Main points to be covered

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

Cum. Inc. Example of recent paper using both cumulative incidence and rates to analyze longitudinal data. McDonald et al., NEJM 2004 Inc. rates

Cross-Sectional Study Design

Measuring Association in a Cross-Sectional Study • Simplest case is to have a dichotomous outcome and dichotomous predictor variable • Everyone in the sample is classified as diseased or not and having the risk factor or not, making a 2 x 2 table • The proportions with disease are compared among those with and without the risk factor

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 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 a a Yes b a + b PR = Exposure c c d c + d No

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

A Note on RR • RR 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 types of ratios • There can be substantial difference in the association of a risk factor with prevalent versus incident disease

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 a a Yes b a + b PR = Exposure c c d c + d No So a/a+b and c/c+d = probabilities of disease and PR is ratio of two probabilities

Probability and Odds • Odds just another way to express the probability of an event • If an event occurs 1 out of 5 times (1/5 = probability), then out of the 5 times 1 time will be the event and 4 times will be the non-event (1/4 = odds) • More formally, Odds = p / (1 - p) and Probability = odds / (1 + odds)

Odds • Most familiar use is in gambling because the ratio of the event to the non-event tells you what the pay out has to be for a fair bet • If you wager 1 dollar 5 times on an event with a 1/5 probability, you need to be paid 4 dollars the one time the event occurs to balance out the 4 dollars you lose the other 4 times • The odds are ¼, usually expressed as a ratio

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? What are odds of disease in exposed? And the same for the un-exposed? 2 Yes 3 5 Exposure 1 4 5 No 7 10 3

Probability and odds ratios in a 2 x 2 table Disease Yes No PR = 2/5 1/5 = 2 2 3 Yes 5 0R = 2/3 1/4 Exposure = 2.67 1 4 5 No 7 10 3

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 Formula of p / 1-p in exposed / p / 1-p in unexposed

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

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 a Yes No a + c a b a Yes 1 - a + c OR = Exposure b d c b + d No b 1 - b + d

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

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

Difference vs. Ratio Measures • 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 = 0.26 / 0.15 = 1.7

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

Relative Measures and Strength of Association with a Risk Factor • In practice many risk factors have a relative measure (prevalence, risk, 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 • If disease incidence is very low, can have very strong association on relative measure but absolute difference is small • Example: TB recurrence is 1.4% in patients with > 6 months rifampin treatment and 4% in patients with < 3 months rifampin. • Risk ratio = 0.04 / 0.014 = 2.9 • Risk difference = 0.04 - 0.014 = 2.6%

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 reporting incidence difference Stiel et al., NEJM, 2004

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

Kaplan-Meier and Risk Ratio • Most risk ratios in the medical literature are the ratio of two cumulative incidences by Kaplan-Meier method (or life table) in a cohort with censoring • A risk ratio is the cumulative incidence in exposed over incidence in unexposed with or without censoring • Standard error for statistical test is calculated differently if there is censoring

Kaplan-Meier and Risk Ratio in Cohort • A single 2 x 2 table of those left at the end of follow-up would leave out those censored and would not be valid • Every time an event occurs, can form a 2 x 2 table of proportion with event in exposed and in unexposed • log rank test is the usual statistic in K-Msurvival analysis (tests association in series of 2 x 2 tables weighted by sample size)

Example of KM estimates of risk ratio Fawzi et al., NEJM, 2004

Risk ratio and Odds ratio Odds is always > probability because odds is p divided by (1 - p) = < 1 But ratios can be the same if incidence in exposed and unexposed is the same: RR = 1 and OR = 1 If RR = 1, OR will be farther from 1 than RR

Risk ratio and Odds ratio If RR > 1, then OR farther from 1 than RR: 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 RR < 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

Important property of odds ratio #2 • OR approximates RR 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: RR = 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, 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

“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 RR, 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) = 4.4 0.07 (1- 0.07) OR(non-event)= 0.07 (1- 0.07) = 0.23 0.25 (1- 0.25) Reciprocal: 1/4.46 = 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)

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

Main points to be covered