Introductory Biostatistics

These results are necessary in an e¤ort to identify important prognostic or
risk factors. 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 the correlated explanatory variables. The uses 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 alin-
ear regression modelin which, for example, the relative risk due to a 1-unit
increase in the value of a continuousXi(Xi¼xþ1 versusXi¼x) is indepen-
dent ofx. Therefore, if thislinearityseems to be violated, incorporation of
powers ofXishould be considered seriously. 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 the patient if
one or more covariate values are missing.

11.4.2 Testing Hypotheses in Multiple Regression

Once we have fit a multiple proportional hazards regression model and ob-
tained estimates for the various parameters of interest, we want to answer
questions about the contributions of various factors to the prediction of the
future of patients. 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 survivor-

ship?

2.Test for the value of a single factor. Does the addition of one particular

factor of interest add significantly to the prediction of survivorship over

and above that achieved by other factors?

3.Test for contribution of a group of variables. Does the addition of a group

of factors add significantly to the prediction of survivorship over and

above that achieved by other factors?

Overall Regression Tests We now consider the first question stated above
concerning an overall test for a model containgkfactors: say,

l½tjX¼ðx 1 ;x 2 ;...;xkފ¼l 0 ðtÞeb^1 x^1 þb^2 x^2 þþbkxk

The null hypothesis for this test may stated as: ‘‘Allkindependent variables
considered togetherdo not explain the variation in survival times.’’ In other
words,

H 0 :b 1 ¼b 2 ¼¼bk¼ 0

MULTIPLE REGRESSION AND CORRELATION 397