Introductory Biostatistics

Both statistics are provided by most standard computer programs, such as
SAS, and they are asymptotically equivalent, yielding identical statistical deci-
sions most of the times.

Example 9.4 Refer to the data set on prostate cancer of Example 9.1 (Table
9.1). With all five covariates, we have the following test statistics for the global
null hypothesis:

1. Likelihood test:

wLR^2 ¼ 22 :126 with 5 df; p¼ 0 : 0005

2. Score test:

wS^2 ¼ 19 :451 with 5 df; p¼ 0 : 0016

Note:An SAS program would include these instructions:

PROC LOGISTIC DESCENDING
DATA = CANCER;
MODEL NODES = X-RAY, GRADE, STAGE, AGE, ACID;

where CANCER is the name assigned to the data set, NODES is the variable
name for nodal involvement, and X-RAY, GRADE, STAGE, AGE, and
ACID are the variable names assigned to the five covariates.

Tests 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 other factors
already present in the model. The null hypothesis for this test may stated as:
‘‘FactorXidoes not have any value added to the prediction of the response
given that other factors are already included in the model.’’ In other words,

H 0 :bi¼ 0

To test such a null hypothesis, one can perform a likelihood ratio chi-
squared test, with 1 df, similar to that for the global hypothesis above:

w^2 LR¼ 2 ½lnLðbb^;allX’sÞ
lnLðbb^;all otherX’s withXideletedފ

A much easier alternative method is using

zi¼

bb^i

SEðbb^iÞ

330 LOGISTIC REGRESSION