# Analysis of Case Control Studies - University of Calgary in ... Analysis of Case Control Studies E exposure to asbestos D disease: bladder cancer S strata: smoking status 2X2 Table p = Pr(Exposure) p may depend on D and/or S log( p /(1 p )) 0 1 D for D 0;

0 for D 1; 0 1 so 1 log( p1 /(1 p1 )) log( p0 /(1 p0 )) 1 is the difference between 2 log odds A difference between 2 log odds The log of the odds of exposure for those

with disease minus the log of the odds of exposure for those without disease Now. Brace yourself The exponent of a difference is a ratio! YAY! So what you say? Lets take the exponent of: 1 log( p1 /(1 p1 )) log( p0 /(1 p0 )) e 1 exp(1 ) exp(log( p1 /(1 p1 )) log( p0 /(1 p0 ))) p1 /(1 p1 )

p0 /(1 p0 ) The odds ratio: OR Remember: exp and log are inverses of one another Exp and Log Exp(Log(A)) = A = Log(Exp(A)) Exp(A-B) = Exp(A)Exp(-B) =Exp(A)/Exp(B) So the exponent of a difference is a ratio of exponents

Log(A/B) = Log(A) Log(B) So the logarithm of a ratio is a difference between logarithms Stratified analysis via logistic regression Lets try: log( p /(1 p)) 0 1 D 2 S 3 DS if S 0; 0 1 D if S 1;

( 0 2 ) ( 1 3 ) D so then : e 1 exp(1 ) is the OR for strata 0 (call it OR 0 ) and e 1 3 exp( 1 3 ) is the OR for strata 1 (call it OR 1 ) exp( 3 ) is the ratio of the 2 ORs (OR 1 / OR 0 ) 3 is the difference between 2 log odds ratios (log(OR1 ) - log(OR 0 )) 3 can also be described as a difference between 2 differences (write out what log(OR) is!) Effect modification

Test: 3 0 This is the same null hypothesis as the test for homogeneity in a classical analysis. Evidence against this null hypothesis indicates that there is evidence that the stratum specific odds ratios are different If there is no evidence against. assess confounding just like linear regression

log( p /(1 p )) 0 1adj D 2 S is compared with log( p /(1 p )) 0 1cr D exp(1adj ) OR adj is compared with OR cr exp(1cr ) Since ORs are ratios, ratios of ORs are usually considered (as opposed to differences)