Biostatistics

436 CHAPTER 9 SIMPLE LINEAR REGRESSION AND CORRELATION
Y
Y
Y
X
X
X
(a)
(b)
(c)
FIGURE 9.4.6 Scatter diagrams showing (a) direct linear relationship, (b) inverse linear
relationship, and (c) no linear relationship between X and Y.
that there is an inverse linear relationship between X and Y. When there is no linear
relationship between X and Y, ^b 1 is equal to zero. These three situations are illustrated in
Figure 9.4.6.
The Test Statistic
known is
For testing hypotheses about b 1 the test statistic when s 2 yjx is
z ¼ ^b 1 ðb 1 Þ 0
(9.4.8)
s ^b 1
where ðb 1 Þ 0
is the hypothesized value of b 1 . The hypothesized value of b 1 does not have
to be zero, but in practice, more often than not, the null hypothesis of interest is that
b 1 ¼ 0.
As a rule s 2 yjx
is unknown. When this is the case, the test statistic is
t ¼ ^b 1 ðb 1 Þ 0
(9.4.9)
s^b 1
where s^b 1
is an estimate of s ^b 1
and t is distributed as Student’s t with n 2 degrees of
freedom.
If the probability of observing a value as extreme as the value of the test statistic
computed by Equation 9.4.9 when the null hypothesis is true is less than a=2 (since we have
a two-sided test), the null hypothesis is rejected.
EXAMPLE 9.4.2
Refer to Example 9.3.1. We wish to know if we can conclude that the slope of the
population regression line describing the relationship between X and Y is zero.
Solution:
1. Data. See Example 9.3.1.
2. Assumptions. We presume that the simple linear regression model and
its underlying assumptions are applicable.