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Keppel, G. & Wickens, T. D. Design and Analysis
Chapter 21: The Three-Factor Design: The Overall Analysis of Variance
• Appropriately, K&W focus on the three-way interaction in this chapter. It’s really the only
new concept in a three-way design.
21.1 Components of the Three-Way Design
• Just as you could think of a two-way (AxB) design as the melding of two one-way (A and B)
designs, so too can you think of the three-way (AxBxC) design as the melding of three oneway (A, B, and C) designs. Thus, you will be able to analyze the main effects of each of the
three factors. In addition, however, you will gain information about interactions among pairs
of factors (AxB, AxC, and BxC) as you would expect from the two-way design. The big
difference in the three-way design is that you will also be able to assess the three-way
(AxBxC) interaction. The three-way interaction is a slippery little critter, so we’ll spend some
time in trying to understand what it’s all about.
• Just as was true for the single-factor and two-factor independent groups designs, you need
to be concerned about the assumption of homogeneity of variance (so a Brown-Forsythe or
Levene test for homogeneity of variance would make sense). You also need to be concerned
about the independence of observations. So, all of their earlier discussions of assumptions
apply to this design.
21.2 The Three-way Interaction
• First, get a load of the jargon. You can think of the “new” effect in a three-way design as a
three-way interaction, as a second order interaction, as a triple, or as an AxBxC interaction.
• You could think of the three-way design as the combination of the two-way designs that
you’d find at each of the levels of the third factor. Thus, if you look at the 2x4x3 design that
K&W illustrate in Table 21.3, you will notice that one way to think of the three-way design
is that you have a 3x4 two-way design (e.g., factors B and C) at each of the two levels of A.
Another way to think of the three-way design is that you have a 2x3 two-way design (e.g.,
factors A and C) at each of the four levels of B. These internal interactions could be called
simple interactions or conditional interactions. To my eye, it’s much easier to think of the
data as existing in the smallest number of groups, but that’s just my own bias, the data could
be organized in any of the three ways that K&W illustrate in Table 21.1.
• K&W define the three-way interaction as:
A three-way interaction is present when the simple interactions of two
variables differ with levels of the third variable.
• Thus, when the three-way interaction is not significant, the simple interactions are all
essentially the same. Similarity can be thought of in terms of either form or size of effect. A
three-way interaction can occur if all the simple interactions are similar in effect size, but
differ in form. But a three-way interaction may also occur is all the simple interactions are
the same form, but differ in effect size.
• A three-way interaction can be present when the simple interactions are significant, but
differ in form (or effect). A three-way interaction can also be present with one simple
interaction is not significant, but another (or other) simple interactions are significant. It’s
K&W 21 - 1
also possible for a three-way interaction to arise when none of the simple interactions are
significant, but their forms differ. (Though it’s harder for me to imagine interpreting that
three-way interaction.)
Examples of Three-Way Interactions
• In an earlier edition of this text, Keppel described a three-way design in an experiment by
Wallace and Underwood (1964). Note that instead of being a pain in the butt, the three-way
interaction was actually important to the researchers. Factor A was the degree of strength of
word associations (high = e.g. fruit words like apple, peach, etc. and low = unrelated words
like fly, saw, snow, etc.). Factor B was the type of learning task (free recall vs. paired-word
task). Crossing these two factors resulted in 4 lists of words for testing. In one list, the highly
related (high strength of association) words were presented one at a time for later free recall.
In another list the highly related words were presented in pairs, to be learned as a pair. In the
third list, the unrelated (low strength of association) words were presented in a single list for
later free recall. In the final list, the unrelated words were presented in pairs. Wallace and
Underwood predicted that the related words would be better learned in a single list compared
to the unrelated words. For words learned in pairs, however, they predicted that the unrelated
words would be easier to learn than the related words. That is, they predicted a two-way
interaction. Oops...that’s only 2 factors. Factor C was a nonmanipulated factor of
intelligence/linguistic competence (college students vs. mental retardates). Thus, the
researchers predicted that the interaction would be present with the college students but not
with the mental retardates (which would be a three-way interaction). The dependent variable
was the number of trials to learn the lists to a criterion. The results are seen below:
The figure above shows that the interaction of Factors A and B is different for C1 (College
Students) and C2 (Mental Retardates), an outcome that illustrates a three-way interaction.
• In the current edition of the text, K&W use a study by Petty et al. (1981) to illustrate the
three-way interaction. Their study was also a 2x2x2 design. What factors would persuade a
student that a comprehensive exam should be given to all graduating seniors? To investigate
this question, Petty et al. used two types of argument (strong or weak). They also used two
levels of involvement (or impact on) the participant [low = another school long in the future
vs. high = their school next year]. Finally, the style of the argument was either regular (a
simple summary) or rhetorical (ending with a question for the reader). With n = 20 there
K&W 21 - 2
were 160 participants. The DV for this study was the participant’s rating of agreement with
the proposal to introduce the comprehensive exam. As in the preceding example, the
researchers were actually interested in producing a three-way interaction. As illustrated in the
figure from the text (seen below), when stated in the regular form, the strength of the
argument had little impact on the students who thought that they were hearing about another
institution in the distant future. However, if they thought that they were hearing an argument
for their own institution in the near future, strong arguments were more effective than weak
arguments. The picture is quite different when rhetorical questions are used. For the students
hearing rhetorical questions, if the arguments didn’t seem to involve them, they were more
likely to agree the introduction of a comprehensive exam when strong arguments were used
compared to weak arguments. For the students who were more directly affected by the
proposed change, however, there was a smaller advantage for the strong arguments over the
weak arguments.
• You may recall that in discussing simple effects for a two-way analysis, K&W noted that a
significant interaction implied that the simple effects must be different. At the same time,
differing simple effects did not imply that a two-way interaction was present. For the threeway analysis, they note that the three-way interaction implies that the simple two-way
interactions must differ. However, different two-way interactions does not imply that the
three-way interaction must be significant. K&W illustrate the principle in Figure 21.4, but
note that the analytic approach in this situation may not differ markedly from that when the
three-way interaction is significant.
K&W 21 - 3
• For additional practice, let’s consider some potential outcomes from a 3x2x2 design. I
won’t place these results within an experimental context, but will consider the three factors
A, B, and C. For each set of data, determine the effects that appear to be significant.
A1
5
6
B1
B2
Effect
C1
A2
2
6
A3
2
9
A1
5
6
Analysis
A
B
C
AxB
AxC
BxC
AxBxC
K&W 21 - 4
C2
A2
2
6
A3
2
6
A1
6
9
B1
B2
Effect
C1
A2
6
9
A3
6
9
A1
2
9
Analysis
A
B
C
AxB
AxC
BxC
AxBxC
K&W 21 - 5
C2
A2
4
7
A3
6
5
A1
1
9
B1
B2
Effect
C1
A2
6
4
A3
6
4
A1
9
1
Analysis
A
B
C
AxB
AxC
BxC
AxBxC
K&W 21 - 6
C2
A2
6
4
A3
6
4
21.3 Computational Procedures
• K&W illustrate (pp. 476-477) how to generate computational formulas for the three-way
ANOVA. My own suggestion would be to use a computer.  On the other hand, I do think
that it’s good to have a sense of the appropriate df involved in the analysis. In general, then,
the source table for the three-way independent groups design would be as follows:
Source
A
B
C
AxB
AxC
BxC
AxBxC
S/ABC
Total
SS
[A] – [T]
[B] – [T]
[C] – [T]
[AB] – [A] – [B] + [T]
[AC] – [A] – [C] + [T]
[BC] – [B] – [C] + [T]
[ABC] – [AB] –[AC] – [BC]
+ [A] + [B] + [C] – [T]
[Y] – [ABC]
[Y] – [T]
df
a–1
b–1
c–1
(a – 1) (b – 1)
(a – 1) (c – 1)
(b – 1) (c – 1)
(a – 1) (b – 1) (c – 1)
(a) (b) (c) (n – 1)
(a) (b) (c) (n) - 1
MS
SSA / dfA
SSB / dfB
SSC / dfC
SSAxB / dfAxB
SSAxC / dfAxC
SSBxC / dfBxC
SSAxBxC /
dfAxBxC
SSS/ABC / dfS/ABC
F
MSA / MSS/ABC
MSB / MSS/ABC
MSC / MSS/ABC
MSAxB / MSS/ABC
MSAxC / MSS/ABC
MSBxC / MSS/ABC
MSAxBxC /
MSS/ABC
• As you can see, the source table bears a striking resemblance to the one-way and two-way
independent groups designs. Each of the main effects stems from the variance of the group
means (and has df that would make sense in that regard). The error term (MSS/ABC) is simply
the average group variance from each of the unique groups. That pooling, of course, means
that it’s important to gain a sense of the extent to which you may have violated the
homogeneity of variance assumption. Thus, the Brown-Forsythe procedure makes sense. The
two-way interactions are conceptually identical to the interactions that you worked on for the
two-way ANOVA. The three-way interaction is different, so we’ll try to make some sense of
that effect.
• K&W provide a numerical example for a 3x3x2 design. Factor A is the type of Feedback
provided (None, Positive, or Negative). Factor B is the type of learning material (lowfrequency words with low emotional content, high-frequency words with low emotional
content, and high-frequency words with high emotional content). [Those two factors should
be familiar to you from a prior problem.] Factor C is the age of the participants (5th Grade
students vs. high school seniors).
• Although it will make the Data View less comprehensible, I’ll use numbers (and Value
Labels) to represent the levels of the factors as follows:
Factor
Levels
A (Feedback)
1 = None, 2 = Pos, 3 = Neg
B (Word Type) 1 = LoFreq/LoEm, 2 = HiFreq/LoEm, 3 = HiFreq/HiEm
C (Age)
1 = 5th Grade, 2 = Senior
K&W 21 - 7
Thus, the data (in part) in SPSS will look like the window below (left):
The analysis will proceed through the General Linear Model->Univariate, which will
produce the window above (right).
Your source table will look quite similar to the one on p. 479:
• To assess the possibility of heterogeneity of variance, you could compute the BrownForsythe test on the data. Because the B-F test is essentially a one-way ANOVA, I first create
a new variable to hold unique grouping information (which I called Group). Thus, each of the
18 unique conditions is labeled uniquely (1 – 18), which collapses the three-way design into
a one-way design. I then find the median for each of the groups and compute a new variable
(ztrans) that is the absolute value of the difference between each recall score and the median
for that group. Next, I compute a one-way ANOVA on the ztrans scores, with Group as the
factor:
K&W 21 - 8
As you can see, there is little concern about heterogeneity of variance. The same conclusion
would emerge from a Levene test on these data, though it’s much simpler to compute that
analysis (simply check the Homogeneity of Variance test option).
Thus, we would use the typical  = .05 for the overall ANOVA and use a pooled variance
estimate for any subsequent analyses. The mean data are seen in the figure below.
• From the source table, it is clear that there is a two-way interaction of Type of Word and
Age. Moreover, there are main effects for Feedback, Type of Word, and Age.
• Subsequent analyses of these data await Ch. 22.
21.4 Effect Size, Sample Size, and Power
Effect Size
• K&W again provide the formula for the partial omega squared, which seems more broadly
useful, in that you would use this approach when all three factors are experimental.
wˆ
2
<effect >
2
sˆ effect
= 2
2
sˆ effect + sˆ error
2
wˆ <effect
> =
(
(21.6)
)
df effect Feffect - 1
(
)
df effect Feffect - 1 + N
K&W 21 - 9
(21.7)
Where N is the total sample size (in this case abcn).
Thus, for the three-way interaction in the above example, your estimate would be:
2
wˆ <AxBxC
> =
4 ´ .324
= .014
(4 ´ .324) + 90
• If some of the factors are not experimental, K&W suggest using the complete effect size
approach or the semipartial effect size approach (pp. 481-482). Of course, the simplest
approach is to allow SPSS to compute partial eta squared for you. 
Sample Size
• The approach to estimating sample size is general, so the formulas in Ch. 11 still apply:
2
1- w <effect
>
2
(21.11)
N = f df effect + 1
2
(
)
w <effect >
Thus, for the three-way interaction here, to achieve power of ~.80 would put  at about 1.6.
Solving the equation would tell you that you needed about 900 total participants, or n = 50.
As K&W note, there are other approaches to raising power, which one might well consider in
this study (and generally).
Power
• The approach to estimating power is also general, which means that the procedures from
earlier chapters apply. As far as power goes, however, I’d simply trust SPSS to compute
power appropriately. In this case, SPSS estimated observed power for the interaction to be
.393.
K&W 21 - 10
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