Introductory Biostatistics

exp½bb^iG 1 :96 SEðbb^iފ

These results are necessary in the e¤ort to identify important risk factors for the
binary outcome. Of course, before such analyses are done, the problem and the
data have to be examined carefully. If some of the variables are highly corre-
lated, 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. The use of products
such asX 1 X 2 and higher power terms such asX 12 may be necessary and can
improve the goodness of fit. It is important to note that we are assuming a
(log)linearregression model, in which, for example, the odds ratio due to a
1-unit increase in the value of a continuousXi(Xi¼xþ1 versusXi¼x)is
independent ofx. Therefore, if thislinearityseems to be violated, the incorpo-
ration of powers ofXishould be seriously considered. The use of products will
help in the investigation of possible e¤ect modifications. Finally, there is the
messy problem of missing data; most packaged programs would delete a sub-
ject if one or more covariate values are missing.

9.2.2 E¤ect Modifications

Consider the model

pi¼

1

1 þexp½
ðb 0 þb 1 x 1 iþb 2 x 2 iþb 3 x 1 ix 2 iފ

i¼ 1 ; 2 ;...;n

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 that bothX 1 andX 2 are binary.

1. ForX 2 ¼1, or exposed, we have

ðodds ratio;not exposed toX 1 Þ¼eb^0 þb^2

ðodds ratio;exposed toX 1 Þ¼eb^0 þb^1 þb^2 þb^3

so that the ratio of these ratios, eb^1 þb^3 , represents the odds ratio as-

sociated withX 1 , exposed versus nonexposed, in the presence ofX 2 ,

whereas

2. ForX 2 ¼0, or not exposed, we have

ðodds ratio;not exposed toX 1 Þ¼eb^0 ði:e:;baselineÞ

ðodds ratio;exposed toX 1 Þ¼eb^0 þb^1

so that the ratio of these ratios,eb^1 , represents the odds ratio associated

with X 1 , exposed versus nonexposed, in the absence ofX 2. In other

words, the e¤ect ofX 1 depends on the level (presence or absence) ofX 2 ,

and vice versa.

MULTIPLE REGRESSION ANALYSIS 327