42 The paper \"Good 5or women, Good for Men ad for People Simpson\'s Paradox and

Question

42 The paper "Good 5or women, Good for Men ad for People Simpson's Paradox and the Impor- ance of Sex-Spedfic Analysis in Olbservatioral Scud es (journal of Women's Health and Gender-Based Medicine t2001b 867-872) decibed he results of a -_-_ stady in which one treatment was shows to be e for men and betner for women than a competing rament. However, if the data fr men and women are combined, it appeas as though the competing unatmen s bener. To see bow this can bappen, coaside the a oanyg dara ables coostructed fom informatien he papes. Sabjects in the scady were given either Trmat ment A or Treatment B, and sarvival was noted. Le Sbe te event that a patient selected at random survives, A be e meat that a patient selected at random moned Trestment A, and B be the event that a pacien selected at andom received Treatment B a. The following table summaiaes dara Sor men and wossen combined Survived DedTo Treatment A Treatment 85 300 59 500 i56 L Find Ps. v. Which treatment appears to be bet Now conaider the summary daa for the men wbo b. purticipated in the arudy Survived Ded Toa B0 200 20? Treacment A 20 140 Tocal Find PisB v. Which treatment appeus to be bett c. Now consider the summary daca for the women who parcipated ia tbe srudyr Survived De To Treatment A Treatment 95 221 316 5 100 ”260 144 i Find PS. Find PSLA) Find PS B) h. Waidh veacent appeans to be ben

Explanation / Answer

The grand total is first computed as: G = 300 + 300 = 600

a) (i) P(S) = 456 / 600 = 0.76

(ii) P(S | A) is computed using bayes theorem as:

P( S | A) = P(S and A) / P(A) = 215 / 300 = 0.7167

(iii) Again same way as previous part, this is computed as:

P( S | B) = 241 / 300 = 0.8033

(iv) From the above 2 parts, we can clearly see that as P(S | B) > P(S | A) , therefore treatment B is better than A.

b) (i) P(S) = 140 / 240 = 0.5833

(ii) P(S | A) = 120 / 200 = 0.6

(iii) P(S | B) = 20 / 40 = 0.5

(iv) Once again from the above 2 parts, we can conclude that treatment A seems better than B

c) (i) P(S) = 316 / 360 = 0.8778

(ii) P(S | A) = 95 / 100 = 0.95

(iii) P(S | B) = 221/ 260 = 0.85

(iv) Treatment A is better than B

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