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2by2 table

# 2by2 table - Data from the Framingham Heart Study Systolic...

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Data from the Framingham Heart Study: Systolic Blood Pressure (mmHg) and 18 year CHD mortality among 45-55 year-old males Association?? 23/61 = 0.38 71/349=0.20 94/410=0.23 0.38>0.20 Proportion developing CHD depends on SBP 23/94 = 0.24 38/316=0.12 61/410=0.15 0.24>0.12 Proportion with SBP≥165 at baseline depends on whether they developed CHD Altered data 14/61 = 0.23 80/349 = 0.23 94/410= 0.23 0.23=0.23 Proportion developing CHD does not depend on SBP 14/94 = 0.15 47/316=0.15 61/410=0.15 0.15=0.15 Proportion with SBP≥165 at baseline does not depend on whether they develop CHD Difference in proportion (Risk Difference) 23/61 – 71/349 = 0.38-0.20 = 0.18 ( 0 if independent) Ratio of proportion (risk ratio or relative risk) (23/61)/(71/349) = 0.8/0.20 = 1.85 (1 if independent) Ratio of odds (Odds Ratio) (23/38) / (71/278) = (23*278) / (71*38) = 2.37 ( 1 if independent) Definitions Two binomial probabilities and p 1 Risk Difference : RD = p 1 - p 0 Risk of Ratio or Relative Risk RR = p 1 /p 0 Odds Ratio : OR = p 1 (1-p 1 )/p 0 (1-p 0 ) We estimate these with sample proportions RD, RR and OR are different ways of measuring association If we divide both proportions by 10, RD is also divided by 10 but RR remains the same (See row 1 and row 4) What to pick is decided by purpose of the disease If disease is common (p=0.20) and risk is double with the exposure, it is important public health problem If life time cumulative incidence is small, but risk is double it would not be public health problem If we want to understand the etiology or biochemical pathway, then double the risk of disease might be important whether the disease is common or not. With different biominal probability can give same RR so, as risk difference changes, RR may not. (row 1 vs row4 vs row8) OR is similar to RR when the prevalence of disease is rare. (row 6 vs row 8) When every one of exposed get disease, OR = (1/1-1)/(.2/.8)= infinity Similarly if none of the unexposed get disease, OR is infinity

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Constant Risk Difference Constant Risk Ratio Constant Odds Ratio Any measure of association in 2*2 tables collapses information rom two numbers (p 0 and p 1 ) into one number (RD, RR, OR). Some information is lost. All pairs p 0 -p 1 consistent with no association are conssisten with no association for all three measures : RD=0, RR=1, OR=1 Null hypothesis H 0 : p 1 =p 0 , H 0 : RD=0, H 0 :RR=1, H 0 : 0R=1 are all equivalent Non null values of each association measure do not identify the same pairs of p1 and p0. The collection of p 1 and p 0 for which OR= 2 do not all yield the same value of RD or of RR Choice of association measure makes a statement about what types of difference between p 1 and p 0 are viewed as equivalent Data resulting in a 2*2 table can come from several sampling schemes: o Cross sectional: Ex: HPV DNA and age, country o Observational Cohort Study; Ex: Framingham Heart Study o Observational case control study: Ex: Contraceptive and Reproductive experiences (CARE) study o Randomized Clinical Trail : Ex; APPROVe trail Interpretation RD, RR or OR?
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