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In-Depth Information
Y
aj
−
A
2
n
−
S
2
a
T
2
(
a
)(
n
)
SS
A
×
S
=
+
=
2, 147
−
2, 069
.
88
−
1, 976
.
60
+
1, 946
.
03
=
46.55 ,
(10.3)
T
2
(
a
)(
n
)
Y
aj
−
SS
T
=
=
2, 147
−
1, 946
.
03
=
200.97
.
(10.4)
As you will see Sections 10.12.3 and 10.12.4 (see Figures 10.13 and 10.14),
the SPSS output for the sum of squares (under
Type III Sum of Squares
)
directly corresponds to our final hand computations. Specifically,
SS
A
and
SS
A
×
S
can be seen in Figure 10.13 under the sources labeled
PREPOST
and
Error (PREPOST)
, respectively. Likewise,
SS
S
can be found in Figure
10.14 under the source labeled
Error
. We will explain these figures in more
detail at the end of this chapter.
10.9.2 CALCULATING DEGREES OF FREEDOM
Below are the formulas for the degrees of freedom associated with each
sum of squares, and the simple computations involved based on our
numerical example:
df
A
=
a
−
1
=
5
−
1
=
4
df
S
=
−
=
−
=
n
1
8
1
7
df
A
×
S
=
(
a
−
1)(
n
−
1)
=
(5
−
1)(8
−
1)
=
(4)(7)
=
28
df
T
=
(
a
)(
n
)
−
1
=
(5)(8)
−
1
=
40
−
1
=
39
.
10.9.3 CALCULATING MEAN SQUARES AND
F
RATIO
We calculate three mean squares by dividing each sum of squares by its
respective degrees of freedom. The
F
ratio is formed by dividing the mean
square treatments by the error component mean square treatments by
subjects. The calculation of these mean squares and the
F
ratio follows:
SS
A
df
A
=
123
.
85
MS
A
=
=
30.96
(10.5)
4
SS
S
df
S
=
30
57
7
.
MS
S
=
=
4.37
(10.6)
SS
A
×
S
df
A
×
S
=
46
55
28
.
MS
A
×
S
=
=
1.66
(10.7)
MS
A
MS
A
×
S
=
30
.
96
F
=
=
18.65
.
(10.8)
1
.
66
