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full factorial design and its constituent components that comprise the
simple effects analyses. In Figure 8.5, the full factorial arrangement of
the two independent variables (residential community size and gender)
is broken down into two single factor analyses of residential community
size, collapsed across or conducted at each level of gender (
b
1
: females and
b
2
: males). Such an analysis is referred to as the simple effects analysis of
Factor
A
at
b
1
and Factor
A
at
b
2
. An alternative approach, which we will
not describe in detail, is to conduct simple effects analyses by collapsing
across levels of Factor
A
.Suchanapproachwouldbereferredtoasthe
simple effects analysis of Factor
B
at
a
1
,Factor
B
at
a
2
, and Factor
B
at
a
3
. Either simple effects approach (
A
at
b
k
or
B
at
a
j
)willprovidethe
researcher with the necessary information to interpret the interaction.
Computational details on the latter approach can be found in Keppel et al.
(1992) and Keppel and Wickens (2004).
Let us now turn our attention to conducting the simple effects of
Factor
A
at
b
1
and
A
at
b
2
based on our previous numerical example
examining loneliness as a function of residential community size and
gender. Note that we will be modifying slightly our formulas for sum of
squares, degrees of freedom, and mean squares because we are collapsing
our analyses across each level of Factor
B
(i.e.,
b
1
and
b
2
)toconducttwo
single-factor ANOVAs.
The following computational steps will be based on the sums found
in the
AB
Matrix of Sums in Table 8.3. We begin with the first row of
sums (
b
1
) to conduct the simple effects of
A
at
b
1
, and then proceed to the
second row of sums (
b
2
) to conduct the simple effects of
A
at
b
2
.
8.8.1 SIMPLE EFFECTS OF
A
AT
b
1
(
AB
j
1
)
2
n
(
B
1
)
2
(
a
)(
n
)
SS
A
at
b
1
=
−
(75)
2
(190)
2
(210)
2
(475)
2
(3)(5)
+
+
=
−
(5)
85,825
5
225,625
15
=
−
=
17,115
−
15,041.67
=
2,123.33
(8.6)
df
A
at
b
1
=
a
−
1
=
3
−
1
=
2
(8.7)
SS
A
at
b
1
df
A
at
b
1
=
2,123.33
2
MS
A
at
b
1
=
=
1,061.67
(8.8)
MS
A
at
b
1
MS
S
/
AB
=
1,061.67
36
F
A
at
b
1
=
=
28.83
.
(8.9)
.
83
(2, 24). The
critical value of
F
(see Appendix C) at the 5 percent level is 3.40. Because
We evaluate our
F
A
at
b
1
with
df
=
df
A
at
b
1
,and
df
S
/
AB
=
