Introductory Biostatistics

finding a satisfactory interpretation for thecurvature e¤ect coe‰cientb 2 is not
easy. Perhaps a rather interesting application would be finding a valuexof the
covariateXso as to maximize or minimize the

lnðodds;X¼xÞ¼b 0 þb 1 xþb 2 x^2

9.2.4 Testing Hypotheses in Multiple Logistic Regression

Once we have fit a multiple logistic regression model and obtained estimates for
the various parameters of interest, we want to answer questions about the con-
tributions of various factors to the prediction of the binary response variable.
There are three types of such questions:

1.Overall test. Taken collectively, does the entire set of explatory or inde-

pendent variables contribute significantly to the prediction of response?

2.Test for the value of a single factor.Does the addition of one particular

variable of interest add significantly to the prediction of response over

and above that achieved by other independent variables?

3.Test for contribution of a group of variables. Does the addition of a group

of variables add significantly to the prediction of response over and above

that achieved by other independent variables?

Overall Regression Tests We now consider the first question stated above
concerning an overall test for a model containgkfactors, say,

pi¼

1

1 þexp½
ðb 0 þ

Pk

j¼ 1 bjxjiފ

i¼ 1 ; 2 ;...;n

The null hypothesis for this test may be stated as: ‘‘Allkindependent vari-
ablesconsidered togetherdo not explain the variation in the responses.’’ In other
words,

H 0 :b 1 ¼b 2 ¼¼bk¼ 0

Two likelihood-based statistics can be used to test thisglobalnull hypothesis;
each has a symptotic chi-square distribution withkdegrees of freedom under
H 0.

1. Likelihood ratio test:

wLR^2 ¼ 2 ½lnLðbb^Þ
lnLð 0 ފ

2. Score test:

w^2 S¼

dlnLð 0 Þ

db

T

d^2 lnLð 0 Þ

db^2

"#
1

dlnLð 0 Þ

db



MULTIPLE REGRESSION ANALYSIS 329