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430 CHAPTER 9 SIMPLE LINEAR REGRESSION AND CORRELATION
TABLE 9.4.1
ANOVA Table for Simple Linear Regression
Source of
Variation SS d.f. MS V.R.
Linear regression SSR 1 MSR SSR/1 MSR/MSE
Residual SSE n 2 MSE SSE/(n 2)
Total SST n 1
4. Test statistic. The test statistic is V.R. as explained in the discussion
that follows.
From the three sums-of-squares terms and their associated degrees
of freedom the analysis of variance table of Table 9.4.1 may be constructed.
In general, the degrees of freedom associated with the sum of
squares due to regression is equal to the number of constants in the regression
equation minus 1. In the simple linear case we have two estimates,
and ; hence the degrees of freedom for regression are 2 - 1 = 1.
b 0
b 1
5. Distribution of test statistic. It can be shown that when the hypothesis
of no linear relationship between X and Y is true, and when the assumptions
underlying regression are met, the ratio obtained by dividing the
regression mean square by the residual mean square is distributed as F
with 1 and n - 2 degrees of freedom.
6. Decision rule. Reject H 0 if the computed value of V.R. is equal to or
greater than the critical value of F.
7. Calculation of test statistic. As shown in Figure 9.3.2, the computed
value of F is 217.28.
8. Statistical decision. Since 217.28 is greater than 3.94, the critical value
of F (obtained by interpolation) for 1 and 107 degrees of freedom, the
null hypothesis is rejected.
9. Conclusion. We conclude that the linear model provides a good fit to
the data.
10. p value. For this test, since 217.28 7 8.25, we have p 6 .005. ■
Estimating the Population Coefficient of Determination The
sample coefficient of determination provides a point estimate of r 2 the population coefficient
of determination. The population coefficient of determination, r 2 has the same
function relative to the population as r 2 has to the sample. It shows what proportion of
the total population variation in Y is explained by the regression of Y on X. When the
number of degrees of freedom is small, r 2 is positively biased. That is, r 2 tends to be
large. An unbiased estimator of r 2 is provided by
r ~2 = 1 - g1y i - yN i 2 2 >1n - 22
g1y i - y2 2 >1n - 12
(9.4.3)