exp½bb^iG 1 :96 SEðbb^iÞThese results are necessary in the e¤ort to identify important risk factors in
matched designs. 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 (unfortunately, it is very di‰cult to tell). It is
important to note that we are assuming alinear regression modelin 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 (again, it is very di‰cult to tell; the only easy way is fitting a
polynomial model as seen in a later example), the incorporation of powers of
Xishould be considered seriously. The use of products will help in the investi-
gation of possible e¤ect modifications. Finally, there is the messy problem of
missing data; most packaged programs would delete a subject if one or more
covariate values are missing.
Testing Hypotheses in Multiple Regression Once we have fit a multiple condi-
tional logistic regression model and obtained estimates for the various param-
eters of interest, we want to answer questions about the contributions of vari-
ous factors to the prediction of the binary response variable using matched
designs. There are three types of such questions:
1.Overall test. Taken collectively, does the entire set of explatory or inde-
pendent variables contribute significantly to the prediction of response?
2.Test for the value of a single factor. Does the addition of one particular
variable of interest add significantly to the prediction of response over
and above that achieved by other independent variables?
3.Test for contribution of a group of variables. Does the addition of a group
of variables add significantly to the prediction of response over and above
that achieved by other independent variables?Overall Regression Test We now consider the first question stated above con-
cerning an overall test for a model containgkfactors. The null hypothesis for
this test may be stated as: ‘‘Allkindependent variablesconsidered togetherdo
not explain the variation in response any more than the size alone.’’ In other
words,
H 0 :b 1 ¼b 2 ¼¼bk¼ 0Three statistics can be used to test thisglobal null hypothesis; each has a
symptotic chi-square distribution withk degrees of freedom underH 0 :the
420 ANALYSIS OF SURVIVAL DATA