Introductory Biostatistics

linear regression, we have

0 aR^2 a 1

andR^2 only assumes the value 0 when allbi¼0.

2. Theerror mean square,

MSE¼

SSE

n
 2

serves as an estimate of the constant variances^2 as stipulated by the

regression model.

8.2.7 Testing Hypotheses in Multiple Linear Regression

Once we have fit a multiple 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 explanatory or

independent variables contribute significantly to the prediction of the

response (or the explanation of variation among responses)?

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. The null hypothesis
for this test may be stated as: ‘‘Allkindependent variablesconsidered together
do not explain the variation in the responses.’’ In other words,

H 0 :b 1 ¼b 2 ¼¼bk¼ 0

Thisglobalnull hypothesis can be tested using theFstatistic in Table 8.6:

F¼

MSR

MSE

anFtest atðk;n
k
1 Þdegrees of freedom.

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

298 CORRELATION AND REGRESSION