1. 1 EPI 5240: Introduction to Epidemiology Measures used to compare groups October 5, 2009 Dr. N. Birkett, Department of Epidemiology & Community Medicine, University of Ottawa

2. 2 Session Overview Methods of Comparing groups –Risk/rate ratios –Odd ratios –Difference measures

3. 3 ONE BIG WARNING!!!!! Some books (e.g. the Greenberg one used in the summer course) rotate their 2X2 tables from the normal approach. That is, they have the outcomes as the rows and the exposure as the columns. BE WARNED. This could cause confusion. My tables use the more common approach.

4. 4 Comparing groups (1) Two main outcome measures –Incidence (either risk or rate) –Prevalence How do you determine if an exposure is related to an outcome? –Need to compare the measure in the two groups. Differences Ratios (we’ll start with this one). –Ratio measures have NO units. –All ratio measures have the same interpretation 1.0 = no effect < 1.0  protective effect > 1.0  increased risk –Values over 2.0 are of strong interest

5. 5 Comparing groups:Cohorts (2) YES NO YES 1,000 9,000 10,000 NO 100 9,900 10,000 1,100 18,900 20,000 Disease Exp RISK RATIO Risk in exposed: = 1000/10000 Risk in Non-exposed= 100/10000 If exposure increases risk, you would expect the risk in the exposed to be larger than risk in the unexposed. How much larger can be assessed by the ratio of one to the other: Risk in exp Risk ratio (RR) = ----------------------- Risk in non-exp = (1000/10000)/(100/10000) = 10.0

6. 6 Comparing groups:Cohorts (3) YES NO YES a b a+b NO c d c+d a+c b+d N Disease Exp RISK RATIO Risk in exposed: = a/(a+b) Risk in Non-exposed= c/(c+d) If exposure increases risk, you would expect a/(a+b) to be larger than c/(c+d). How much larger can be assessed by the ratio of one to the other: Risk in exp Risk ratio (RR) = ----------------------- Risk in non-exp = (a/(a+b))/(c/(c+d) a/(a+b) = -------------- c/(c+d)

7. 7 Comparing groups:Cohorts (4) YES NO High 42 80 122 Low 43302 345 85 382 467 Death Pollutant level Risk in exposed: = 42/122 = 0.344 Risk in Non-exposed= 43/345 = 0.125 Exp risk Risk ratio (RR) = ---------------------- Non-exp risk = 0.344/0.125 = 2.76

8. 8 95% CI’s for CIR (1) For a mean value, the 95% CI is given as: where Assumes mean has a normal (Gaussian) distribution

9. 9 Might try using the same approach to obtain ’95% CI’ for CIR using: BUT: CIR is NOT normally distributed –Range from 0 to +∞ –Null value = 1.0 –Implies a non-symmetric distribution 95% CI’s for CIR (2)

10. 10 Plot of ‘CIR’ distribution when H 0 is true

11. 11 Instead, use ‘log(CIR)’ where the log is taken to the ‘natural’ base ‘e –Often written ln(CIR) ln(CIR) is approximately normally distributed –Range from -∞ to +∞ –Null value = 0.0 95% CI is given by: Need to find formula for ‘se(ln(CIR))’ 95% CI’s for CIR (3)

12. 12 Plot of ‘ln(CIR)’ distribution when H 0 is true

13. 13 if exposed and unexposed are independent 95% CI’s for CIR (4) After some math, this gives the following result (next slide)

14. 14 95% CI’s for CIR (5) YES NO YES a b a+b NO c d c+d a+c b+d N Disease Exp

15. 15 95% CI’s for CIR (6) We’re close now. Just take the ‘anti-logs’ (usually called the ‘exp’ function

16. 16 Comparing groups:Cohorts (5) YES NO High 42 80 122 Low 43302 345 85 382 467 Death Pollutant level Risk ratio (RR) or CIR = 2.76 80 302 var(ln(CIR)) = ------------ + ------------- = 0.03597 42*122 43*345 se(ln(CIR)) = sqrt(0.03597) = 0.190 Upper 95% CI = 2.76 * exp(+1.96*0.190) = 4.00 Lower 95% CI = 2.76 * exp(-1.96*0.190) = 1.90 Conclusion: CIR is: 2.76 (1.90 to 4.00)

17. 17 Comparing groups: Cohorts (6) Hypothesis testing (H 0 : CIR=1) –Much less common than 95% CI’s –Normal approximation test is generally OK

18. 18 Comparing groups:Cohorts (7) YES NO YES 1,000 9,000 10,000 NO 100 9,900 10,000 1,100 18,900 20,000 Disease Exp RISK DIFFERENCE Risk in exposed: = 1000/10000 Risk in Non-exposed= 100/10000 If exposure increases risk, you would expect the risk in the exposed to be larger than risk in the unexposed. How much larger can be assessed by the difference between the two: Risk difference (RD) = (Risk in Exp) – (Risk in Non-exp) 1000 100 900 = ---------- - ----------- = ----------- = 0.90 10,000 10,000 10,000

19. 19 Comparing groups:Cohorts (8) YES NO YES a b a+b NO c d c+d a+c b+d N Disease Exp RISK DIFFERENCE Risk in exposed: = a/(a+b) Risk in Non-exposed= c/(c+d) If exposure increases risk, you would expect a/(a+b) to be larger than c/(c+d). How much larger can be assessed by the difference between the two: Risk difference (RD) = (Risk in Exp) – (Risk in Non-exp) a c = ---------- - ----------- a + b c + d

20. 20 Comparing groups:Cohorts (9) YES NO High 42 80 122 Low 43302 345 85 382 467 Death Pollutant Level Risk in exposed: = 42/122 = 0.344 Risk in Non-exposed= 43/345 = 0.125 Risk difference (RD) = (Risk in Exp) - (Risk in Non-exp) = 0.344 - 0.125 = 0.219

21. 21 We assume that the incidence follows a binomial distribution Can be considered as approximately normal if incidence isn’t too small). 95% CI’s for Risk Diff (1)

22. 22 95% CI’s for Risk Diff (2) YES NO High 42 80 122 Low 43302 345 85 382 467 Death Pollutant level RD = 0.219 42*80 43*302 var(RD) = ------------ + ------------- = 0.00217 122 3 345 3 se(RD) = sqrt(0.00217) = 0.047 Upper 95% CI = 0.219 + 1.96*0.047 = 0.310 Lower 95% CI = 0.219 - 1.96*0.047 = 0.127 Conclusion: RD is: 0.219 (0.127 to 0.310)

23. 23 Comparing groups: Cohorts (10) Which comparative measure do you use? Depends on the circumstances. Risk Ratio  RELATIVE risk measure Risk Difference  ABSOLUTE risk measure Post-menopausal estrogens & endometrial cancer –RR = 2.3 –RD = 2/10,000

24. 24 Comparing groups:Cohorts (11) Disease Person-years YES 1,000 9,500 NO 100 9,950 1,100 19,450 Exp RATE RATIO Rate in exposed: = 1000/9500 Rate in Non-exposed= 100/9950 If exposure increases rate of getting disease, you would expect the rate in exposed to be larger than the rate in unexposed. How much larger can be assessed by the ratio of one to the other: Rate in Exp Rate ratio (RR) = ------------------------ Rate in Non-exp = (1000/9500)/(100/9950) = 10.5

25. 25 Comparing groups: Cohorts (12) DISEASE Person-time YES A Y 1 NO B Y 2 A + B Y 1 + Y 2 Exp RATE RATIO Rate in exposed: = A/Y 1 Rate in Non-exposed= B/Y 2 If exposure increases rate of getting disease, you would expect A/Y 1 to be larger than B/Y 2. How much larger can be assessed by the ratio of one to the other: Rate in Exp Rate ratio (RR) = ------------------------ Rate in Non-exp = (A/Y 1 ))/(B/Y 2 ) A/Y 1 = -------------- B/Y 2

26. 26 Comparing groups:Cohorts (13) Rate in exposed: = 42/101 = 0.416 Rate in Non-exposed= 43/323.5 = 0.133 Rate in Exp Rate ratio (RR) = ------------------------ Rate in Non-exp = 0.416/0.133 = 3.13 Pollutant level Dead Person-years High 42 101 Low 43 323.5 85 424.5

27. 27 Use the same approach to obtain ’95% CI’ for IDR as we used for CIR: BUT: IDR is NOT normally distributed –Range from 0 to +∞ –Null value = 1.0 –Implies a non-symmetric distribution 95% CI’s for IDR (1)

28. 28 Instead, use ‘log(IDR)’ where the log is taken to the ‘natural’ base ‘e –Often written ln(IDR) ln(IDR) is approximately normally distributed –Range from -∞ to +∞ –Null value = 0.0 95% CI is given by: Need to find formula for ‘se(ln(IDR))’ 95% CI’s for IDR (2)

29. 29 if exposed and unexposed are independent 95% CI’s for IDR (3) After some math, this gives the following result (next slide)

30. 30 95% CI’s for IDR (4) DISEASE Person-time YES a Y 1 NO c Y 2 a+c Y 1 + Y 2 Exp DOES NOT DEPEND ON PERSON-TIME!!

31. 31 95% CI’s for IDR (5) We’re close now. Just take the ‘anti-logs’ (usually called the ‘exp’ function

32. 32 Comparing groups:Cohorts (14) Pollutant level Dead Person-years High 42 101 Low 43 323.5 85 424.5 Rate ratio (RR) or IDR = 3.13 1 1 var(ln(IDR)) = ------ + ----- = 0.047 42 43 se(ln(IDR)) = sqrt(0.047) = 0.217 Upper 95% CI = 3.13 * exp(+1.96*0.217) = 4.79 Lower 95% CI = 3.13 * exp( -1.96*0.217) = 2.05 Conclusion: IDR is: 3.13 (2.05 to 4.79)

33. 33 Comparing groups: Cohorts (15) Hypothesis testing (H 0 : IDR=1) –Much less common than 95% CI’s –Normal approximation test is generally OK

34. 34 Comparing groups:Cohorts (16) Disease Person-years YES 1,000 9,500 NO 100 9,950 1,100 19,450 Exp RATE DIFFERENCE Rate in exposed: = 1000/9500 Rate in Non-exposed= 100/9950 If exposure increases rate of getting disease, you would expect the rate in exposed to be larger than the rate in unexposed. How much larger can be assessed by the difference between the two: Rate difference = (Rate in Exp) – (Rate in Non-exp) 1000 100 = --------- - --------- = 0.095 cases/PY 9500 9950

35. 35 Comparing groups:Cohorts (17) DISEASE Person-time YES A Y 1 NO B Y 2 A + B Y 1 + Y 2 Exp RATE DIFFERENCE Rate in exposed: = A/Y 1 Rate in Non-exposed= B/Y 2 If exposure increases rate of getting disease, you would expect A/Y 1 to be larger than B/Y 2. How much larger can be assessed by the difference between the two: Rate difference = (Rate in Exp) – (Rate in Non-exp) A B = ------ - ------- Y 1 Y 2

36. 36 Comparing groups: Cohorts (18) Rate in exposed: = 42/101 = 0.416 Rate in Non-exposed= 43/323.5 = 0.133 Rate difference (RD) = (Rate in Exp) – (Rate in Non-exp) = 0.416 - 0.133 = 0.283 cases/person-year Pollutant level Dead Person-years High 42 101 Low 43 323.5 85 424.5

37. 37 We assume that the incidence follows a Poisson distribution Can be considered as approximately normal if incidence isn’t too small). 95% CI’s for Rate Diff (1)

38. 38 95% CI’s for Rate Diff (2) RD = 0.283 cases/PY 42 43 var(RD) = --------- + ----------- = 0.00453 101 2 323.5 2 se(RD) = sqrt(0.00453) = 0.067 Upper 95% CI = 0.283 + 1.96*0.067 = 0.415 Lower 95% CI = 0.283 - 1.96*0.067 = 0.152 Conclusion: Rate Diff is: 0.283 (0.152 to 0.415) Cases/PY Pollutant level Dead Person-years High 42 101 Low 43 323.5 85 424.5

39. 39 Comparing groups: Cohorts (19) Some Issues What does RR (or RD) mean –Can mean risk or rate ratio. Some people think this is pedantic rather than correct –Need to tell which from context. –Sometimes referred to as Relative Risk (generic term). Are risk differences or ratios preferred? –RR’s are much more common –Both have a role to play.

40. 40 CAN NOT COMPUTE A RISK RATIO! Can not estimate incidence from a case-control study. Can not compute risk differences. Why? We choose the subjects based on their outcome status. Usually, that means making the number of cases and controls equal. Hence, the ‘incidence’ in the case-control study is fixed at 0.50. In real world, it is most likely much lower (1/100,000). Let’s look at an example. Comparing groups:Case-control (1)

41. 41 Comparing groups:Case-control (2) YES NO YES 1,000 9,000 10,000 NO 100 9,900 10,000 1,100 18,900 20,000 Disease Exp RISK RATIO Risk in exposed: = 0.1 Risk in Non-exposed= 0.01 RR = 0.1/0.01 = 10.0 Case Control YES 1,000 524 1,524 NO 100 576 676 1,100 1,100 2,200 Exp ‘RISK RATIO’ ‘Risk’ in exposed: = 0.656 ‘Risk’ in Non-exposed= 0.148 ‘RR’ = 0.656/.148 = 4.44

42. 42 CAN NOT COMPUTE A RISK RATIO! So, what do we do? –Cornfield & Haenzel provided solution in 1960. They looked at the ODDS of exposure. The ratio of the odds of exposure in the cases and controls is almost the same as the RR, if the disease is rare. Comparing groups:Case-control (3)

43. 43 Comparing groups:Case-control (4) YES NO YES 900 400 1,300 NO 100 600 700 1,000 1,000 2,000 Disease Exp ODDS RATIO Odds of exposure in cases = 900/100 Odds of exposure in controls = 400/600 If exposure increases rate of getting disease, you would to find more exposed cases than exposed controls. That is, the odds of exposure for case would be high. How much larger can be assessed by the ratio of one to the other: Exp odds in cases Odds ratio (OR) = ----------------------------- Exp odds in controls = (900/100)/(400/600) = 13.5

44. 44 Comparing groups:Case-control (5) YES NO YES a b a+b NO c d c+d a+c b+d N Disease Exp ODDS RATIO Odds of exposure in cases= a/c Odds of exposure in controls= b/d If exposure increases rate of getting disease, you would to find more exposed cases than exposed controls. That is, the odds of exposure for case would be high (a/c > b/d). How much larger can be assessed by the ratio of one to the other: Exp odds in cases Odds ratio (OR) = ----------------------------- Exp odds in controls = (a/c)/(b/d) ad = ---------- bc

45. 45 Yes No High 42 18 Low 43 67 85 85 Pollutant Level Odds of exp in cases: = 42/43 = 0.977 Odds of exp in controls:= 18/67 = 0.269 Odds ratio (OR) = Odds in cases/odds in controls = 0.977/ 0.269 = (42*67)/(43*18) = 3.64 Comparing groups:Case-control (6) Disease NOTE: Risk ratio = 2.76 Rate ratio = 3.13

46. 46 Use the same approach to obtain ’95% CI’ for OR as we used for CIR/IDR: BUT: OR is NOT normally distributed –Range from 0 to +∞ –Null value = 1.0 –Implies a non-symmetric distribution 95% CI’s for OR (1)

47. 47 Instead, use ‘log(OR)’ where the log is taken to the ‘natural’ base ‘e –Often written ln(OR) ln(OR) is approximately normally distributed –Range from -∞ to +∞ –Null value = 0.0 95% CI is given by: Need to find formula for ‘se(ln(OR))’ 95% CI’s for OR (2)

48. 48 95% CI’s for OR (3) Case Control YES a b NO c d a+c a+d Exp

49. 49 95% CI’s for OR (4) We’re close now. Just take the ‘anti-logs’ (usually called the ‘exp’ function

50. 50 Odds ratio (OR) = 3.63 1 1 1 1 var(ln(OR)) = ----- + ---- + ---- + ---- = 0.118 42 18 43 67 se(ln(OR)) = sqrt(0.118) = 0.343 Upper 95% CI = 3.63 * exp(+1.96*0.343) = 7.11 Lower 95% CI = 3.63 * exp( -1.96*0.343) = 1.85 Conclusion: OR is: 3.63 (1.85 to 7.11) Comparing groups:Case-control (6) Yes No High 42 18 Low 43 67 85 85 Pollutant Level Disease

51. 51 Comparing groups:Case-Control (7) Hypothesis testing (H 0 : OR=1) –Much less common than 95% CI’s –Normal approximation test is generally OK JUST USE THE STANDARD Chi-square TEST!

52. 52 You can compute an OR for a cohort. Why would you do so? –OR’s are the key outcome measure for logistic regression, one of the most common analysis methods used in epidemiology –Unless disease is common, the OR and the RR from the cohort will be very similar. But, where possible, rate ratios are preferred. Comparing groups:Case-control (8)

53. 53 Cohort studies –Relative risk –Relative rate –Risk/rate differences Case-control study –Odds-ratio Summary: comparisons