The odds ratio that controls for all four poten-
tial confounders plus retained interaction
terms is given by the expression shown here.
This expression gives a formula for calculating
numerical values for the odds ratio. This for-
mula contains the coefficientsb^;^d 1 ;^d 2 , and^d 4 ,
but also requires specification of three effect
modifiers – namely,V 1 ,V 2 , andV 4 , which are in
the model as product terms withE.The numerical value computed for the odds
ratio will differ depending on the values speci-
fied for the effect modifiersV 1 ,V 2 , andV 4. This
should not be surprising because the presence
of interaction terms in the model means that
the value of the odds ratio differs for different
values of the effect modifiers.The above odds ratio is thegold standardodds
ratio expression for our example. This odds
ratio controls for all potential confounders
being considered, and it provides baseline
odds ratio values to which all other odds ratio
computations obtained from dropping candi-
date confounders can be compared.The odds ratio that controls for previously
retained variables but excludes the control of
V 3 is given by the expression shown here. Note
that this expression is essentially of the same
form as the gold standard odds ratio. In partic-
ular, both expressions involve the coefficient of
the exposure variable and the same set of effect
modifiers.However, the estimated coefficients for this
odds ratio are denoted with an asterisk (*)to
indicate that these estimates may differ from
the corresponding estimates for the gold stan-
dard. This is because the model that excludes
V 3 contains a different set of variables and,
consequently, may result in different estimated
coefficients for those variables in common to
both models.In other words, because the gold standard
model containsV 3 , whereas the model for the
asterisked odds ratio does not containV 3 ,itis
possible thatb^will differ from^b*, and that the^d
will differ from the^d*.EXAMPLE (continued)
dORV 1 ;V 2 ;V 3 ;V 4 ¼exp^bþ^d 1 V 1 þ^d 2 V 2 þ^d 4 V 4 ;
where^d 1 ;^d 2 , and^d 4 are coefficients of
EV 1 ,EV 2 , andEV 4 ¼EV 1 V 2
dOR differs for different specifications
ofV 1 ,V 2 ,V 4
Gold standardOR:d
Controls for all potential
confounders
Gives baselineORd
dOR¼expð^bþ^d 1 V 1 þ^d 2 V 2 þ^d* 4 V 4 Þ;
where^b;^d 1 ;^d 2 ;^d 4 are coefficients in
model withoutV 3
Model withoutV 3 :
E;V 1 ;V 2 ;V 4 ;EV 1 ;EV 2 ;EV 4Model withV 3 :
E, V 1 , V 2 , V 3 , V 4 , EV 1 , EV 2 , EV 4
Possible that
^b 6 ¼^b;^d 16 ¼^d 1 ;^d 26 ¼^d 2 ;^d 46 ¼^d 4
Presentation: IV. Confounding Assessment with Interaction 219