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 the response 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 deletedÞAs 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 the response; these variables
have some trait in common. Another application would be a collection of
powersand=orproduct terms (referred to asinteraction variables). It is often of
interest to assess the interaction e¤ects collectively before trying to consider
individual interaction terms in a model as suggested previously. 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 rate, which may be inflated due to
multiple testing.
Example 9.7 Refer to the data set on prostate cancer of Example 9.1 (Table
9.1) with all five covariates. We consider, collectively, these four interaction
terms: acidx-ray, acidstage, acidgrade, and acidage. The basic idea
is to see ifanyof the other variable would modify the e¤ect of the level of acid
phosphatase on the response.
- With the original five variables, we obtained lnL¼ 24 :063.
- With all nine variables, five original plus four products, we obtained
lnL¼ 20 :378.
Therefore,
wLR^2 ¼ 2 ½lnLðbb^;nine variablesÞlnLðbb^;five original variablesÞ
¼ 7 : 371 ;4df; 0 : 05 ap-valuea 0 : 10In other words, all four interaction terms,considered together, are marginally
significant (0: 05 ap-valuea 0 :10); there may be some weak e¤ect modification
332 LOGISTIC REGRESSION