Lecture 24. Ingo Ruczinski. November 24, Department of Biostatistics Johns Hopkins Bloomberg School of Public Health Johns Hopkins University

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1 Department of Biostatistics Johns Hopkins Bloomberg School of Public Health Johns Hopkins University November 24, 2015

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3 1 Odds ratios for retrospective studies 2 Odds ratios approximating the prospective RR 3

4 Lung cancer Smoker Cases Controls Total Yes No Case status obtained from records Cannot estimate P(Case Smoker) Can estimate P(Smoker Case)

5 Continued Can estimate odds ratio b/c Odds(case smoker) Odds(case smoker c ) = Odds(smoker case) Odds(smoker case c )

6 Proof C - case, S - smoker Odds(case smoker) Odds(case smoker c ) = P(C S)/P( C S) P(C S)/P( C S) = P(C, S)/P( C, S) P(C, S)/P( C, S) = P(C, S)P( C, S) P(C, S)P( C, S) Exchange C and S and the result is obtained

7 Notes Sample OR is n 11n 22 n 12 n 21 Sample OR is unchanged if a row or column is multiplied by a constant Invariant to transposing Is related to RR

8 Notes continued OR = P(S C)/P( S C) P(S C)/P( S C) = P(C S)/P( C S) P(C S)/P( C S) = P(C S) P( C S) P(C S) P( C S) = RR 1 P(C S) 1 P(C S) OR approximate RR if P(C S) and P(C S) are small (or if they are nearly equal)

9 Disease Exposure Yes No Total Yes No Cross-sectional data ˆ P(D) = 10/ ˆ OR = (9 999)/(1 1) = 8991 ˆ RR = (9/10)/(1/1000) = 900 D is rare in the sample D is not rare among the exposed

10 OR = 1 implies no association OR > 1 positive association OR < 1 negative association Notes For retrospective CC studies, OR can be interpreted prospectively For diseases that are rare among the cases and controls, the OR approximates the RR Delta method SE for log OR is 1 n n n n 22

11 1 Example Lung cancer Smoker Cases Controls Total Yes No OR ˆ = = 3.0 SE ˆ 1 log OR ˆ = =.26 CI = log(3.0) ± = [.59, 1.61] The estimated odds of lung cancer for smokers are 3 times that of the odds for non-smokers with an interval of [exp(.59), exp(1.61)] = [1.80, 5.00] 1 Data from Agresti, Categorical Data Analysis, second edition

12 the OR Lung cancer Smoker Cases Controls Total Yes No X the number of smokers for the cases Y the number of smokers for the controls Calculate an exact CI for Have to eliminate a nuisance parameter

13 Notation logit(p) = log{p/(1 p)} is the log-odds Differences in logits are log-odds ratios logit{p(smoker Case)} = δ P(Smoker Case) = e δ /(1 + e δ ) logit{p(smoker Control)} = δ + θ P(Smoker Control) = e δ+θ /(1 + e δ+θ ) θ is the log-odds ratio δ is the nuisance parameter

14 Notation X is binomial with n 1 trials and success probability e δ /(1 + e δ ) Y is binomial with n 2 trials and success probability e δ+θ /(1 + e δ+θ ) P(X = x) = = ( n1 x ( n1 x ) { } e δ x { e δ 1 + e δ ) { } 1 n1 e xδ 1 + e δ } n1 x

15 P(X = x) = P(Y = z x) = ( n2 z x ( n1 x P(X + Y = z) = u ) { } 1 n1 e xδ 1 + e δ ) { e (z x)δ+(z x)θ e δ+θ P(X = u)p(y = z u) } n2 P(X = x X + Y = z) = P(X = x)p(y = z x) P(X = u)p(y = z u) u

16 Non-central hypergeometric distribution P(X = x X + Y = z; θ) = u ( n1 x ( n1 u ) ( n2 z x ) ( n2 z u ) e xθ ) e uθ θ is the log odds ratio This distribution is used to calculate exact hypothesis tests for H 0 : θ = θ 0 Inverting exact tests yields exact confidence intervals for Simplifies to the hypergeometric distribution for θ = 0