Douglas C. Montgomery - Introduction to Linear Regression Analysis

These expected mean squares indicate that if the observed value of F 0is large, then it is likely that the slope β 1≠ 0. Appendix C.3also shows that if β 1≠ 0, then F 0follows a noncentral F distribution with 1 and n − 2 degrees of freedom and a non-centralityparameter of

This noncentrality parameter also indicates that the observed value of F 0should be large if β 1≠ 0. Therefore, to test the hypothesis H 0: β 1= 0, compute the test statistic F 0and reject H 0if

The test procedure is summarized in Table 2.4.

TABLE 2.4 Analysis of Variance for Testing Significance of Regression

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	F 0
Regression		1	MS R	MS R/ MS Res
Residual		n − 2	MS Res
Total	SS T	n − 1

We will test for significance of regression in the model developed in Example 2.1for the rocket propellant data. The fitted model is , SS T= 1,693,737.60, and Sxy = −41,112.65. The regression sum of squares is computed from Eq. (2.34)as

The analysis of variance is summarized in Table 2.5. The computed value of F 0is 165.21, and from Table A.4, F 0.01,1,18= 8.29. The P value for this test is 1.66 × 10 −10. Consequently, we reject H 0: β 1= 0.

The Minitab output in Table 2.3 also presents the analysis-of-variance test significance of regression. Comparing Tables 2.3 and 2.5, we note that there are some slight differences between the manual calculations and those performed by computer for the sums of squares. This is due to rounding the manual calculations to two decimal places. The computed values of the test statistics essentially agree.

We noted in Section 2.3.2that the t statistic

(2.37)

could be used for testing for significance of regression. However, note that on squaring both sides of Eq. (2.37), we obtain

(2.38)

Thus, in Eq. (2.38)is identical to F 0of the analysis-of-variance approach in Eq. (2.36). For example; in the rocket propellant example t 0= −12.5, so . In general, the square of a t random variable with f degrees of freedom is an F random variable with one and f degrees of freedom in the numerator and denominator, respectively. Although the t test for H 0: β 1= 0 is equivalent to the F test in simple linear regression, the t test is somewhat more adaptable, as it could be used for one-sided alternative hypotheses (either H 1: β 1< 0 or H 1: β 1> 0), while the F test considers only the two-sided alternative. Regression computer programs routinely produce both the analysis of variance in Table 2.4 and the t statistic. Refer to the Minitab output in Table 2.3.

TABLE 2.5 Analysis-of-Variance Table for the Rocket Propellant Regression Model

The real usefulness of the analysis of variance is in multiple regression models. We discuss multiple regression in the next chapter.

Finally, remember that deciding that β 1= 0 is a very important conclusion that is only aidedby the t or F test. The inability to show that the slope is not statistically different from zero may not necessarily mean that y and x are unrelated. It may mean that our ability to detect this relationship has been obscured by the variance of the measurement process or that the range of values of x is inappropriate. A great deal of nonstatistical evidence and knowledge of the subject matter in the field is required to conclude that β 1= 0.

In this section we consider confidence interval estimation of the regression model parameters. We also discuss interval estimation of the mean response E ( y ) for given values of x . The normality assumptions introduced in Section 2.3continue to apply.

In addition to point estimates of β 0, β 1, and σ 2, we may also obtain confidence interval estimates of these parameters. The width of these confidence intervals is a measure of the overall quality of the regression line. If the errors are normally and independently distributed, then the sampling distribution of both and is t with n − 2 degrees of freedom. Therefore, a 100(1 − α ) percent confidence interval (CI) on the slope β 1is given by