Main points to be covered

1. Main points to be covered • Measures of association compare measures of disease occurrence between levels of a predictor variable (eg, exposed/unexposed) • Disease incidence and risk in a cohort study • Absolute risk vs. relative risk • Properties of the 2 X 2 table: Relative risk vs. odds ratio

2. Measuring Association in a Cohort Study • Simplest case is to have a dichotomous exposure • Everyone in cohort is classified as exposed or unexposed • Incidence of the outcome is measured in the two groups of exposed and unexposed • Two incidences are compared

3. Following two groups by exposure status within a cohort: Equivalent to following two cohorts defined by exposure

4. Analysis of Disease Incidence in a Cohort

5. Difference vs. Ratio • Two basic ways to compare two incidence measures: • difference: subtract one from the other • ratio: form a ratio of one over the other • Example: if cumulative incidence is 26% in exposed and 15% in unexposed, • risk difference = 26% - 15% = 11% • risk ratio = relative risk = 26/15 = 1.7

6. Why use difference vs. ratio? • Risk difference gives an absolute measure of the association of exposure on disease occurrence • public health implication is clearer with absolute measure: how much disease might eliminating the exposure prevent? • Risk ratio gives a relative measure • relative measure gives better sense of strength of an association between exposure and disease for etiologic inferences

7. Example of Absolute vs. Relative Measure of Risk • If disease incidence is very low, can have very strong association on relative measure but absolute difference is small • Example: incidence in unexposed is 0.3% and incidence in exposed is 1.5% • relative risk = 1.5/0.3 = 5.0 • risk difference = 1.5 - 0.3 = 1.2%

8. Relative Risk • Relative risk sometimes used to mean either the ratio of two cumulative incidences (incidence proportions) or the ratio of two incidence rates • Text distinguishes risk and rate and so distinguishes relative risk from relative rate

9. Relative Risk vs. Relative Rate • Risk is based on proportion of persons with disease = cumulative incidence • Relative risk = ratio of 2 cumulative incidence estimates • Rate is based on events per person-time = incidence rate • Relative rate = ratio of 2 incidence rates

10. Relative Risk in Cohort • Relative risk = ratio of two cumulative incidences in a cohort • To simplify the presentation, in text ratio of two cumulative incidences assumes no censoring (and no confounding) • Allows presentation of relative risk and odds ratio in the setting of a 2 X 2 table

11. Relative Risk in Cohort • Ratio of two cumulative incidences by Kaplan-Meier method in a cohort with censoring is still a relative risk • Standard error of RR is calculated differently if there is censoring • Significance testing of RR cannot be done with one 2 x 2 table when censoring present • most common statistic for testing difference between two K-M incidences is log rank test (series of 2 x 2 tables weighted by sample size)

12. 2 x 2 table for association of disease and exposure Disease Yes No Yes a + b b a Exposure c + d c d No N = a+b+c+d a + c b + d

13. 2 x 2 table translated into a cohort with no losses to follow-up Disease No disease (a + b) a b Exposed time Disease No disease c d (c + d) Unexposed Relative risk = disease proportion in exposed / disease proportion in unexposed

14. Relative risk of disease in exposed and unexposed Disease Yes No a b a Yes a + b RR = Exposure c d c c + d No

15. Odds versus Probability • Odds based on probability; expresses probability (p) as ratio: odds = p / (1 - p) • odds is always > p because divided by < 1 • For example, if probability of dying = 1/5, then odds of dying = 1/5 / 4/5 = 1/4 • Thinking of odds as 2 outcomes, the numerator is the # of times of one outcome and the denominator the # of times of the other • P = odds / (1 + odds), so 1/4 / 1 + 1/4 = 1/5

16. Odds versus Probability • Less intuitive than probability (probably wouldn’t say “my odds of dying are 1/4”) • No less legitimate mathematically, just not so easily understood • Used in epidemiology primarily because the log of the ratio of two odds is given by the coefficients in logistic regression equations

17. Odds ratio of disease in a cohort • Since odds = p / 1- p, odds of disease in exposed = cumulative incidence in exposed / 1 - cumulative incidence in exposed • And odds in unexposed = cumulative incidence in unexposed / 1 - cumulative incidence in unexposed • Ratio of two odds is the odds ratio (OR)

18. Odds ratio of disease in exposed and unexposed Disease a Yes No a + b a b a Yes 1 - a + b OR = Exposure c d c c + d No c 1 - c + d

19. Odds ratio of disease in exposed and unexposed a a + b b a + b c c + d d c + d a a b c d a + b a 1 - a + b ad bc = OR = = = c c + d c 1 - c + d ad bc is called the cross-productof a 2 x 2 table Better to calculate two odds than cross-product

20. Odds ratio of exposure in diseased and not diseased Disease a Yes No a + c a b a Yes 1 - a + c OR = Exposure b d c b + d No b 1 - b + d

21. Important characteristic of odds ratio a a + c c a + c b b + d d b + d a a c b d a + c a 1 - a + c ad bc = = = ORexp = b b + d b 1 - b + d OR for disease = OR for exposure

22. Relative risk and Odds ratio If incidence in exposed and unexposed is the same, RR = 1 and OR = 1 Odds is always > probability because odds is p divided by (1 - p) = < 1 If RR = 1, OR will be farther from 1 than RR For example: RR=0.4/0.2=2 then OR=0.67/0.25=2.7 and RR=0.2/0.3=0.7 then OR=0.25/0.43=0.6

23. Relative risk and Odds ratio If risk of disease is low in both exposed and unexposed, RR and OR approximately = Text example: incidence of MI risk in high bp group is 0.018 and in low bp group is 0.003: RR = 0.018/0.003 = 6.0 OR = 0.01833/0.00301 = 6.09

24. Relative risk and Odds ratio If risk of disease is high in either or both exposed and unexposed, RR and OR differ Example, if risk in exposed is 0.6 and 0.1 in unexposed: RR = 0.6/0.1 = 6.0 OR = 0.6/0.4 / 0.1/0.9 = 13.5 OR approximates RR only if incidence is low in both exposed and unexposed group

25. “Bias” in OR as estimate of RR • Text refers to “bias” in OR as estimate of RR (OR = RR x (1-incid.unexp)/(1-incid.exp)) • not “bias” in usual sense because both OR and RR are mathematically valid and use the same numbers • Simply that OR cannot be thought of as a surrogate for the RR unless incidence is low

26. Symmetry of OR versus non-symmetry of RR OR of non-event is 1/OR of event RR of non-event = 1/RR of event Example: If cum. inc. in exp. = 0.2529 and cum. inc. in unexp. = 0.0705, then RR (event)= 0.2529 / 0.0705 = 3.59 RR(non-event)= 0.0705 / 0.2529 = 0.8 Not reciprocal: 1/3.59 = 0.279 = 0.8

27. Symmetry of OR versus non-symmetry of RR Example continued: OR(event)= 0.2529/1- 0.2529 / 0.0705/ 1- 0.0705 = 4.46 OR(non-event)= 0.0705/1- 0.0705 / 0.2529/1- 0.2529 = 0.22 Reciprocal: 1/4.46 = 0.22

28. Confidence interval for RR Calculated on log scale (See Appendix A.3) Text example: RR=6.0 and log 6.0= 1.792 SE(log RR) =  b/a(a+b) + d/c(c+d) = 0.197 95% CI (RR) = exp {log RR + [1.96 x  b/a(a+b) + d/c(c+d)]} 1.96 x 0.197 = 0.386 95% CI = exp [1.792 + 0.386] = exp(1.406) and exp(2.178) = 95% CI = 4.08 to 8.83

29. RRHypothesis Testing H0: RR = 1 Equivalent to testing that proportion with outcome in exposed equals proportion with outcome in unexposed Statistic: Chi-square or Fisher’s exact

30. Prevalence Ratios • Text refers to Point Prevalence Rate Ratio, but avoiding rate with prevalence, just use prevalence ratio (PR) • Analogous to incidence ratio: prevalence in exposed (+) divided by prevalence in unexposed (-) • Analogous ratio exists for odds ratio called prevalence odds ratio • prevalenceexp/1-prevalenceexp / prevalenceunexp/1- prevalenceunexp

31. Summary points • Cohort with no loss to follow-up can be displayed as a 2 x 2 table • Risk difference gives absolute difference; risk ratio gives relative difference • Both RR and OR calculated from 2 x 2 • OR farther from 1.0 than RR • OR approximates RR if incidence low in both exposed and unexposed