the men who participated in the study.

	Survived	Died	Total
Treatment A	120	80	200
Treatment B	20	20	40
Total	140	100

(i)

Find

P(S).

(ii)

Find

P(S|A).

(iii)

Find

P(S|B).

(iv)

Which treatment appears to be better?

Treatment ATreatment B    

(c)

Now consider the summary data for the women who participated in the study.

	Survived	Died	Total
Treatment A	92	8	100
Treatment B	227	33	260
Total	319	41

(i)

Find

P(S).

(ii)

Find

P(S|A).

(iii)

Find

P(S|B).

(iv)

Which treatment appears to be better?

Treatment ATreatment B    

(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?)

The results are distorted in favor of Treatment A, as women respond to both treatments better than men, but Treatment A was given to far more women than men.The results are distorted in favor of Treatment B, as women respond to these treatments better than men, but Treatment A was given to far more women than men.    The results are distorted in favor of Treatment A, as women respond to these treatments better than men, but Treatment B was given to far more women than men.The results are distorted in favor of Treatment B, as women respond to both treatments better than men, but Treatment B was given to far more women than men.

Answer 1

a) Probability = Number of people for that event/Total number of people

I) P(S) = Total people survived/Total sample = 459/600 = 0.765

ii) P(S|A) = P(S and A)/P(A) (from the Bayes theorem)

= Total people who received Treatment A and survived/Total people who received Treatment A = 212/300 = 0.7067

III) P(S|B) = P(S and B)/P(B) = 247/300 = 0.8233

(Similarly like part ii)

iv) Since probability of survival given treatment b is better than probability of survival given treatment a as we can see that 0.8233 >0.7067, so clearly Treatment B is better.