• 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
K&W 21 - 1

interaction is not significant, but another (or other) simple interactions are significant. It’s 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 C
1
(College
Students) and C
2
(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
K&W 21 - 2

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 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.
C1
10
8
6 b2
4
2
0 a1
B1
B2
Effect
B
A
C
AxB
AxC
BxC
AxBxC
A1
5
6 a2
A a3
C1
A2
2
6 b1
A3
2
9
6
4
2
0
10
8 a1
A1
5
6
Analysis
C2
A2
2
6
C2 b2 b1 a2
A
A3
2
6 a3
K&W 21 - 4

K&W 21 - 5

C1
10
8
6 b2 b1
4
2
0 a1
B1
B2
Effect
B
A
C
AxB
AxC
BxC
AxBxC
A1
6
9 a2
A a3
C1
A2
6
9
A3
6
9
C2
10
8
6
4
2
0 b1 b2
A1
2
9 a1
C2
A2
4
7 a2
A a3
A3
6
5
Analysis
K&W 21 - 6

K&W 21 - 7

C1
10
8
6
4
2
0 a1
B1
B2
Effect
B
A
C
AxB
AxC
BxC
AxBxC
A1
1
9 a2
A
C1
A2
6
4 a3 b1 b2
A3
6
4
A1
9
1
C2
6
4
2
0
10
8 b1 b2
C2
A2
6
4 a1 a2
A a3
A3
6
4
Analysis
K&W 21 - 8

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
[ A ] – [ T ]
SS
[ B ] – [ T ]
[ C ] – [ T ]
[ AB ] – [ A ] – [ B ] + [ T ]
AxC
BxC [
[ AC
BC
] – [
] – [
A
B
] – [
] – [
C
C
] + [
] + [
T
T
]
]
AxBxC [ ABC ] – [ AB ] –[ AC ] – [ BC ]
+ [ A ] + [ B ] + [ C ] – [ T ] a – 1 df MS
SS
A
/ df
A
F
MS
A
/ MS
S/ABC
(
(
( b c a
– 1
– 1
– 1) ( b – 1)
( a – 1) ( c – 1)
( b – 1) ( c – 1)
( a – 1) ( b – 1) ( c – 1) SS
AxBxC df
AxBxC a a
) (
) ( b b
) (
) ( c c
) (
) ( n n
– 1)
) - 1
SS
B
/ df
B
SS
C
/ df
C
SS
AxB
/ df
AxB
SS
SS
AxC
BxC
/ df
/ df
/
AxC
BxC
SS
S/ABC
/ df
S/ABC
MS
B
MS
C
/ MS
S/ABC
/ MS
MS
AxBxC
MS
S/ABC
/
S/ABC
MS
AxB
/ MS
S/ABC
MS
AxC
/ MS
S/ABC
MS
BxC
/ MS
S/ABC
S/ABC [ Y ] – [ ABC ]
Total [ Y ] – [ T ]
• 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 ( MS
S/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 (5 th Grade students vs. high school seniors).
• Although it will make the output less comprehensible, I’ll use numbers 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 = 5 th Grade, 2 = Senior
K&W 21 - 9

Thus, the data (in part) in PASW 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 Brown-
Forsythe 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 - 10

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.

2
 effect


 2
 2 effect effect
  2 error
(21.6)


2
 effect

 df df effect effect


F effect
F effect

1

1



N
(21.7)
K&W 21 - 11


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

AxBxC


4

.324
(4

.324)

90

.014
• If some of the factors are not experimental, K&W suggest using the complete effect size approach is to allow PASW 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:
N
 
2
 df effect

1
 1



2
2
 effect

 effect

(21.11)
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 PASW to compute power appropriately. In this case, PASW estimated observed power for the interaction to be
.393.
K&W 21 - 12