the additional assumption that the model errors εi are normally distributed. Thus, the complete assumptions are that the errors are normally and independently distributed with mean 0 and variance σ2, abbreviated NID(0, σ2). In Chapter 4 we discuss how these assumptions can be checked through residual analysis.
2.3.1 Use of t Tests
Suppose that we wish to test the hypothesis that the slope equals a constant, say β10. The appropriate hypotheses are
where we have specified a two-sided alternative. Since the errors εi are NID(0, σ2), the observations yi are NID(β0 + β1xi, σ2). Now
is distributed N(0, 1) if the null hypothesis H0: β1 = β10 is true. If σ2 were known, we could use Z0 to test the hypotheses (2.23). Typically, σ2 is unknown. We have already seen that MSRes is an unbiased estimator of σ2. Appendix C.3 establishes that (n − 2)MSRes/σ2 follows a
follows a tn−2 distribution if the null hypothesis H0: β1 = β10 is true. The degrees of freedom associated with t0 are the number of degrees of freedom associated with MSRes. Thus, the ratio t0 is the test statistic used to test H0: β1 = β10. The test procedure computes t0 and compares the observed value of t0 from Eq. (2.24) with the upper α/2 percentage point of the tn−2 distribution (tα/2,n−2). This procedure rejects the null hypothesis if
(2.25)
Alternatively, a P-value approach could also be used for decision making.
The denominator of the test statistic, t0, in Eq. (2.24) is often called the estimated standard error, or more simply, the standard error of the slope. That is,
(2.26)
Therefore, we often see t0 written as
(2.27)
A similar procedure can be used to test hypotheses about the intercept. To test
(2.28)
we would use the test statistic
(2.29)
where
2.3.2 Testing Significance of Regression
A very important special case of the hypotheses in Eq. (2.23) is
(2.30)
These hypotheses relate to the significance of regression. Failing to reject H0: β1 = 0 implies that there is no linear relationship between x and y. This situation is illustrated in Figure 2.2. Note that this may imply either that x is of little value in explaining the variation in y and that the best estimator of y for any x is
Alternatively, if H0: β1 = 0 is rejected, this implies that x is of value in explaining the variability in y. This is illustrated in Figure 2.3. However, rejecting H0: β1 = 0 could mean either that the straight-line model is adequate (Figure 2.3a) or that even though there is a linear effect of x, better results could be obtained with the addition of higher order polynomial terms in x (Figure 2.3b).
Figure 2.2 Situations where the hypothesis H0: β1 = 0 is not rejected.
