Читать книгу Introduction to Linear Regression Analysis - Douglas C. Montgomery - Страница 37

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, 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 . 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 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 β₁ = 0.