#### An alternative to the odds ratio and the difference in probabilities is the relative risk, which is a common statistic for comparing disease rates between risk groups

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# An alternative to the odds ratio and the difference in probabilities is the relative risk, which is a common statistic for comparing disease rates between risk groups. If Tu and ny are the probabilities of disease for unvaccinated and vaccinated subjects, the relative risk (due to not vaccinating) is p = Ju/Tv. (So if the relative risk is 2, the probability of disease is twice as large for unvaccinated as for vaccinated individuals.) (a) Show that the odds ratio is very close to the relative risk when very small proportions are involved. (You may do this by showing that the two quantities are quite similar for a few illustrative choices of small values of Ty and Ju.) (b) Calculate the relative risk of not vaccinating for the situation introduced at the beginning of Exercise 12. (c) Calculate the relative risk of not vaccinating for the situation described in Exercise 12(e). (d) Here is one situation with a relative risk of 50: The unvaccinated probability of disease is 0.0050 and the vaccinated probability of disease is 0.0001 (the probability of disease is 50 times greater for unvaccinated subjects than for vaccinated ones). Suppose in another situation the vaccinated probability of disease is 0.05. What unvaccinated probability of disease would imply a relative risk of 50? (Answer: None. This shows that the range of relative risk possibilities depends on the baseline disease rate. This is an undesirable feature of relative risk.)12, Suppose that the probability of a disease is 0.00369 in a population of unvaccinated subjects and that the probability of the disease is 0.001 in a population of vaccinated subjects. (a) What are the odds of disease without vaccine relative to the odds of disease with vaccine? (b) How many people out of 100,000 would get the disease if they were not treated? (c) How many people out of 100,000 would get the disease if they were vaccinated? (d) What proportion of people out of 100,000 who would have gotten the disease would be spared from it if all 100,000 were vaccinated? (This is called the protection rate.) (e) Follow the steps in parts (a)-(d) to derive the odds ratio and the protection rate if the unvac- cinated probability of disease is 0.48052 and the vaccinated probability is 0.2. (The point is that the odds ratio is the same in the two situations, but the total benefit of vaccination also depends on the probabilities.) 