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

REFERENCE

multiplicative interaction vs.
additive interaction
Epidemiologic Research, Chap. 19

Logistic model variables:

X 1 ¼Að 0 , 1 Þ

X 2 ¼Bð 0 , 1 Þ

)

main effects

X 3 ¼AB interaction effect
variable

logit P(X)¼aþb 1 Aþb 2 B
þb 3 AB,

where

PðXÞ¼risk givenAandB

¼RAB

b 3 ¼lne

OR 11

OR 10 OR 01



Note that in determining whether or not the no

interaction equation is satisfied, the left- and

right-hand sides of the equation do not have to

be exactly equal. If the left-hand side is approx-

imately equal to the right-hand side, we can

conclude that there is no interaction. For

instance, if the left-hand side is 14.5 and the

right-hand side is 14, this would typically be

close enough to conclude that there is no inter-

action on a multiplicative scale.

A more complete discussion of interaction,

including the distinction between multipli-

cative interactionand additive interaction,is

given in Chap. 19 ofEpidemiologic Research

by Kleinbaum, Kupper, and Morgenstern

(1982).

We now define a logistic model that allows

the assessment of multiplicative interaction

involving two (0, 1) indicator variablesAand

B. This model contains three independent vari-

ables, namely,X 1 equal toA, X 2 equal toB, and

X 3 equal to the product termAtimesB. The

variablesAandBare called main effect vari-

ables and the product term is called an interac-

tion effect variable.

The logit form of the model is given by the

expression logit of P(X) equalsaplusb 1 times

Aplusb 2 timesBplusb 3 timesAtimesB.P(X)

denotes the risk for developing the disease

given values ofAandB, so that we can alterna-

tively write P(X)asRAB.

For this model, it can be shown mathemati-

cally that the coefficient b 3 of the product

term can be written in terms of the three odds

ratios we have previously defined. The formula

isb 3 equals the natural log of the quantity OR 11

divided by the product of OR 10 and OR 01 .We

can make use of this formula to test the null

hypothesis of no interaction on a multiplicative

scale.

EXAMPLE (continued)

Note: “¼” means approximately equal

()

e.g., 14.514.0)no interaction

Presentation: III. Assessing Multiplicative Interaction 53