Now consider the deviation of the observed responses from their fitted values based on
the regression model:
i
i
i
i
i
e
X
b
b
Y
Y
Y
=
+

=

)
(
1
0
^
. When these terms are squared and
summed up, this is referred to as the
error sum of squares (SSE)
. We’ve already
encounterd this quantity and used it to estimate the error variance.
∑
=

=
n
i
i
i
Y
Y
SSE
1
2
^
)
(
When the observed responses fall close to the regression line, SSE will be small. When
the data are not near the line, SSE will be large.
Finally, there is a third quantity, representing the deviations of the predicted values from
the mean. Then these deviations are squared and summed up, this is referred to as the
regression sum of squares
(
SSR)
.
∑
=

=
n
i
i
Y
Y
SSR
1
2
^
)
(
The error and regression sums of squares sum to the total sum of squares:
SSE
SSR
SSTO
+
=
which can be seen as follows:
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SSR
SSE
Y
Y
Y
Y
Y
Y
Y
Y
e
Y
X
e
b
e
b
Y
Y
Y
Y
Y
X
b
b
e
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
SSTO
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
n
i
i
n
i
i
i
n
i
i
n
i
i
i
n
i
n
i
i
i
i
n
i
i
n
i
i
n
i
i
i
n
i
i
i
n
i
i
n
i
i
i
n
i
i
i
i
n
i
i
n
i
i
i
n
i
i
i
i
i
i
i
n
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
+
=

+

=
+

+

=

+
+

+

=

+
+

+

=


+

+

=


+

+

=

=
⇒


+

+

=

+

=

⇒

+

=

+

=

∑
∑
∑
∑
∑
∑
∑
∑
∑
∑
∑
∑
∑
∑
∑
∑
∑
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
1
2
^
1
2
^
1
2
^
1
2
^
1
1
1
1
0
1
2
^
1
2
^
1
1
0
1
2
^
1
2
^
1
^
^
1
2
^
1
2
^
1
^
^
2
^
2
^
1
2
^
^
2
^
2
^
2
^
^
2
^
^
^
^
)
(
)
(
)
0
(
2
)
(
)
(
2
)
(
)
(
)
(
2
)
(
)
(
)
)(
(
2
)
(
)
(
)
)(
(
2
)
(
)
(
)
(
)
)(
(
2
)
(
)
(
)]
(
)
[(
)
(
)
(
)
(
The last term was 0 since
∑
∑
=
=
0
i
i
i
X
e
e
,
Each sum of squares has associated with
degrees of freedom
. The total degrees of
freedom is
df
T
=
n
1. The error degrees of freedom is
df
E
=
n
2. The regression degrees of
freedom is
df
R
= 1. Note that the error and regression degrees of freedom sum to the total
degrees of freedom:
)
2
(
1
1

+
=

n
n
.
Mean squares are the sums of squares divided by their degrees of freedom:
2
1

=
=
n
SSE
MSE
SSR
MSR
Note that
MSE
was our estimate of the error variance, and that we don’t compute a total
mean square. It can be shown that the expected values of the mean squares are:
∑
=

+
=
=
n
i
i
X
X
MSR
E
MSE
E
1
2
2
1
2
2
)
(
}
{
}
{
β
σ
σ
Note that these expected mean squares are the same if and only if
β
1
=0.
The Analysis of Variance is reported in tabular form:
Source
df
SS
MS
F
Regression
1
SSR
MSR=SSR/1
F=MSR/MSE
Error
n
2
SSE
MSE=SSE/(
n
2)
C Total
n
1
SSTO
F
Test of
β
1
= 0 versus
β
1
≠
0
As a result of Cochran’s Theorem (stated on page 76 of text book), we have a test of
whether the dependent variable
Y
is linearly related to the predictor variable
X
. This is a
very specific case of the
t
test described previously. Its full utility will be seen when we
consider multiple predictors. The test proceeds as follows:
•
Null hypothesis:
0
:
1
0
=
β
H
•
Alternative (Research) Hypothesis:
0
:
1
≠
β
A
H
•
Test Statistic:
MSE
MSR
F
TS
=
*
:
•
Rejection Region:
)
2
,
1
;
1
(
*
:


≥
n
F
F
RR
α
•
P
value:
*}
)
2
,
1
(
{
F
n
F
P
≥

Critical values of the
F
distribution (indexed by numerator and denominator degrees’ of
freedom) are given in Table B.4, pages 13401345.