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have been summed and squared and means and standard deviations were
computed in the usual manner. Two additional matrices are needed to
provide the ingredients for the computations involved in producing sums
of squares. An
AB
matrix of sums is generated by collapsing subjects' scores
across each
AB
treatment combination. An
AS
matrix is produced by col-
lapsing each subject's total score across the levels of the within-subjects
variable (Factor
B
). An
AB
matrix of means provides the researcher
with an efficient way of scrutinizing the main effect and interaction
dynamics.
13.4.1 SUM OF SQUARES
There are six sums of squares that we will calculate for a simple mixed
design ANOVA, all of which are produced by manipulating the scores or
sums in the
ABS
,
AB
,and
AS
matrices.
Sum of squares between groups (
SS
A
).
Sum of squares subjects within the
A
treatments (
SS
S
/
A
). This pro-
vides the error component to evaluate the
A
main effect or between-
subject portion.
Sum of squares within groups (
SS
B
).
Sum of squares
A
by
B
interaction effect (
SS
A
×
B
).
Sum of squares repeated measures error term (
SS
B
×
S
/
A
). (Read
“sum of squares
B
by
S
over
A.
”ItreflectstheFactor
B
by sub-
ject interaction at each level of Factor
A
; see Keppel, 1991.) This is
theerrortermforevaluatingthe
B
main effect and
A
×
B
interaction
effect.
Sum of squares total (
SS
T
).
The formulas for these six sums of squares and the summary of their
calculations based on data from Table 13.1 are as follows:
A
2
(
b
)(
n
)
−
T
2
(
a
)(
b
)(
n
)
SS
A
=
(35)
2
(66)
2
(32)
2
(133)
2
(3)(2)(5)
+
+
=
−
(2)(5)
=
660
.
50
−
589
.
63
=
70.87
(13.1)
AS
2
b
A
2
(
b
)(
n
)
SS
S
/
A
=
−
(5)
2
+
(8)
2
+···+
(5)
2
+
(6)
2
(35)
2
+
(66)
2
+
(32)
2
=
−
2
(2)(5)
=
675
.
50
−
660
.
50
=
15.00
(13.2)
