8.2.2 Meaning of Regression Parameters
Similar to the univariate case,birepresents one of the following:
- Theincrease(ordecrease,ifbiis negative) in the mean ofYassociated
with the exposure ifXi is binary (exposedXi¼1 versus unexposed
Xi¼0),assumingthat other independent variables are fixed; or - Theincrease(ordecrease,ifbiis negative) in the mean ofYassociated
with a 1-unit increase in the value ofXi,Xi¼xþ1 versusXi¼x. For an
m-unit increase in the value ofXi,sayXi¼xþmversusXi¼x, the
corresponding increase (or decrease) in the mean ofYismbiassuming
that other independent variables are fixed. In other words,birepresents
theadditional contributionofXiin the explanation of variation amongy
values. Of course, before such analyses are done, the problem and the
data have to be examined carefully. If some of the variables are highly
correlated, one or fewer of the correlated factors are likely to be as good
predictors as all of them; information from similar studies also has
to be incorporated so as to drop some of these correlated explanatory
variables.
8.2.3 E¤ect Modifications
Consider a multiple regression model involvingtwoindependent variables:
Yi¼b 0 þb 1 x 1 iþb 2 x 2 iþb 3 x 1 ix 2 iþeiIt can be seen that the meaning ofb 1 andb 2 here is not the same as that given
earlier because of the cross-product termb 3 x 1 x 2. Suppose, for simplicity, that
bothX 1 andX 2 are binary; then:
- ForX 2 ¼1 or exposed, we have
my¼b 0 þb 1 þb 3 if exposed toX 1
b 0 if not exposed toX 1so that the increase (or decrease) in the mean ofYdue to an exposure to
X 1 isb 1 þb 3 , whereas- ForX 2 ¼0 or not exposed, we have
my¼b 0 þb 1 if exposed toX 1
b 0 if not exposed toX 1so that the increase (or decrease) in the mean ofYdue to an exposure to
X 1 isb 1.MULTIPLE REGRESSION ANALYSIS 295