is (r

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1]. We wish to test the hypotheses

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H0: H1:

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0 0 (12-27)

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where 0 denotes a vector of zeroes. The model may be written as y X X1

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(12-28)

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where X1 represents the columns of X associated with associated with 2.

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and X2 represents the columns of X

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CHAPTER 12 MULTIPLE LINEAR REGRESSION

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For the full model (including both 1 and 2), we know that 1X X2 addition, the regression sum of squares for all variables including the intercept is SSR 1 2 and MSE y y n

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X y X y

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1 degrees of freedom2

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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 H0: 1 0 to be true. The reduced model is found from Equation 12-28 as y The least squares estimate of SSR 1

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(12-29)

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is 2

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X y

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1X X2 2 1X y, and 2 2 1p r degrees of freedom2

(12-30)

The regression sum of squares due to SSR 1

given that SSR 1 2

is already in the model is SSR 1

(12-31)

This sum of squares has r degrees of freedom. It is sometimes called the extra sum of squares due to 1. Note that SSR 1 1 0 2 2 is the increase in the regression sum of squares due to including the variables x1, x2, p , xr in the model. Now SSR 1 1 0 2 2 is independent of MSE, and the null hypothesis 1 0 may be tested by the statistic SSR 1

1 | 22 r MSE

(12-32)

If the computed value of the test statistic f0 f ,r,n p, we reject H0, concluding that at least one of the parameters in 1 is not zero and, consequently, at least one of the variables x1, x2, p , xr in X1 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 individual regressor xj as if it were the last variable added to the model by computing SSR 1

j0 0, 1,

k 2,

1, 2, p , k

This is the increase in the regression sum of squares due to adding xj to a model that already includes x1, . . . , 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 variables to use in the model.

12-2 HYPOTHESIS TESTS IN MULTIPLE LINEAR REGRESSION

EXAMPLE 12-5

Consider the wire bond pull strength data in Example 12-1. We will investigate the contribution of the variable x2 (die height) to the model using the partial F-test approach. That is, we wish to test H0: H1:

To test this hypothesis, we need the extra sum of squares due to SSR 1

20 1, 0 2

SSR 1 SSR 1

1, 2, 0 2 1, 2 0

SSR 1 1, 0 2 SSR 1 1 0 0 2

In Example 12-3 we have calculated SSR 1

1, 2 0 02

X y

a a yi b

1two degrees of freedom2

and from Example 11-8, where we fit the model Y SSR 1 Therefore, SSR 1

20 1, 0 2 10 02

, we can calculate

12.9027212027.71322 5885.8521 1one degree of freedom2

S 1 xy

5990.7712 5885.8521 104.9191 1one degree of freedom2

This is the increase in the regression sum of squares due to adding x2 to a model already containing x1. To test H0: 2 0, calculate the test statistic f0 SSR 1

20 1, 0 2

104.9191 1 5.2352

Note that the MSE from the full model, using both x1 and x2, is used in the denominator of the test statistic. Since f0.05,1,22 4.30, we reject H0: 2 0 and conclude that the regressor die height (x2) contributes significantly to the model. Table 12-4 shows the Minitab regression output for the wire bond pull strength data. Just below the analysis of variance summary in this table the quantity labeled Seq SS shows the sum of squares obtained by fitting x1 alone (5885.9) and the sum of squares obtained by fitting x2 after x1. Notationally, these are referred to above as SSR 1 1 0 0 2 and SSR 1 2 0 1, 0 2 . Since the partial F-test in the above example involves a single variable, it is equivalent to the t-test. To see this, recall from Example 12-5 that the t-test on H0: 2 0 resulted in the test statistic t0 4.4767. Furthermore, the square of a t-random variable with degrees of freedom is an F-random variable with one and degrees of freedom, and we note that t 2 0 (4.4767)2 20.04 f0.

12-2.3

More About the Extra Sum of Squares Method (CD Only)