158 CHAPTER 4 Probability 44. Relative Risk and Odds Ratio In a clinical trial of 2103 subjects treated with Nasonex, 26 reported headaches. In a control group of 1671 subjects given a placebo, 22 reported headaches. Denoting the proportion of headaches in the treatment group by pt and denoting the proportion of headaches in the control (placebo) group by pc, the relative risk is pt>pc. The relative risk is a measure of the strength of the effect of the Nasonex treatment. Another such measure is the odds ratio, which is the ratio of the odds in favor of a headache for the treatment group to the odds in favor of a headache for the control (placebo) group, found by evaluating the following: pt>11 - pt2 pc>11 - pc2 The relative risk and odds ratios are commonly used in medicine and epidemiological studies. Find the relative risk and odds ratio for the headache data. What do the results suggest about the risk of a headache from the Nasonex treatment? Key Concepts In this section we present the addition rule as a tool for finding P1A or B2, which is the probability that either event A occurs or event B occurs (or they both occur) as the single outcome of a procedure. To find P1A or B2, we begin by adding the number of ways that A can occur and the number of ways that B can occur, but add without double counting. The word “or” in the addition rule is associated with the addition of probabilities. This section also presents the basic multiplication rule used for finding P1A and B2, which is the probability that event A occurs and event B occurs. If the outcome of event A somehow affects the probability of event B, it is important to adjust the probability of B to reflect the occurrence of event A. The rule for finding P1A and B2 is called the multiplication rule because it involves the multiplication of the probability of event A and the probability of event B (where, if necessary, the probability of event B is adjusted because of the outcome of event A). The word “and” in the multiplication rule is associated with the multiplication of probabilities. In Section 4-1 we considered only simple events, but in this section we consider compound events. 4-2 Addition Rule and Multiplication Rule Addition Rule Notation for Addition Rule P1A or B2 = P1in a single trial, event A occurs or event B occurs or they both occur2 The word “or” used in the preceding notation is the inclusive or, which means either one or the other or both. The formal addition rule is often presented as a formula, but blind use of formulas is not recommended. Instead, understand the spirit of the rule and use that understanding, as in the intuitive addition rule that follows. DEFINITION A compound event is any event combining two or more simple events.
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