Please turn off cell phones, pagers, etc. The lecture will begin shortly.

Exam 2 summary Exam 2 had 40 items. The test results indicated that there were three bad items. The grades were adjusted: The test was also longer and more difficult than anticipated. adjusted score = (# items correct × 2.5 ) + 12

Distribution of adjusted scores n = 310 mean = 82.7median = 85 min = 35max = 107SD = 14.7

Lecture 21 This lecture will finish topics from Chapter Review probability and odds (Section 12.2) 2. Measures of association in 2×2 tables (Section 12.2) 3. Choosing the baseline (Section 12.3) Due to time constraints, there will not be a quiz today. There will be a quiz on Friday with odds ∞.

1. Review of probability and odds Probability and odds are numerical measures of how likely an event is to occur. Probability is a number between 0 and 1 Odds is a number between 0 and ∞ oddsprob oddsprob

Conversion formulas Given one, you can find the other by these formulas: odds = probability 1 - probability probability = odds 1 + odds Examples prob =.2corresponds to odds =.2/.8 =.25 prob =.75corresponds to odds =.75/.25 = 3 odds =.5corresponds to prob =.5/1.5 =.333 odds = 9corresponds to prob = 9/10 =.9

Estimating probabilities and odds Measure a binary variable for a sample of n subjects and summarize the results as a frequency table Yes No a b Freq Totaln Estimated probability of yes = a n Yes Yes + No Estimated odds of yes = a b Yes No

Example Twenty likely voters were asked, “Do you approve of the President’s job performance?” YYNYNNYYNNNNYNYNNYNYYYNYNNYYNNNNYNYNNYNY Approve Disapprove 9 11 Freq Estimate the probability and odds of approval. Estimated probability = 9/20 =.45 Estimated odds = 9/11 =.82

2. Measures of association in 2×2 tables Recall that with two continuous variables, a useful measure of association is the correlation coefficient. For two binary variables, the most common measures of association are Relative risk Odds ratio The relative risk is a ratio of probabilities. The odds ratio is a ratio of odds.

Estimating the relative risk Compute the proportion for each row Divide one proportion by the other 11,037 11,034 Total Aspirin Placebo ,933 10,845 YesNo Heart attack? 29321,77822,071Total Aspirin Placebo 104 / 11,037 =.0094 Proportion with heart attack 189 / 11,034 =.0171 Example The estimated relative risk is.0094 /.0171 = 0.55

Estimating the odds ratio Compute the odds for each row Divide one odds by the other 11,037 11,034 Total Aspirin Placebo ,933 10,845 YesNo Heart attack? 29321,77822,071Total Aspirin Placebo 104 / 10,933 =.0095 Estimated odds of heart attack 189 / 10,845 =.0174 Example The estimated odds ratio is.0095 /.0174 = 0.55

Easier way to estimate the odds ratio If the frequencies in the 2×2 table are ab cd then the estimated odds ratio is (a×d) / (b×c). 11,037 11,034 Total Aspirin Placebo ,933 10,845 YesNo Heart attack? 29321,77822,071Total Example 104 × 10,845 10,933 × 189 = 0.55 The estimated odds ratio is

Interpreting the relative risk A relative risk of 1.0 means that the proportion of “yes” in row 1 is the same as the proportion of “yes” in row Total Vitamin C Placebo YesNo Cold this year? Total This means that there is no evidence of a relationship between the explanatory variable and the response variable. Example Vitamin C Placebo 45 / 100 =.45 Proportion with cold 54 / 120 =.45 RR =.45 /.45 = 1.0 Vitamin C appears to be no more effective than a placebo for preventing the common cold.

A relative risk greater than 1.0 means that the proportion of “yes” in row 1 is greater than the proportion of “yes” in row Total Smoker Nonsmoker YesNo Throat cancer? Total RR = 1.2 means that the proportion is 20% greater Example Smoker Nonsmoker 15 / 2000 =.0075 Rate of throat cancer 13 / 2500 =.0052 RR =.0075 /.0052 = 1.44 The rate of throat cancer among smokers is 44% higher than among nonsmokers. RR = 1.5 means that the proportion is 50% greater RR = 2.0 means that the rate is doubled RR = 3.0 means that the rate is tripled

A relative risk less than 1.0 means that the proportion of “yes” in row 1 is lower than the proportion of “yes” in row 2. Example The rate of heart attack for those who took aspirin was 45% lower than for those who took a placebo. RR = 0.90 means that the rate is 10% lower RR = 0.75 means that the rate is 25% lower RR = 0.50 means that the rate is 50% lower RR = 0.20 means that the rate is 80% lower 11,037 11,034 Total Aspirin Placebo ,933 10,845 YesNo Heart attack? 29321,77822,071Total Aspirin Placebo 104 / 11,037 =.0094 Rate of heart attack 189 / 11,034 =.0171 RR =.0094 /.0171 = 0.55

Interpreting the odds ratio The interpretation of an odds ratio is very similar to the interpretation of a relative risk. The only difference is that an odds ratio expresses the increase or decrease in terms of odds rather than rates. OR = 1.0 means that there is no evidence of a relationship OR = 1.2 means that the odds of “yes” in row 1 are 20% higher than the odds of “yes” in row 2 OR = 0.60 means that the odds of “yes” in row 1 are 40% lower than the odds of “yes” in row 2

Example Total Men Women YesNo Legal abortion for any reason Total From the 2002 General Social Survey “Should it be possible for a pregnant woman to obtain a legal abortion if the woman wants it for any reason?” OR = 215 × × 269 = 1.13 Based on this sample, men appear to be slightly more likely than women to support legalized abortion for any reason. The estimated odds of support are 13% higher among men than among women.

Based on the data from the last example, can we really conclude that the level of support of legalized abortion “for any reason” is greater among men than among women? Is it real? Perhaps not. The odds ratio of 1.13 is only an estimate, and it is not far from 1.0. How far away from 1.0 does an estimate need to be for us to conclude that the effect is real, and not just due to random chance? That depends on the margins of error, which in turn depend on the sample sizes in the two groups (men and women). Techniques for judging whether the effect is real or not will be discussed next week.

3. Choosing the baseline The relative risk is a ratio of proportions: RR = proportion of “yes” in one row proportion of “yes” in the other row In our examples thus far, we have used Row 1 of the 2×2 table for the numerator Row 2 of the 2×2 table for the denominator But we are free to use either row as the numerator or denominator, as long as we interpret the result correctly.

Total Smoker Nonsmoker YesNo Throat cancer? Total Example Smoker Nonsmoker 15 / 2000 =.0075 Rate of throat cancer 13 / 2500 =.0052 If we use “nonsmoker” as the numerator and “smoker” as the denominator, we get RR =.0052 /.0075 = 0.72 Interpretation: The estimated rate of throat cancer is 28% lower among non-smokers than among smokers. If we use “smoker” as the numerator and “nonsmoker” as the denominator, we get RR =.0075 /.0052 = 1.44 Interpretation: The estimated rate of throat cancer is 44% higher among smokers than among non-smokers.

Which way is better? Both ways of presenting the relative risk are correct. But the second way (RR=1.44) is a little easier to understand. In this example, “non-smoking” is the normative condition, and “smoking” is the condition that is potentially hazardous. If one of the two groups (either row 1 or row 2) can be regarded as normative behavior a control group (e.g., “placebo” or “nothing”) treatment as usual the majority then it makes sense to use that group as the denominator. The group in the denominator becomes the baseline for assessing the risk level of the group in the numerator

Another example In this example, the placebo group is the control group, and those who are taking aspirin are receiving the “new” or “novel” treatment. 11,037 11,034 Total Aspirin Placebo ,933 10,845 YesNo Heart attack? 29321,77822,071Total So it makes sense to use the placebo group as the baseline. Aspirin Placebo 104 / 11,037 =.0094 Rate of heart attack 189 / 11,034 =.0171 RR =.0094 /.0171 = 0.55