likelihood ratio test, Wald’s test, and the score test. All three statistics are pro-
vided by most standard computer programs such as SAS, and they are asymp-
totically equivalent (i.e., for very large sample sizes), yielding identical statisti-
cal decisions most of the time. However, Wald’s test is used much less often
than the other two.
Example 11.13 Refer to the data for low-birth-weight babies in Example
11.11 (Table 11.14). With all four covariates, we have the following test statis-
tics for the global null hypothesis:
(a) Likelihood test:w^2 LR¼ 9 :530 with 4 df;p¼ 0 : 0491(b) Wald’s test:wW^2 ¼ 6 :001 with 4 df;p¼ 0 : 1991(c) Score test:w^2 S¼ 8 :491 with 4 df;p¼ 0 : 0752The results indicates a weak combined explanatory power; Wald’s test is not
even significant. Very often, this means implicitly that perhaps only one or two
covariates are associated significantly with the response of interest (a weak
overall correlation).
Test for a Single Variable Let us assume that we now wish to test whether the
addition of one particular independent variable of interest adds significantly to
the prediction of the response over and above that achieved by factors already
present in the model (usually after seeing a significant result for the global
hypothesis above). The null hypothesis for this single-variable test may be
stated as: ‘‘FactorXidoes not have any value added to the prediction of the
responsegiven that other factors are already included in the model.’’ In other
words,
H 0 :bi¼ 0To test such a null hypothesis, one can 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
CONDITIONAL LOGISTIC REGRESSION 421