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There will be a quiz today.

There will be a quiz today. Please have a sheet of paper ready. The quiz will cover relative risk and odds ratios, which we will briefly review at the beginning of today’s lecture. Lecture 22. This lecture will cover. 1. Review relative risk, odds ratios.

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There will be a quiz today.

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  1. There will be a quiz today. Please have a sheet of paper ready. The quiz will cover relative risk and odds ratios, which we will briefly review at the beginning of today’s lecture.

  2. Lecture 22 This lecture will cover 1. Review relative risk, odds ratios 2. 2×2 tables and third variables(Section 12.4) 3. Testing for independence in 2×2 tables (Chapter 13)

  3. proportion of “yes” in one group odds of “yes” in one group RR = OR = proportion of “yes” in the other group odds of “yes” in the other group 1. Relative risk and odds ratios Relative risk and odds ratio measure the association between two binary variables.

  4. Yes No A B Group 1 C D Group 2 A / (A + B) A × D RR = OR = C / (C + D ) B × C Computing RR and OR Decide which of the two responses will be treated as “yes,” and put that response in the left column. Decide which of the two groups is the “normative” or “baseline” group, and put that group in the bottom row.

  5. Interpretating RR and OR RR = 1 indicates that the proportion/probability/rate of “yes” is the same in both groups. RR > 1 indicates that the proportion/probability/rate of yes in the first group is higher than the baseline. 1.3 means 30% higher; 1.8 means 80% higher; 2 means that it has doubled; and so on. RR < 1 indicates that the proportion/probability/rate of yes in the first group is lower than the baseline. 0.9 means 10% lower; 0.7 means 30% higher; 0.5 means 50% lower; and so on. OR is interpreted in the same way, except that we replace “proportion/probability/rate” with “odds.”

  6. Laid off? Yes No Total African-American 130 1,382 1,512 White 87 2,813 2,900 Total 217 4,195 4,412 130 / 1512 .0860 .0860 = = 130 × 2813 87 / 2900 .0309 .0300 1382 × 87 Example • Layoff is the “yes,” and it is already in the left column • Let’s use “White” as the baseline; it’s already in the bottom row RR = = 2.87 OR = = 3.04 Both of these statistics indicate that African-American employees were laid off at a much higher rate than Whites.

  7. 2. 2×2 Tables and Third Variables Several weeks ago, we learned that the correlation between two continuous variables can be distorted if they are both related to a third variable (called a confounder). This can also happen with binary variables. A confounder may cause a spurious relationship to appear. It may cause a RR or OR to appear stronger (further from 1.0) or weaker (closer to 1.0) than it should be. In extreme cases, it may cause a RR or OR greater than 1.0 to become less than 1.0, or a RR or OR less than 1.0 to become greater than 1.0. This is called Simpson’s paradox.

  8. Delinquent? Yes No Scout 19 239 258 Total Non-scout 44 228 232 Example Scouting and delinquency Rates of delinquency 19/258 = .0736 44/232 = .190 RR = .0736 / .190 = 0.39 OR = (19 × 228) / (239 ×44) = 0.42 Scouting appears to reduce the rate of delinquency by 61% and the odds of delinquency by 58%. But we cannot conclude that the effect is causal, because this is an observational study. There are many possible confounders.

  9. Delinquent? Delinquent? 11 / (11 + 43) 8 / (8 + 196) Yes Yes No No RR = RR = = 1.02 = 1.20 2 / (2 + 59) 42 / (42 + 169) Scout Scout 11 8 43 196 Non-scout Non-scout 42 2 169 59 11 × 169 8 × 59 OR = OR = = 1.20 = 1.03 43 × 42 2 × 196 Suppose we control for income by dividing the sample into two income groups: Low income High income

  10. In the low income group, scouts have a slightly higher rate of delinquency than non-scouts. (This is not “statistically significant” and could be due to chance.) In the high income group, scouts again have a higher rate of delinquency than non-scouts. (Again, this is not statistically significant, as we will show next week.) Yet, when the two income groups are combined, scouts have a much lower rate of delinquency than non-scouts. How can this be? Income—and perhaps many other demographic and socioeconomic factors—is confounding the relationship between scouting and delinquency.

  11. Low income High income Delinquent? Yes No Delinquent? Scout 11 43 Yes No Non-scout 42 169 Scout 8 196 Total Total 53 10 255 212 Non-scout 2 59 Look at the overall rate of delinquency in each income group. In the low income group, the rate of delinquency is 53 / (53 + 212) = 0.20 = 20% In the high income group, the rate of delinquency is 10 / (10 + 255) = 0.038 = 3.8%

  12. Low income High income Delinquent? Yes No Delinquent? Scout 11 43 Yes No 54 204 Non-scout 42 169 Scout 8 196 211 61 Non-scout 2 59 Total Total Now look at the overall rate of scouting in each income group. In the low income group, the rate of scouting is 54 / (54 + 211) = 0.20 = 20% In the high income group, the rate of scouting is 204/ (204 + 61) = 0.77 = 77%

  13. In this example, we have seen that • The low income group has a much higher rate • of delinquency than the high income group • The low income group has a much lower rate • of scouting than the high income group The strong relationships • between income and delinquency • between income and scouting combine to create an apparently strong association between scouting and delinquency. But that association “disappears” when we control for income.

  14. Another example Suppose that we compare the batting averages of two baseball players. It can sometimes happen that • Player A has a higher average than B • in the first half of the season • Player A also has a higher average than B • in the second half of the season Yet Player B has a higher average than A for the entire season!

  15. First half Second half Hit? Yes No Hit? Hit? Player A 4 6 Yes Yes No No Player B 35 65 Player A Player A 25 29 81 75 Player B Player B 2 37 8 73 Overall Player A: 4 / 10 = .400 Player B: 35 / 100 = .350 Player A: 25 / 100 = .250 Player B: 2 / 10 = .200 Player A: 29 / 110 = .264 Player B: 37 / 110 = .336

  16. Who is better? In this case, Player B is clearly better. The fact that A outhit B in the first half is a fluke due to the fact that A hardly played at all. The fact that A outhit B in the second half is also a fluke due to the fact that B hardly played at all. Conclusion: The overall association between two variables can be quite different from the association within subgroups.

  17. Heart attack? Rate of heart attack Yes No Total Aspirin 104 / 11,037 = .0094 Aspirin 104 10,933 11,037 Placebo 189 / 11,034 = .0171 Placebo 189 10,845 11,034 Total 293 21,778 22,071 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. 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 Example RR = .0094 / .0171 = 0.55 The rate of heart attack for those who took aspirin was 45% lower than for those who took a placebo.

  18. 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

  19. Legal abortion for any reason Yes No Total 215 × 244 Men 215 269 484 OR = = 1.13 Women 172 244 416 172 × 269 Total 387 513 900 Example 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?” 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.

  20. Is it real? 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? 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.

  21. proportion of “yes” in one row RR = proportion of “yes” in the other row 3. Choosing the baseline The relative risk is a ratio of proportions: 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.

  22. Throat cancer? Rate of throat cancer Yes No Total Smoker 15 / 2000 = .0075 15 1985 2000 Smoker Nonsmoker 13 / 2500 = .0052 13 2487 2500 Nonsmoker Total 28 4472 4500 Example 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.

  23. 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

  24. Heart attack? Rate of heart attack Yes No Total Aspirin 104 10,933 11,037 Placebo 189 10,845 11,034 Aspirin 104 / 11,037 = .0094 Placebo 189 / 11,034 = .0171 Total 293 21,778 22,071 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. So it makes sense to use the placebo group as the baseline. RR = .0094 / .0171 = 0.55

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