Three likelihood-based statistics can be used to test thisglobalnull hypoth-
esis; each has a symptotic chi-square distribution withkdegrees of freedom
underH 0 : the likelihood ratio test, Wald’s test, and the score test. All three chi-
square statistics are provided by most standard computer programs.
Tests for a Single Variable Let us assume that we now wish to test whether
the addition of one particular factor of interest adds significantly to the predic-
tion of survivorship over and above that achieved by other factors already
present in the model. The null hypothesis for this test may be stated as: ‘‘Factor
Xidoes not have any value added to the prediction of survivorshipgiven that
other factors are already included in the model.’’ In other words,
H 0 :bi¼ 0To test such a null hypothesis, one can perform a likelihood ratio chi-square
test, with 1 df, similar to that for the global hypothesis above:
wLR^2 ¼ 2 ½lnLðbb^;allX’sÞlnLðbb^;all otherX’s withXideletedÞA much easier alternative method is to use
zi¼bb^
i
SEðbb^iÞwherebb^iis the corresponding estimated regression coe‰cient and SEðbb^iÞis the
estimate of the standard error ofbb^i, both of which are printed by standard
packaged programs. In performing this test, we refer the value of thezstatistic
to percentiles of the standard normal distribution. This is equivalent to Wald’s
chi-square test as applied to one parameter.
Contribution of a Group of Variables This testing procedure addresses the
more general problem of assessing the additional contribution of two or more
factors to the prediction of survivorship over and above that made by other
variables already in the regression model. In other words, the null hypothesis is
of the form
H 0 :b 1 ¼b 2 ¼¼bm¼ 0To test such a null hypothesis, one can perform a likelihood ratio chi-square
test withmdf:
wLR^2 ¼ 2 ½lnLðbb^;allX’sÞlnLðbb^;all otherX’s withX’s under investigation deletedÞ398 ANALYSIS OF SURVIVAL DATA