Introductory Biostatistics

This phenomenon is callede¤ect modification(i.e., one factor modifies the
e¤ect of the other. The cross-product termx 1 x 2 is called aninteraction term;
The use of these products will help in the investigation of possible e¤ect mod-
ifications. Ifb 3 ¼0, the e¤ect of two factors acting together, as measured by
the odds ratio, is equal to the combined e¤ects of two factors acting separately,
as measured by the product of two odds ratios:

eb^1 þb^2 ¼eb^1 eb^2

This fits the classic definition ofno interactionon a multiplicative scale.

9.2.3 Polynomial Regression

Consider the model

pi¼

1

1 þexp½
ðb 0 þb 1 xiþb 2 xi^2 ފ

i¼ 1 ; 2 ;...;n

whereXis a continuous covariate. The meaning ofb 1 here is not the same as
that given earlier because of the quadratic termb 2 x^2 i. We have, for example,

lnðodds;X¼xÞ¼b 0 þb 1 xþb 2 x^2

lnðodds;X¼xþ 1 Þ¼b 0 þb 1 ðxþ 1 Þþb 2 ðxþ 1 Þ^2

so that after exponentiating, the di¤erence leads to

OR¼

ðodds;X¼xþ 1 Þ

ðodds;X¼xÞ

¼exp½b 1 þb 2 ð 2 xþ 1 ފ

a function ofx.
Polynomial models with an independent variable present in higher powers
than the second are not often used. The second-order or quadratic model has
two basic type of uses: (1) when the true relationship is a second-degree poly-
nomial or when the true relationship is unknown but the second-degree
polynomial provides a better fit than a linear one, but (2) more often, a qua-
dratic model is fitted for the purpose of establishing the linearity. The key item
to look for is whetherb 2 ¼0.
The use of polynomial models is not without drawbacks. The most potential
drawback is that multicollinearity is unavoidable. Especially if the covariate is
restricted to a narrow range, the degree of multicollinearity can be quite high;
in this case the standard errors are often very large. Another problem arises
when one wants to use the stepwise regression search method. In addition,

328 LOGISTIC REGRESSION