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

Interaction:dOR¼exp

^

bþ~^djWj



^band^djnonzero

no interaction:dOR¼expðb^Þ

Coefficients change when potential
confounders dropped:

 Meaningful change?

 Subjective?

If the model contains interaction terms, the

first step is difficult in practice. The odds

ratio expression, as shown here, involves two

or more coefficients, including one or more

nonzero^d. In contrast, when there is no inter-

action, the odds ratio involves the single co-

efficient^b.

It is likely that at least one or more of the^band

^d coefficients will change somewhat when

potential confounders are dropped from the

model. To evaluate how much of a change is a

meaningfulchange when considering the col-

lection of coefficients in the odds ratio formula

is quitesubjective. This will be illustrated by the

example.

As an example, suppose our initial model con-

tains E, four Vs, namely, V 1 , V 2 , V 3 , and

V 4 ¼V 1 V 2 , and fourEVs, namely,EV 1 ,EV 2 ,

EV 3 , andEV 4. Note thatEV 4 alternatively can

be considered as a three-factor product term as

it is of the formEV 1 V 2.

Suppose also that becauseEV 4 is a three-factor

product term, it is tested first, after all the other

variables are forced into the model. Further,

suppose that this test is significant, so that

the termEV 4 is to be retained in all further

models considered.

Because of the hierarchy principle, then, we

must retainEV 1 andEV 2 in all further models

as these two terms are components ofEV 1 V 2.

This leavesEV 3 as the only remaining two-fac-

tor interaction candidate to be dropped if not

significant.

To test forEV 3 , we can do either a likelihood

ratio test or a Wald test for the addition ofEV 3

to a model afterE, V 1 ,V 2 ,V 3 ,V 4 ¼V 1 V 2 ,EV 1 ,

EV 2 , andEV 4 are forced into the model.

Note that all four potential confounders –V 1

throughV 4 – are forced into the model here

because we are at the interaction stage so far,

and we have not yet addressed confounding in

this example.

EXAMPLE

Variables in initial model:

E;V 1 ;V 2 ;V 3 ;V 4 ¼V 1 V 2

EV 1 ;EV 2 ;EV 3 ;EV 4 ¼EV 1 V 2

SupposeEV 4 (¼EV 1 V 2 )

significant

Hierarchy principle:

EV 1 andEV 2 retained in all further

models

EV 3 candidate to be dropped

Test forEV 3 (LR or Wald test)

V 1 ,V 2 ,V 3 ,V 4 (allpotential

confounders) forced into model

during interaction stage

Presentation: IV. Confounding Assessment with Interaction 217