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

DEFINITION

OR 11 = OR 10 × OR 01

no interaction
on a
multiplicative
scale

multiplication

No interaction:

effect of

AandB

acting

together

0

B

B

@

1

C

C

A ¼

combined

effect of

A and B

acting

separately

0

B

B

B

B@

1

C

C

C

CA

""

OR 11 OR 10 OR 01

multiplicative

scale

no interaction formula:

OR 11 ¼OR 10 OR 01

Now that we have defined appropriate odds

ratios for the two independent variables situa-

tion, we are ready to provide an equation for

assessing interaction. The equation is stated as

OR 11 equals the product of OR 10 and OR 01 .If

this expression is satisfied for a given study

situation, we say that there is “no interaction

on amultiplicativescale.” In contrast, if this

expression is not satisfied, we say that there is

evidence of interaction on a multiplicative

scale.

Note that the right-hand side of the “no inter-

action” expression requiresmultiplication of

two odds ratios, one corresponding to the com-

bination 10 and the other to the combination

1. Thus, the scale used for assessment of inter-
   action is called multiplicative.

When the no interaction equation is satisfied,

we can interpret the effect of both variablesA

andBacting together as being the same as

the combined effect of each variable acting

separately.

The effect of both variables acting together is

given by the odds ratio OR 11 obtained whenA

andBare both present, that is, whenAequals 1

andBequals 1.

The effect ofAacting separately is given by the

odds ratio forAequals 1 andBequals 0, and

the effect ofBacting separately is given by the

odds ratio forAequals 0 andBequals 1. The

combined separate effects ofAandBare then

given by the product OR 10 times OR 01.

Thus, when there is no interaction on a multi-

plicative scale, OR 11 equals the product of

OR 10 and OR 01.

Presentation: III. Assessing Multiplicative Interaction 51