Tests on sets of regression parameters

In the previous section we developed a hypothesis test for individual regression parameters. There are situations, where we are interested in the effect of the combination of several variables. Now we will perform a test of hypothesis with the null hypothesis that the coefficients of all independent variables in the regression model are equal to zero and the alternative hypothesis that the coefficients of all independent variables are not zero. For the multiple regression model

the two hypotheses for such test are written as

at least one

A test of hypothesis for this case is performed by using the F distribution. The F distribution has 2 degrees of freedom- for numerator and - for denominator. Table 6 in Appendix lists the values of F for F distribution. The value of the test statistic can be obtained from the computer solution, or it can also be calculated by using formula

or

where MSR stands for the mean square regression and MSE for the mean square error.

;

In the end, to test the null hypothesis

against the alternative hypothesis

at least one

the decision rule for a significance level is

reject if

where is the number for which and follows an F distribution with numerator degrees of freedom K and denominator degrees of freedom .

Example:

Using 5% significance level, can you conclude that the coefficients of all independent variables in the example 4.1 are equal to zero? Use the MINITAB solution shown in Figure 4.1

Solution:

The two hypotheses are

at least one

The portion of the solution is reproduced below

Analysis of Variance

Source DF SS MS F P

Regression 2 4824.5 2412.3 12.72 0.005

Residual Error 7 1327.9 189.7

Total 9 6152.4

From the portion of MINTAB solution we obtain

;

and the value of the test statistic is

Because the value of the test statistic greater than , it falls in the rejection region. Consequently, we reject the null hypothesis and conclude that at least one of the two ’s is different from zero.

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