Let D be the event that a person develops a certain disease, and C be the event that the

Chapter 2, Problem 59

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Let D be the event that a person develops a certain disease, and C be the event that the person was exposed to a certain substance (e.g., D may correspond to lung cancer and C may correspond to smoking cigarettes). We are interested in whether exposure to the substance is related to developing the disease (and if so, how they are related). The odds ratio is a very widely used measure in epidemiology of the association between disease and exposure, defined as OR = odds(D|C) odds(D|Cc) , where conditional odds are defined analogously to unconditional odds: odds(A|B) = P (A|B) P (Ac|B) . The relative risk of the disease for someone exposed to the substance, another widely used measure, is RR = P(D|C) P(D|Cc) . The relative risk is especially easy to interpret, e.g., RR = 2 says that someone exposed to the substance is twice as likely to develop the disease as someone who isnt exposed (though this does not necessarily mean that the substance causes the increased chance of getting the disease, nor is there necessarily a causal interpretation for the odds ratio). (a) Show that if the disease is rare, both for exposed people and for unexposed people, then the relative risk is approximately equal to the odds ratio. (b) Let pij for i = 0, 1 and j = 0, 1 be the probabilities in the following 2 2 table. D Dc C p11 p10 Cc p01 p00 For example, p10 = P(C, Dc). Show that the odds ratio can be expressed as a crossproduct ratio, in the sense that OR = p11p00 p10p01 . (c) Show that the odds ratio has the neat symmetry property that the roles of C and D can be swapped without changing the value: OR = odds(C|D) odds(C|Dc). This property is one of the main reasons why the odds ratio is so widely used, since it turns out that it allows the odds ratio to be estimated in a wide variety of problems where relative risk would be hard to estimate well.