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

Moreover, because the odds ratio expression

involves the exponential of a linear function

of the four coefficients, these coefficients are

on a log odds ratio scale rather than an odds

ratio scale. Using a log scale to judge the mean-

ingfulness of a change is not as clinically rele-

vant as using the odds ratio scale.

For example, a change in ^b from12.69

to12.72 and a change in^d 1 from 0.0692 to

0.0696 are not easy to interpret as clinically

meaningful because these values are on a log

odds ratio scale.

A more interpretable approach, therefore, is to

view such changes on the odds ratio scale. This

involves calculating numerical values for the

odds ratio by substituting into the odds ratio

expression different choices of the values for

the effect modifiersWj.

Thus, to calculate an odds ratio value from the

gold standard formula shown here, which con-

trols for all four potential confounders, we

would need to specify values for the effect

modifiers V 1 , V 2 , and V 4 , where V 4 equals

V 1 V 2. For different choices ofV 1 andV 2 ,we

would then obtain different odds ratio values.

This information can be summarized in a table

or graph of odds ratios, which consider the

different specifications of the effect modifiers.

A sample table is shown here.

To assess confounding on an odds ratio scale,

we would then compute a similar table or

graph, which would consider odds ratio values

for a model that drops one or more eligibleV

variables. In our example, because the only

eligible variable isV 3 , we, therefore, need to

obtain an odds ratio table or graph for the

model that does not contain V 3. A sample

table of OR*values is shown here.

Thus, to assess whether we need to control for

confounding fromV 3 , we need to compare two

tables of odds ratios, one for the gold standard

and the other for the model that does not con-

tainV 3.

EXAMPLE (continued)

dOR¼expð^bþ^d 1 V 1 þ^d 2 V 2 þ^d 4 V 4 Þ
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linear function

^b;^d 1 ;^d 2 ;^d 4 on log odds ratio scale, but
odds ratio scale is clinically relevant

Log odds ratio scale:
^b¼ 12 : 69 vs:^b¼ 12 : 72
^d 1 ¼ 0 : 0692 vs:^d 1 ¼ 0 : 0696

Odds ratio scale:

CalculateORd¼exp



b^þ~^djWj



for different choices ofWj

Gold standard OR:
dOR¼expð^bþ^d 1 V 1 þ^d 2 V 2 þ^d 4 V 4 Þ;

whereV 4 ¼V 1 V 2.

SpecifyV 1 andV 2 to get OR:

V 1 ¼ 20 V 1 ¼ 30 V 1 ¼ 40

V 2 ¼ 100 dOR dOR ORd
V 2 ¼ 200 dOR dOR ORd

Model withoutV 3 :
dOR¼expð^bþ^d 1 V 1 þ^d 2 V 2 þ^d* 4 V 4 Þ

V 1 ¼ 20 V 1 ¼ 30 V 1 ¼ 40

V 2 ¼ 100 dOR dOR ORd
V 2 ¼ 200 dOR dOR ORd

Compare tables of
dORs vs: dOR*s
gold standard model withoutV 3

Presentation: IV. Confounding Assessment with Interaction 221