Applied Statistics and Probability for Engineers

12-2 HYPOTHESIS TESTS IN MULTIPLE LINEAR REGRESSION 429

We should reject H 0 if the computed value of the test statistic in Equation 12-18, f 0 , is greater than

f ,k,np. The procedure is usually summarized in an analysis of variance table such as Table 12-9.

We can find a computing formula for SSEas follows:

Substituting into the above, we obtain

SSEy¿yˆ¿X¿y (12-19)

eyyˆyXˆ

SSE a

n

i 1

1 yiyˆi 22  a

n

i 1

ei^2 e¿e

H 1 : jZ 0 for at least one j (12-17)

H 0 :  1  2 # # #k 0

F 0  (12-18)

SSRk

SSE 1 np 2



MSR

MSE

Rejection of implies that at least one of the regressor variables

x 1 , x 2 , p, xkcontributes significantly to the model.

The test for significance of regression is a generalization of the procedure used in simple

linear regression. The total sum of squares SSTis partitioned into a sum of squares due to re-

gression and a sum of squares due to error, say,

SSTSSR SSE

Now if is true, is a chi-square random variable with k

degrees of freedom. Note that the number of degrees of freedom for this chi-square random

variable is equal to the number of regressor variables in the model. We can also show the

SSE    2 is a chi-square random variable with npdegrees of freedom, and that SSEand SSR

are independent. The test statistic for H 0 :  1  2 pk 0 is

H 0 :  1  2 pk 0 SSR 2

H 0 :  1  2 pk 0

Table 12-9 Analysis of Variance for Testing Significance of Regression in Multiple Regression

Source of Degrees of

Variation Sum of Squares Freedom Mean Square F 0

Regression SSR kMSR MSRMSE

Error or residual SSE npMSE

Total SST n 1

appropriate hypotheses are

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