Applied Statistics and Probability for Engineers

12-2 HYPOTHESIS TESTS IN MULTIPLE LINEAR REGRESSION 433

The main diagonal element of the matrix corresponding to is C 22 0.0000015,

so the t-statistic in Equation 12-24 is

Note that we have used the estimate of 2 reported to four decimal places in Table 12-10. Since

t0.025,222.074, we reject H 0 :  2 0 and conclude that the variable x 2 (die height) con-

tributes significantly to the model. We could also have used a P-value to draw conclusions.

The P-value for t 0 4.4767 is P0.0002, so with = 0.05 we would reject the null hy-

pothesis. Note that this test measures the marginal or partial contribution of x 2 given that x 1 is

in the model. That is, the t-test measures the contribution of adding the variable x 2 die

height to a model that already contains x 1 wire length. Table 12-4 shows the value of the

t-test computed by Minitab. The Minitab t-test statistic is reported to two decimal places. Note

that the computer produces a t-test for each regression coefficient in the model. These t-tests

indicate that both regressors contribute to the model.

There is another way to test the contribution of an individual regressor variable to the

model. This approach determines the increase in the regression sum of squares obtained by

adding a variable xj(say) to the model, given that other variables xi(ij) are already included

in the regression equation.

The procedure used to do this is called the general regression significance test, or the

extra sum of squares method.This procedure can also be used to investigate the contribution

of a subsetof the regressor variables to the model. Consider the regression model with k

regressor variables

(12-25)

where yis (n 1), Xis (n p), is (p 1), is (n 1), and pk1. We would like to

determine if the subset of regressor variables x 1 , x 2 ,..., xr(rk) as a whole contributes sig-

nificantly to the regression model. Let the vector of regression coefficients be partitioned as

follows:

(12-26)

where  1 is (r 1) and  2 is [(pr) 1]. We wish to test the hypotheses

c

 1

 2

d

yX

t 0 

ˆ 2

2 ˆ^2 C 22



0.01253

21 5.2352 21 0.0000015 2

4.4767

1 X¿X 2 ^1 ˆ 2

where 0 denotes a vector of zeroes. The model may be written as

(12-28)

where X 1 represents the columns of Xassociated with  1 and X 2 represents the columns of X

associated with  2.

yXX 1  1 X 2  2 

H 1 :  1  0 (12-27)

H 0 :  1  0

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