Applied Statistics and Probability for Engineers

434 CHAPTER 12 MULTIPLE LINEAR REGRESSION

For the full model(including both  1 and  2 ), we know that In

addition, the regression sum of squares for all variables including the intercept is

and

SSR() is called the regression sum of squares due to . To find the contribution of the terms

in  1 to the regression, fit the model assuming the null hypothesis H 0 :  1  0 to be true. The

reduced modelis found from Equation 12-28 as

(12-29)

The least squares estimate of  2 is and

(12-30)

The regression sum of squares due to  1 given that  2 is already in the model is

(12-31)

This sum of squares has rdegrees of freedom. It is sometimes called the extra sum of squares

due to  1. Note that is the increase in the regression sum of squares due to

including the variables x 1 , x 2 , p, xrin the model. Now is independent of MSE, and

the null hypothesis  1  0 may be tested by the statistic

SSR 1  10  22

SSR 1  10  22

SSR 1  10  22 SSR 1  2 SSR 1  22

SSR 1  22 ˆ¿ 2 X 2 ¿y 1 pr degrees of freedom 2

ˆ 2  1 X¿ 2 X 22 ^1 X¿ 2 y,

yX 2  2 

MSE

y¿yˆX¿y

np

SSR 1  2 ˆ¿X¿y 1 pk1 degrees of freedom 2

ˆ 1 X¿X 2 ^1 X¿y.

If the computed value of the test statistic f 0 f ,r,np, we reject H 0 , concluding that at least one

of the parameters in  1 is not zero and, consequently, at least one of the variables x 1 , x 2 , p, xr

in X 1 contributes significantly to the regression model. Some authors call the test in Equation

12-32 a partial F-test.

The partial F-test is very useful. We can use it to measure the contribution of each indi-

vidual regressor xjas if it were the last variable added to the model by computing

This is the increase in the regression sum of squares due to adding xjto a model that already

includes x 1 ,... , xj 1 , xj 1 ,... , xk. The partial F-test is a more general procedure in that we

can measure the effect of sets of variables. In Section 12-6.3 we show how the partial F-test

plays a major role in model building—that is, in searching for the best set of regressor vari-

ables to use in the model.

SSR 1 j 0  0 ,  1 , p , j 1 , j 1 , p , k 2 , j1, 2, p , k

F 0  (12-32)

SSR 1  1 |  22 r

MSE

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