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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-1/13j/b/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 13J / b / i / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

Because the second model is saturated, its fitted worsening probabilities are the observed proportions:

p_{C} = \frac{101}{5728 + 101} = 0.01733, p_{L} = \frac{3}{1364 + 3} = 0.002195, p_{S} = \frac{27}{4413 + 27} = 0.006081.

Thus the drug efficacy as a risk reduction is

E_{L} = 1 - \frac{p _{L}}{p _{C}} = 1 - 0.1266 = 87.3%,

and

E_{S} = 1 - \frac{p _{S}}{p _{C}} = 1 - 0.3509 = 64.9% .

These are reasonably similar to the published estimates of

90.0%

and

62.1%

.

The interaction coefficients provide the same quick comparison on the odds scale:

e^{- 2.08156} = 0.125 and e^{- 1.05847} = 0.347.

They are treatment-versus-control odds ratio. Since worsening is rare in every group, each odds ratio is close to its risk ratio, giving the approximate efficacies

87.5%

and

65.3%

.

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