Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

We refer to this estimate as thegold standard

estimateof effect because we consider it the

best estimate we can obtain, which controls

forallthe potential confounders, namely, the

fiveVs, in our model.

We can nevertheless obtain other estimated

odds ratios by dropping some of theVs from

the model. For example, we can dropV 3 ,V 4 ,

andV 5 from the model and then fit a model

containingE,V 1 , andV 2. The estimated odds

ratio for this “reduced” model is also given by

the expression e to theb^, where^bis the coeffi-

cient ofEin the reduced model. This estimate

controls for onlyV 1 andV 2 rather than all five

Vs.

Because the reduced model is different from

the gold standard model, the estimated odds

ratio obtained for the reduced model may be

meaningfully different from the gold standard.

If so, then we say that the reduced model does

not control for confounding because it does not

give us the correct answer (i.e., gold standard).

For example, suppose that the gold standard

odds ratio controlling for all five Vs is 2.5,

whereas the odds ratio obtained when

controlling for onlyV 1 andV 2 is 5.2. Then,

because these are meaningfully different odds

ratios,wecannotusethereducedmodelcontain-

ingV 1 andV 2 because the reduced model does

not properly control for confounding.

Now although use of onlyV 1 andV 2 may not

control for confounding, it is possible that

some other subset of theVs may control for

confounding by giving essentially the same

estimated odds ratio as the gold standard.

For example, perhaps when controlling forV 3

alone, the estimated odds ratio is 2.7 and when

controlling forV 4 andV 5 , the estimated odds

ratio is 2.3. The use of either of these subsets

controls for confounding because they give

essentially the same answer as the 2.5 obtained

for the gold standard.

EXAMPLE (continued)

Gold standard estimate:

Controls for all potential

confounders (i.e., all fiveVs)

Other OR estimates:

Drop someVs

e.g., dropV 3 ,V 4 ,V 5

Reduced model:

logit PðXÞ¼aþbEþg 1 V 1 þg 2 V 2

dOR¼eb^

controls forV 1 andV 2 only

Reduced model 6 ¼gold standard

model

correct answer

dORðreducedÞ¼?dORðgold standardÞ

If different, then reduced modeldoes

notcontrol for confounding

Suppose:

Gold standard (all five Vs)

meaningfully

different

does not control

for confounding

reduced model (V 1 and V 2 )

OR = 2.5

OR = 5.2

dOR some other

subset ofVs

!

¼?dOR

gold

standard

!

If equal, then subset controls

confounding

dORðV 3 aloneÞ¼ 2 : 7

dORðV 4 andV 5 Þ¼ 2 : 3

dORðgold standardÞ¼ 2 : 5

All three estimates are “essentially”

the same as the gold standard

212 7. Modeling Strategy for Assessing Interaction and Confounding