Applied Statistics and Probability for Engineers

432 CHAPTER 12 MULTIPLE LINEAR REGRESSION

If H 0 : j0 is not rejected, this indicates that the regressor xjcan be deleted from the model.

The test statistic for this hypothesis is

The adjustedR^2 statistic essentially penalizes the analyst for adding terms to the model. It is

an easy way to guard against overfitting,that is, including regressors that are not really useful.

Consequently, it is very useful in comparing and evaluating competing regression models. We

will use R^2 adjfor this when we discuss variable selectionin regression in Section 12-6.3.

12-2.2 Tests on Individual Regression Coefficients

and Subsets of Coefficients

We are frequently interested in testing hypotheses on the individual regression coefficients. Such

tests would be useful in determining the potential value of each of the regressor variables in the re-

gression model. For example, the model might be more effective with the inclusion of additional

variables or perhaps with the deletion of one or more of the regressors presently in the model.

Adding a variable to a regression model always causes the sum of squares for regression

to increase and the error sum of squares to decrease (this is why R^2 always increases when a

variable is added). We must decide whether the increase in the regression sum of squares is

large enough to justify using the additional variable in the model. Furthermore, adding an

unimportant variable to the model can actually increase the error mean square, indicating that

adding such a variable has actually made the model a poorer fit to the data (this is why R^2 adjis

a better measure of global model fit then the ordinary R^2 ).

The hypotheses for testing the significance of any individual regression coefficient, say

j, are

where Cjjis the diagonal element of corresponding to Notice that the denominator

of Equation 12-24 is the standard error of the regression coefficient. The null hypothesis H 0 :

j0 is rejected if This is called a partialor marginal testbecause the

regression coefficient depends on all the other regressor variables xi(i j) that are in the

model. More will be said about this in the following example.

EXAMPLE 12-4 Consider the wire bond pull strength data, and suppose that we want to test the hypothesis that

the regression coefficient for x 2 (die height) is zero. The hypotheses are

H 1 :  2

0

H 0 :  2  0

ˆj

0 t 00 t (^) 2,np.
ˆj
1 X¿X 2 ^1 ˆj.
H 1 : j
0 (12-23)
H 0 : j 0
T 0  (12-24)
ˆj
2 ˆ^2 Cjj

ˆj
se 1 ˆj 2
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