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and
b
2
only appear with
a
1
(brief psychotherapy), locations
b
3
and
b
4
only
appear with
a
2
(cognitive-behavioral), and locations
b
5
and
b
6
only appear
with
a
3
(psychoanalytic).
Nested designs are commonly found in educational and animal
research, where subjects form small groups or blocks (e.g., classrooms
or cages) and the particular treatment is given to all members of the
block. In such a case, the classroom or cage is said to be nested in the
particular treatment being studied.
Keppel and Wickens (2004) note two important considerations con-
cerning nested designs. First, in a nested design it is not possible to inves-
tigate the
A
B
interactioneffectbecausenotallofthelevelsofFactor
B
occur under all of the levels of Factor
A
. Second, nested factors are
typically random factors and therefore require a different error term in
computing the
F
ratio than fixed factors. For more comprehensive discus-
sions of nested designs, we recommend Honeck, Kibler, and Sugar (1983),
Keppel (1991), Keppel and Wickens (2004), and Winer et al. (1991).
×
17.4 LATIN SQUARES
The term
Latin square
is said to be derived from an ancient word puzzle
game that focused on placing Latin letters in a matrix such that each letter
appeared only once in each row and column (Kirk, 1995). In practice,
Latin squares have two fundamental uses (Cardinal & Aitken, 2006). The
first use is to employ a Latin square as a research method or experimental
design technique to ensure that a nuisance variable (i.e., an uncontrolled
or incidental factor) such as time of day or type of experimenter is not
confounded with the treatment(s) under study. The second use controls
for the possible confounding influence of these nuisance variable(s) by
incorporating this information into the statistical analysis through an
adjustment of the subsequent error term.
Figure 17.4 displays a simple 4
4 Latin square, which we will use
to illustrate the design. Each Latin letter in the square denotes a sepa-
rate treatment condition (i.e.,
A
×
a
4
). Notice
how the square is configured such that every letter appears only once
in each row and column. This Latin square provides a researcher with a
“roadmap” for
counterbalancing
(or systematically balancing) the effects
of a nuisance variable in a within-subjects or repeated measures design.
For example, suppose that we are working with four adults in examining
the effects of type of music on subsequent recall. Participants listened to
either rock, pop, classical, or jazz music during four, five-minute study
=
a
1
,
B
=
a
2
,
C
=
a
3
,
D
=
A
B
C
D
Figure 17.4
Example of a 4
×
4 Latin square
B
D
A
C
C
A
D
B
design.
D
C
B
A
