Introduction to Linear Regression Analysis. Douglas C. Montgomery

Introduction to Linear Regression Analysis

F_1,n−2 distribution. Appendix C.3 also shows that the expected values of these mean squares are

These expected mean squares indicate that if the observed value of F₀ is large, then it is likely that the slope β₁ ≠ 0. Appendix C.3 also shows that if β₁ ≠ 0, then F₀ follows a noncentral F distribution with 1 and n − 2 degrees of freedom and a non-centrality parameter of

This noncentrality parameter also indicates that the observed value of F₀ should be large if β₁ ≠ 0. Therefore, to test the hypothesis H₀: β₁ = 0, compute the test statistic F₀ and reject H₀ if

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

Example 2.4 The Rocket Propellant Data

We will test for significance of regression in the model developed in Example 2.1 for the rocket propellant data. The fitted model is in28-1 , 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 F₀ is 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 H₀: β₁ = 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.

More About the t Test

We noted in Section 2.3.2 that 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, in28-2 in Eq. (2.38) is identical to F₀ of the analysis-of-variance approach in Eq. (2.36). For example; in the rocket propellant example t₀ = −12.5, so in28-3 . 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 H₀: β₁ = 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 H₁: β₁ < 0 or H₁: β₁ > 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

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	F ₀	P value
Regression	1,527,334.95	1	1,527,334.95	165.21	1.66 × 10⁻¹⁰
Residual	166,402.65	18	9,244.59
Total	1,693,737.60	19

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 β₁ = 0 is a very important conclusion that is only aided by the t or F test. The inability to show that the slope is not statistically different from zero may not necessarily mean that y

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