Fundamentals of epidemiology - an evolving text - Are you looking ...

Moreover, the logistic model, we will see, corresponds to a multiplicative model, which we saw earlier is
the model that is implied by stratified analysis based on the OR or the risk ratio. Furthermore, the
coefficients that we estimate using logistic regression can be converted into OR's, so that we now have
a ratio measure of association.
It is easy to discover what the logistic coefficients are. Since the logit is the logarithm of the odds, then
the difference of two logits is the logarithm of an OR (because subtraction of logs corresponds to
division of their arguments – see the appendix to the chapter on Measures of Frequency and Extent).
Suppose that X3 is a dichotomous (0-1) variable indicating absence (0) or presence (1) of an exposure.
First write the model with the exposure "present" (X3=1), and underneath write the model with the
exposure "absent" (X3=0).
logit(D=1|X1,X2,X3=1) = α + β1X1 + β2X2 + β3 (X3 =1, present)
– logit(D=1|X1,X2,X3=0) = α + β1X1 + β2X2 + 0 (X3 = 0, absent)
_______________________________________________________
When we subtract the second model from the first, all the terms on the right are removed except the
coefficient for X3. On the left, we have the (rather messy) difference of the two logits, one for X3
present and the other for X3 absent:
logit(D=1|X1,X2,X3=1) – logit(D=1|X1,X2,X3=0) = β3
Spelling out the logits:
ln(odds(D=1|X1,X2,X3=1)) – ln(odds(D=1|X1,X2,X3=0)) = β3
and, since a difference of logarithms is the logarithm of a ratio:
odds(D=1|X1,X2,X3=1)
ln [ ———————————— ] = β3
odds(D=1|X1,X2,X3=0)
A ratio of odds is simply an OR, in this case, the OR for the disease with respect to the exposure
represented by X3:
ln [ OR ] = β3
exp (ln [ OR ] ) = exp(β3)
________________________________________________________________________________________________
www.sph.unc.edu/courses/EPID 168, © Victor J. Schoenbach 13. Multicausality ― analysis approaches ― 446
rev. 10/28/1999, 11/16/2000, 4/2/2001