In other words, the e¤ect ofX 1 depends on the level (presence or absence) of
X 2 , and vice versa. This phenomenon is callede¤ect modification(i.e., one fac-
tor modifies the e¤ect of the other). The cross-product termx 1 x 2 is called an
interaction term. Use of these products will help in the investigation of possible
e¤ect modifications. Ifb 3 ¼0, the e¤ect of two factors acting together (repre-
sented by b 1 þb 2 ), is equal to the combined e¤ects of two factors acting
separately. Ifb 3 >0, we have a synergistic interaction; ifb 3 <0, we have an
antagonistic interaction.
8.2.4 Polynomial Regression
Consider the multiple regression model involvingoneindependent variable:
Yi¼b 0 þb 1 xiþb 2 x^2 iþeior it can be written as a multiple model:
Yi¼b 0 þb 1 x 1 iþb 2 x^22 iþeiwithX 1 ¼XandX 2 ¼X^2 , whereXis a continuous independent variable. The
meaning ofb 1 here is not the same as that given earlier because of the quadratic
termb 2 xi^2. We have, for example,
my¼b 0 þb 1 xþb 2 x^2 whenX¼x
b 0 þb 1 ðxþ 1 Þþb 2 ðxþ 1 Þ^2 whenX¼xþ 1so that the di¤erence is
b 1 þb 2 ð 2 xþ 1 Þa 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 poly-
nomial provides a better fit than a linear one, but (2) more often, a quadratic
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, however, is not
without drawbacks. The most potential drawback is thatXandX^2 are strongly
related, especially ifXis restricted to a narrow range; in this case the standard
errors are often very large.
8.2.5 Estimation of Parameters
To findgoodestimates of thekþ1 unknown parametersb 0 andbi’s, statis-
ticians use the same method ofleast squaresdescribed earlier. For each subject
with data values (Yi;Xi’s), we consider the deviation from the observed value
296 CORRELATION AND REGRESSION