As with thez testabove, thismultiple contribution procedureis very useful
for assessing the importance of potential explanatory variables. In particular it
is often used to test whether a similar group of variables, such asdemographic
characteristics, is important for the prediction of survivorship; these vari-
ables have some trait in common. Another application would be a collection
of powersand/orproduct terms (referred to as interaction variables). It is
often of interest to assess the interaction e¤ects collectively before trying to
consider individual interaction terms in a model. In fact, such use reduces the
total number of tests to be performed, and this, in turn, helps to provide better
control of overall type I error rates, which may be inflated due to multiple
testing.
Stepwise Procedure In applications, our major interest is to identify important
prognostic factors. In other words, we wish to identify from many available
factors a small subset of factors that relate significantly to the length of survival
time of patients. In that identification process, of course, we wish to avoid a
type I (false positive) error. In a regression analysis, a type I error corresponds
to including a predictor that has no real relationship to survivorship; such an
inclusion can greatly confuse interpretation of the regression results. In a
standard multiple regression analysis, this goal can be achieved by using a
strategy that adds into or removes from a regression model one factor at a time
according to a certain order of relative importance. The details of this stepwise
process for survival data are similar to those for logistic regression given in
Chapter 9.
Stratification The proportional hazards model requires that for a covariate
X—say, an exposure—the hazards functions at di¤erent levels,lðt;exposedÞ
andlðt;nonexposedÞare proportional. Of course, sometimes there are factors
the di¤erent levels of which produce hazard functions that deviate markedly
from proportionality. These factors may not be under investigation themselves,
especially those of no intrinsic interest, those with a large number of levels,
and/or those where interventions are not possible. But these factors may act
as important confounders which must be included in any meaningful analysis
so as to improve predictions concerning other covariates. Common examples
include gender, age, and neighborhood. To accommodate such confounders, an
extension of the proportional hazards model is desirable. Suppose there is a
factor that occurs onqlevels for which the proportional hazards model may be
violated. If this factor is under investigation as acovariate, the model and sub-
sequent analyses are not applicable. However, if this factor isnotunder inves-
tigation and is considered only as a confounder so as to improve analyses and/
or predictions concerning other covariates,we can treat it as a stratification
factor. By doing that we will get no results concerning this factor (which are not
wanted), but in return we do not have to assume that the hazard functions
corresponding to di¤erent levels are proportional (which may be severely vio-
lated). Suppose that the stratification factorZhasqlevels; this factor is not
MULTIPLE REGRESSION AND CORRELATION 399