2.2 General linear tests  REDUCED vs. FULL MODEL
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HYONJUNG KIM, 2017
If you want to test whether
β
3
is needed, given that
β
1
and
β
2
are already in the model, then
the
extra sums of squares
are defined as
R(
X
3

X
1
, X
2
) = SSE(
X
1
, X
2
)
−
SSE(
X
1
, X
2
, X
3
).
The “R” stands for the reduction in the error sums of squares due to fitting the additional
term. The
F
∗
test for
β
3
= 0 is
F
∗
=
R(
X
3

X
1
, X
2
)
/
(
df
red
−
df
full
)
MSE(full)
=
(SSE(reduced)
−
SSE(full))
/
(
df
red
−
df
full
)
MSE(full)
which is the same test as we had before but expressed in different notation.
The extra sums of squares can also be written as a difference of regression sums of squares.
For example,
R(
X
3

X
1
, X
2
) = SSE(
X
1
, X
2
)
−
SSE(
X
1
, X
2
, X
3
)
= [SST
−
SSR(
X
1
, X
2
)]
−
[SST
−
SSE(
X
1
, X
2
, X
3
) ]
= SSR(
X
1
, X
2
, X
3
)
−
SSR(
X
1
, X
2
)
Sequential Sums of Squares
or “Type I sums of squares” (in SAS)
They are extra sums of squares for adding each term given that preceding terms in the model
statement are already in the model. For the model with 3 independent variables: the Type
I sums of squares are (in SAS MODEL Y=
X
1
X
2
X
3
;) These sequential sums of squares
SOURCE
df
SS
X
1
1
R(
X
1
)
X
2
1
R(
X
2

X
1
)
X
3
1
R(
X
3

X
1
, X
2
)
error
n
−
4
SSE(
X
1
, X
2
, X
3
)
add up to SSR(
X
1
, X
2
, X
3
).
R(
X
1
)+ R(
X
2

X
1
)+ R(
X
3

X
1
, X
2
)
=SSR(
X
1
)+ SSR(
X
1
, X
2
)
−
SSR(
X
1
)+ SSR(
X
1
, X
2
, X
3
)
−
SSR(
X
1
, X
2
) =SSR(
X
1
, X
2
, X
3
)
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