Information Technology Reference
In-Depth Information
The next sum of squares (
SS
A
×
B
×
S
) is derived from all four of the matrices
in Table 11.2.
−
AB
2
n
−
AS
2
b
−
BS
2
a
A
2
(
b
)(
n
)
+
B
2
(
a
)(
n
)
Y
2
SS
A
×
B
×
S
=
+
S
2
(
a
)(
b
)
−
T
2
(
a
)(
b
)(
n
)
+
(1)
2
(1)
2
(5)
2
(3)
2
=
+
+···+
+
(6)
2
+
(24)
2
+
(45)
2
+
(8)
2
+
(27)
2
+
(30)
2
−
6
(11)
2
+
(13)
2
+···+
(11)
2
+
(6)
2
−
3
(3)
2
(2)
2
(12)
2
(9)
2
(75)
2
(65)
2
(3)(6)
+
+···+
+
+
−
+
2
(14)
2
(51)
2
(75)
2
+
+
+
(2)(6)
(22)
2
(27)
2
(21)
2
(13)
2
(140)
2
(2)(3)(6)
+
+···+
+
+
−
(2)(3)
=
.
3.28
(11.7)
Lastly, from the
ABS
matrix of Table 11.2, we derive the
SS
T
.
T
2
(
a
)(
b
)(
n
)
Y
2
SS
T
=
−
(140)
2
(2)(3)(6)
(1)
2
(1)
2
(5)
2
(3)
2
=
+
+···+
+
−
=
233.56
.
(11.8)
All of these sums of squares have been conveniently entered into an
ANOVA summary table (see Table 11.3).
11.6.2 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
=
2
−
1
=
1
df
B
=
−
=
−
=
b
1
3
1
2
df
A
×
B
=
(
a
−
1)(
b
−
1)
=
(2
−
1)(3
−
1)
=
(1)(2)
=
2
df
S
=
n
−
1
=
6
−
1
=
5
