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Measures of Disease Association • Measuring occurrence of new outcome events can be an aim by itself, but usually we want to look at the relationship between an exposure (risk factor, predictor) and the outcome • The type of measure showing an association between an exposure and an outcome event is linked to the study design Main points to be covered • Measures of association compare measures of disease between levels of a predictor variable • Prevalence ratio versus risk ratio • Probability and odds • The 2 X 2 table • Properties of the odds ratio • Absolute risk versus relative risk • Disease incidence and risk in a cohort study Cross-Sectional Study Design: A Prevalent Sample Measures of Association in a Cross-Sectional Study • Simplest case is to have a dichotomous outcome and dichotomous exposure variable • Everyone in the sample is classified as diseased or not and having the exposure or not, making a 2 x 2 table • The proportions with disease are compared among those with and without the exposure • NB: Exposure=risk factor=predictor 2 x 2 table for association of disease and exposure Disease Yes No Yes a b a+b No c d c+d a+c b+d N = a+b+c+d Note: data may not always come to you arranged as above. STATA puts exposure across the top, disease on the side. Prevalence ratio of disease in exposed and unexposed Disease Yes No Yes a b a a+b PR = c No c d c+d Prevalence Ratio • Text refers to Point Prevalence Rate Ratio in setting of cross-sectional studies • We like to keep the concepts of rate and prevalence separate, and so prefer to use prevalence ratio Prevalence ratio (STATA output) Exposed Unexposed | Total --------------------------------------------------- Cases | 14 388 | 402 Noncases | 17 248 | 265 --------------------------------------------------- Total | 31 636 | 667 | | Risk | .4516129 .6100629 | .6026987 Point estimate [95% Conf. Interval] --------------------------------------------- Risk ratio .7402727 | .4997794 1.096491 ----------------------------------------------- chi2(1) = 3.10 Pr>chi2 = 0.0783 STATA calls it a risk ratio by default Prevalence ratio of disease in exposed and unexposed Disease Yes No Yes a b a a+b PR = c No c d c+d So a/a+b and c/c+d = probabilities of disease and PR is ratio of two probabilities Probability and Odds • Odds another way to express probability of an event • Odds = # events # non-events • Probability = # events # events + # non-events = # events # subjects Probability and Odds • Probability = # events # subjects • Odds = # events # subjects = probability # non-events (1 – probability) # subjects • Odds = p / (1 - p) [ratio of two probabilities] Probability and Odds • If event occurs 1 of 5 times, probability = 0.2. • Out of the 5 times, 1 time will be the event and 4 times will be the non-event, odds = 0.25 • To calculate probability given the odds: probability = odds / 1+ odds 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 because the measure of association available in case-control design is the odds ratio • Also important because the log odds of the outcome is given by the coefficient of a predictor in a logistic regression Odds ratio • As odds are just an alternative way of expressing the probability of an outcome, odds ratio (OR), is an alternative to the ratio of two probabilities (prevalence or risk ratios) • Odds ratio = ratio of two odds Probability and odds in a 2 x 2 table Disease Yes No What is p of disease in exposed? Yes 2 3 5 What are odds of disease in exposed? No 1 4 5 And the same for the un-exposed? 3 7 10 Probability and odds ratios in a 2 x 2 table Disease PR = 2/5 1/5 Yes No =2 Yes 2 3 5 0R = 2/3 1/4 = 2.67 No 1 4 5 3 7 10 Odds ratio of disease in exposed and unexposed Disease Yes No a a+b Yes a b a 1- a+b OR = c No c d c+d c 1- c+d Formula of p / 1-p in exposed / p / 1-p in unexposed Odds ratio of disease in exposed and unexposed a a a+b a+b a b a 1- a+b a+b b ad OR = c = = = c c bc c+d c c+d d 1- c+d d c+d Important Property of Odds Ratio #1 • The odds ratio of disease in the exposed and unexposed equals the odds ratio of exposure in the diseased and the not diseased – Important in case-control design Odds ratio of exposure in diseased and not diseased Disease Yes No a a+c Yes a b a 1- a+c OR = b No c d b+d b 1- b+d Important characteristic of odds ratio a a a+c a+c a c a 1- a+c a+c c ad ORexp = = = = b b b bc b+d b+d b d 1- b+d d b+d OR for disease = OR for exposure Measures of Association Using Disease Incidence • With cross-sectional data we can calculate a ratio of the probability or of the odds of prevalent disease in two groups, but we cannot measure incidence • A cohort study allows us to calculate the incidence of disease in two groups Measuring Association in a Cohort Following two groups by exposure status within a cohort: Equivalent to following two cohorts defined by exposure Analysis of Disease Incidence in a Cohort • Measure occurrence of new disease separately in a sub-cohort of exposed and a sub-cohort of unexposed individuals • Compare incidence in each sub-cohort • How compare incidence in the sub-cohorts? Relative Risk vs. Relative Rate • Risk is based on proportion of persons with disease = cumulative incidence • Risk ratio = ratio of 2 cumulative incidence estimates = relative risk • Rate is based on events per person-time = incidence rate • Rate ratio = ratio of 2 incidence rates = relative rate • We prefer risk ratio, rate ratio, odds ratio A Note on RR or “Relative Risk” • Relative risk or RR is very common in the literature, but may represent a risk ratio, a rate ratio, a prevalence ratio, or even an odds ratio • We will try to be explicit about the measure and distinguish the different types of ratios • There can be substantial difference in the association of a risk factor with prevalent versus incident disease Difference vs. Ratio Measures • Two basic ways to compare measures: – difference: subtract one from the other – ratio: form a ratio of one over the other • Can take the difference of either an incidence or a prevalence measure but rare with prevalence • Example using incidence: cumulative incidence 26% in exposed and 15% in unexposed, – risk difference = 26% - 15% = 11% – risk ratio = 0.26 / 0.15 = 1.7 Summary of Measures of Association Ratio Difference Cross-sectional prevalence ratio prevalence difference odds ratio odds difference Cohort risk ratio risk difference rate ratio rate difference odds ratio odds difference Why use difference vs. ratio? • Risk difference gives an absolute measure of the association between exposure and 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 inferences about causes of disease Relative Measures and Strength of Association with a Risk Factor • In practice many risk factors have a relative measure (prevalence, risk, rate, or odds ratio) in the range of 2 to 5 • Some very strong risk factors may have a relative measure in the range of 10 or more – Asbestos and lung cancer • Relative measures < 2.0 may still be valid but are more likely to be the result of bias – Second-hand smoke relative risk < 1.5 Example of Absolute vs. Relative Measure of Risk TB No TB Total recurrence recurrence Treated > 6 mos 14 986 1000 Treated < 3 mos 40 960 1000 Risk ratio = 0.04/0.014 = 2.9 Risk difference = 0.04 – 0.014 = 2.6% If incidence is very low, relative measure can be large but difference measure small Reciprocal of Absolute Difference ( 1/difference) • Number needed to treat to prevent one case of disease • Number needed to treat to harm one person • Number needed to protect from exposure to prevent one case of disease • TB rifampin example: 1/0.026 = 38.5, means that you have to treat 38.5 persons for 6 mos vs. 3 mos. to prevent one case of TB recurrence Example of study reporting risk difference Table 2. Survival and Functional Outcomes from the Two Study Phases Return of Risk Spontaneous Difference Study Phase Circulation p-value (95% CI) Rapid Defibrillation -- 12.9% -- (N=1391) Advanced Life Support 18.0% 5.1% (3.0-7.2) <0.001 (N=4247) Risk difference = 0.051; number needed to treat = 1/0.051 = 20 Stiel et al., NEJM, 2004 Risk Ratio Diarrheal Disease Yes No Total Ate potato salad 54 16 70 Did not eat potato 2 26 28 salad Total 56 42 98 Probability of disease, ate salad = 54/70 = 0.77 Probability of disease, no salad = 2/28 = 0.07 Risk ratio = 0.77/0.07 = 11 Illustrates risk ratio in cohort with complete follow-up Risk Ratio in a Cohort with Censoring Choose a time point for comparing two cumulative incidences: At 6 years, % dead in low CD4 group = 0.70 and in high CD4 group = 0.26. Risk ratio at 6 years = 0.70/0.26 = 2.69 Comparing two K-M Curves Risk ratio would be different for different follow-up times. Entire curves are compared using log rank test (or other similar tests). OR compared to Risk Ratio If Risk Ratio = 1.0, OR = 1.0; otherwise OR farther from 1.0 0 1 ∞ Stronger effect Stronger effect OR Risk Ratio Risk Ratio OR Risk ratio and Odds ratio If Risk Ratio > 1, then OR farther from 1 than Risk Ratio: RR = 0.4 = 2 0.2 OR = 0.4 0.6 = 0.67 = 2.7 0.2 0.25 0.8 Risk ratio and Odds ratio If Risk Ratio < 1, then OR farther from 1 than RR: RR = 0.2 = 0.67 0.3 OR = 0.2 0.8 = 0.25 = 0.58 0.3 0.43 0.7 Odds ratio (STATA output) Exposed Unexposed | Total --------------------------------------------------- Cases | 14 388 | 402 Noncases | 17 248 | 265 --------------------------------------------------- Total | 31 636 | 667 | | Risk | .4516129 .6100629 | .6026987 Point estimate [95% Conf. Interval] --------------------------------------------- Risk ratio .7402727 | .4997794 1.096491 Odds ratio .5263796 | .2583209 1.072801 ----------------------------------------------- chi2(1) = 3.10 Pr>chi2 = 0.0783 Important property of odds ratio #2 • OR approximates Risk Ratio only if disease incidence is low in both the exposed and the unexposed group Risk ratio and Odds ratio If risk of disease is low in both exposed and unexposed, RR and OR approximately equal. Text example: incidence of MI risk in high bp group is 0.018 and in low bp group is 0.003: Risk Ratio = 0.018/0.003 = 6.0 OR = 0.01833/0.00301 = 6.09 Risk ratio and Odds ratio If risk of disease is high in either or both exposed and unexposed, Risk Ratio 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 Risk Ratio only if incidence is low in both exposed and unexposed group “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 Important property of odds ratio #3 • Unlike Risk Ratio, OR is symmetrical: OR of event = 1 / OR of non-event Symmetry of odds ratio versus non-symmetry of risk ratio OR of non-event is 1/OR of event RR of non-event = 1/RR of event Example: If cum. inc. in exp. = 0.25 and cum. inc. in unexp. = 0.07, then RR (event) = 0.25 / 0.07 = 3.6 RR (non-event) = 0.75 / 0.93 = 0.8 Not reciprocal: 1/3.6 = 0.28 = 0.8 Symmetry of OR Example continued: OR(event) = 0.25 (1- 0.25) = 4.43 0.07 (1- 0.07) OR(non-event) = 0.07 (1- 0.07) = 0.23 0.25 (1- 0.25) Reciprocal: 1/4.43 = 0.23 Important property of odds ratio #4 • Coefficient of a predictor variable in logistic regression is the log odds of the outcome (e to the power of the coefficient = OR) – Logistic regression is the method of multivariable analysis used most often in cross-sectional and case-control studies 3 Useful Properties of Odds Ratios • Odds ratio of disease equals odds ratio of exposure – Important in case-control studies • Odds ratio of non-event is the reciprocal of the odds ratio of the event (symmetrical) • Regression coefficient in logistic regression equals the log of the odds ratio Summary points • Cross-sectional study gives a prevalence ratio • Risk ratio should refer to incident disease • Relative ratios show strength of association • Risk difference gives absolute difference indicating number to treat/prevent exposure • Properties of the OR important in case-control studies – OR for disease = OR for exposure – Logistic regression coefficient gives OR