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significant, we would then perform one-way ANOVAs to further explicate
this simple interaction. That is, we would compare the loneliness means
of large and small communities for females, and then repeat that anal-
ysis for males. Because there would be only two groups involved in any
one analysis, a statistically significant
F
ratio would indicate that the two
means were reliably different.
As Keppel et al. (1992) note, “most large factorial experiments can
be profitably transformed into a number of interaction contrasts, each
one focusing on a different aspect of the
A
×
B
interaction” (p. 307).
×
For example, in the present study, our full 3
2 factorial can be broken
into three simpler 2
×
2 analyses:
A
(female, male)
×
B
(large, small),
B
(large,
rural). Each of these minifactorial designs provides a separate piece of
the interaction mosaic puzzle. Under a strategy of performing the 2
A
(female, male)
×
B
(small, rural), and
A
(female, male)
×
2
interaction contrasts, we would then perform one-way ANOVAs following
up on any statistically significant interaction effects to determine where
the group differences were.
The computational steps involved in calculating the necessary sums
of squares, degrees of freedom, mean squares, and
F
ratios are beyond
the scope of the present chapter, but are fairly straightforward and are
introduced in Keppel et al. (1992). For more advanced and comprehensive
treatment of this topic, see Boik (1979), Keppel (1991, Chapter 12), Keppel
and Wickens (2004, Chapter 13), and Winer et al. (1992, Chapter 8).
×
17.2 FIXED AND RANDOM FACTORS
Throughout this text, we have been focusing on various ANOVA designs
that can be construed as
completely randomized designs
; that is, equal num-
bers of participants (
n
) are randomly assigned to the various treatments
or treatment combinations. These designs are typically assessed with the
within-groups mean square (
MS
S
/
A
or
MS
S
/
AB
, etc.) as the error term
in the
F
ratio. The use of such an error term is based on a statistical
model called the fixed effects model. This model is sometimes contrasted
with other statistical models, less often used in the social and behav-
ioral sciences, called random effects and mixed effects models. We briefly
mentioned in Section 10.16 that
SAS Enterprise Guide
treated the subject
identification variable as a random effect; here we expand somewhat on
that discussion.
The distinction among fixed effects, random effects, and mixed effects
models is based on how a researcher identifies the levels of the indepen-
dent variables in an experiment or research project. We consider each in
turn.
17.2.1 FIXED EFFECTS
Fixed effects
or
factors
uselevelsoftheindependentvariablethatare
selected purposefully, rationally, and systematically. For example, fixed
