Introduction to Linear Regression Analysis. Douglas C. Montgomery
estimate of the slope is
Therefore, the test statistic is
If we choose α = 0.05, the critical value of t is t0.025,18 = 2.101. Thus, we would reject H0: β1 = 0 and conclude that there is a linear relationship between shear strength and the age of the propellant.
Minitab Output
The Minitab output in Table 2.3 gives the standard errors of the slope and intercept (called “StDev” in the table) along with the t statistic for testing H0: β1 = 0 and H0: β0 = 0. Notice that the results shown in this table for the slope essentially agree with the manual calculations in Example 2.3. Like most computer software, Minitab uses the P-value approach to hypothesis testing. The P value for the test for significance of regression is reported as P = 0.000 (this is a rounded value; the actual P value is 1.64 × 10−10). Clearly there is strong evidence that strength is linearly related to the age of the propellant. The test statistic for H0: β0 = 0 is reported as t0 = 59.47 with P = 0.000. One would feel very confident in claiming that the intercept is not zero in this model.
2.3.3 Analysis of Variance
We may also use an analysis-of-variance approach to test significance of regression. The analysis of variance is based on a partitioning of total variability in the response variable y. To obtain this partitioning, begin with the identity
Squaring both sides of Eq. (2.31) and summing over all n observations produces
Note that the third term on the right-hand side of this expression can be rewritten as
since the sum of the residuals is always zero (property 1, Section 2.2.2) and the sum of the residuals weighted by the corresponding fitted value
The left-hand side of Eq. (2.32) is the corrected sum of squares of the observations, SST, which measures the total variability in the observations. The two components of SST measure, respectively, the amount of variability in the observations yi accounted for by the regression line and the residual variation left unexplained by the regression line. We recognize
Equation (2.32) is the fundamental analysis-of-variance identity for a regression model. Symbolically, we usually write
Comparing Eq. (2.33) with Eq. (2.18) we see that the regression sum of squares may be computed as
The degree-of-freedom breakdown is determined as follows. The total sum of squares, SST, has dfT = n − 1 degrees of freedom because one degree of freedom is lost as a result of the constraint
We can use the usual analysis-of-variance F test to test the hypothesis H0: β1 = 0. Appendix C.3 shows that (1) SSRes = (n − 2)MSRes/σ2 follows a
follows