
187Regression
When MS
E
(which is minimized by the least squares procedure) is very small, then R
2
approaches1. Note that R
2
is the fraction (or percent) of variance of Y explained by the regression
model. Notethat
1
2
2
−=
+
=R
M
MS
s
E
Y
=
MS
MS
MS
MS S
E
T
E
EM
(6.47)
The difference of R
2
with respect to 1 is the fraction of variance of Y unexplained by the regression.
In addition, it is important to realize that
R
MS
M
2
2
01
2
2
11
2
2
11
= =
//
s
nbbX Y
s
nYbX bX Y
s
YY Y
() ()+−
=
−+−
(6.48)
The sample mean of Y cancels, b
1
is a common factor, and recognizing the sample variance of X
we obtain
R
2
=
() () () ()11
1
22
2
1
2
1
2
nbXX
s
bn XX
s
bs
s
X
Y
−
=
−
=
(6.49)
And by reca