R Programming Odds & Odds Ratios 1

Session 3 Overview 1.Odds 2.Odds Ratio (OR) 3.Confidence Intervals for OR’s 4.Inference based on OR’s 2

Introduction In the last session, we learned about the Relative Risk statistic that allowed us to compare the likelihood of an event between 2 groups. An Odds Ratio (OR) is another way to make this comparison. 3

Odds The odds of an event compares the chance that the event occurs to the chance that it does not. Ex: If there is a 60% chance of rain, then there is a 40% chance that it will not rain. The odds of it raining are 60 to 40, or 3 to 2. 4

Odds Odds are often expressed using the structure “a to b”, so a ratio is implied but often not actually computed. 5

Odds: Smoking Example For the nonsmokers, 374 divorced and 810 had not. Therefore, the odds of divorce for nonsmokers is 374 to 810 or,.46 to 1 (divide both numbers by 810). 6

Odds: Smoking Example Often times, it is easier to interpret an odds > 1 than an odds that is < 1. In order to facilitate interpretation, one can compute the odds of the non-event (i.e., remaining married) instead of finding the odds of the event (divorce). 7

Odds: Smoking Example Inversely, we could compute the odds of remaining married for nonsmokers, odds(married) = 810/374 = Therefore, nonsmokers are approximately 2 times more likely to remained married than to get a divorce. 8

Odds: Smoking Example For the smokers, 238 divorced and 247 had not. Therefore, the odds of divorce for smokers is 238 to 247 or,.96 to 1. Furthermore, the odds of remaining married for smokers is 247 to 238 or,

Properties of Odds 1.Odds are nonnegative. 2.When odds = 4, a success is 4 times as likely as failure (or we expect to observe 4 successes for every failure). 3.When odds = ¼, a failure is 4 times as likely as a success. 10

Odds Ratio The odds ratio (OR) compares the odds of an event of 2 different groups. 11

Odds Ratio: Smoking Ex. Therefore, the odds of divorcing to remaining married for smokers are double the odds for nonsmokers. 12

Properties of Odds Ratio 1.The OR is nonnegative. 2.The independence value (or null value) is 1.0. Therefore, when OR = 1, the 2 groups have the same odds of success. 3.When OR > 1, group 1 has higher odds of success. 4.When OR < 1, group 2 has higher odds of success. 13

Inference based on OR 1.The OR of a sample is a sample statistic that measures the association between success (event)/failure (non-event) and the grouping variable. 14

Inference based on OR 2.Like most sample statistics, we can compute confidence intervals for the OR to make inferences about the larger population based on the sample. 15

Inference based on OR 3.The formula for the CI for an OR is complicated and involves a natural log transformation, therefore, we will rely on R for the computations. 16

Inference based on OR 4.If the CI for an OR does not include 1, then we reject the H 0 of independence and conclude that differences in the odds of success do exist in the population. This is equivalent to concluding that the grouping variable is associated or related to the event. 17

OR: Smoking Example. The odds of divorce for smokers are approximately double the odds for nonsmokers, OR = Furthermore, a 95% CI for the OR does not include the null value of 1, (1.68, 2.59). Therefore, we conclude that an association between smoking behavior and marital status exists in the population. 18