22Chapter Three-Way ANOVA - NYU Psychology

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688 Chapter 22 • Three-Way ANOVA
Three-Way ANOVA
You will need to use the following from previous chapters:
Symbols
22 k:Number of independent groups in a one-way ANOVA
c:Number of levels (i.e., conditions) of an RM factor
n:Number of subjects in each cell of a factorial ANOVA
N :Total number of observations in an experiment
T
Formulas
Formula 16.2: SS (by subtraction) also Formulas 16.3, 16.4, 16.5
inter
Formula 14.3: SS or one of its components
bet
C h a p t e r Concepts
Advantages and disadvantages of the RM ANOVA
SScomponents of the one-way RM ANOVA
SScomponents of the two-way ANOVA
Interaction of factors in a two-way ANOVA
A
SofarIhavecoveredtwotypesoftwo-wayfactorialANOVAs:two-wayinde-
pendent(Chapter14)andthemixeddesignANOVA(Chapter16).Thereisonly
onemoresimpletwo-wayANOVAtodescribe:thetwo-wayrepeatedmeasures
CONCEPTUAL design. [There are other two-way designs, such as those including random-
FOUNDATION effectsornestedfactors,buttheyarenotcommonlyused—seeHays(1994)for
a description of some of these.] Just as the one-way RM ANOVA can be
describedintermsofatwo-wayindependent-groupsANOVA,thetwo-wayRM
ANOVAcanbedescribedintermsofathree-wayindependent-groupsANOVA.
Thisgivesmeareasontodescribethelatterdesignnext.Ofcourse,thethree-
wayfactorialANOVAisinterestinginitsownright,anditsfrequentuseinthe
psychologicalliteraturemakesitanimportanttopictocover,anyway.Iwilldeal
withthethree-wayindependent-groupsANOVAandthetwo-wayRMANOVA
inthissectionandthetwotypesofthree-waymixeddesignsinSectionB.
Computationally, the three-way ANOVA adds nothing new to the proce-
dure you learned for the two-way; the same basic formulas are used a greater
number of times to extract a greater number of SS components from SS
total
(eight SSs for the three-way as compared with four for the two-way). However,
anytime you include three factors, you can have a three-way interaction, and
that is something that can get quite complicated, as you will see. To give you a
manageable view of the complexities that may arise when dealing with three
factors, I’ll start with a description of the simplest case: the 2 ×2 ×2 ANOVA.
A Simple Three-Way Example
At the end of Section B in Chapter 14, I reported the results of a published
study, which was based on a 2 × 2 ANOVA. In that study one factor con-
trasted subjects who had an alcohol-dependent parent with those who did
not. I’ll call this the alcohol factor and its two levels, at risk (of codepen-
dency) and control. The other factor (the experimenter factor) also had two
levels; in one level subjects were told that the experimenter was an exploitive
person, and in the other level the experimenter was described as a nurturing
person. All of the subjects were women. If we imagine that the experiment
688 was replicated using equal-sized groups of men and women, the original
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two-way design becomes a three-way design with gender as the third factor.
We will assume that all eight cells of the 2 ×2 ×2 design contain the same
number of subjects. As in the case of the two-way ANOVA, unbalanced three-
way designs can be difficult to deal with both computationally and concep-
tually and therefore will not be discussed in this chapter (see Chapter 18,
section A). The cell means for a three-factor experiment are often displayed
in published articles in the form of a table, such as Table 22.1.
Nurturing Exploitive Row Mean Table 22.1
Control: Men 40 28 34
Women 30 22 26
Mean 35 25 30
At risk: Men 36 48 42
Women 40 88 64
Mean 38 68 53
Column mean 36.5 46.5 41.5
Graphing Three Factors
The easiest way to see the effects of this experiment is to graph the cell
means. However, putting all of the cell means on a single graph would not be
an easy way to look at the three-way interaction. It is better to use two
graphs side by side, as shown in Figure 22.1. With a two-way design one has
to decide which factor is to be placed along the horizontal axis, leaving the
other to be represented by different lines on the graph. With a three-way
design one chooses both the factor to be placed along the horizontal axis and
the factor to be represented by different lines, leaving the third factor to be
represented by different graphs. These decisions result in six different ways
that the cell means of a three-way design can be presented.
Let us look again at Figure 22.1. The graph for the women shows the two-
way interaction you would expect from the study on which it is based. The
graph for the men shows the same kind of interaction, but to a considerably
lesser extent (the lines for the men are closer to being parallel). This difference
Figure 22.1
Women Men
At risk
Graph of Cell Means for
80 80 Data in Table 22.1
70 70
60 60
50 50
At risk
40 40
30 30
Control
Control
20 20
0 Nurturing Exploitive 0 Nurturing Exploitive
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690 Chapter 22 • Three-Way ANOVA
in amount of two-way interaction for men and women constitutes a three-way
interaction. If the two graphs had looked exactly the same, the Fratio for the
three-way interaction would have been zero. However, that is not a necessary
condition. A main effect of gender could raise the lines on one graph relative
to the other without contributing to a three-way interaction. Moreover, an
interaction of gender with the experimenter factor could rotate the lines on
one graph relative to the other, again without contributing to the three-way
interaction. As long as the difference in slopes (i.e., the amount of two-way
interaction) is the same in both graphs, the three-way interaction will be zero.
Simple Interaction Effects
Athree-wayinteractioncanbedefinedintermsofsimpleeffectsinawaythat
isanalogoustothedefinitionofatwo-wayinteraction.Atwo-wayinteraction
isadifferenceinthesimplemaineffectsofoneofthevariablesasyouchange
levelsoftheothervariable(ifyoulookatjustthegraphofthewomeninFig-
ure 22.1, each line is a simple main effect). In Figure 22.1 each of the two
graphscanbeconsideredasimpleeffectofthethree-waydesign—morespecif-
ically,asimpleinteractioneffect.Eachgraphdepictsthetwo-wayinteraction
ofalcoholandexperimenteratonelevelofthegenderfactor.Thethree-way
interactioncanbedefinedasthedifferencebetweenthesetwosimpleinterac-
tioneffects.Ifthesimpleinteractioneffectsdiffersignificantly,thethree-way
interactionwillbesignificant.Ofcourse,itdoesn’tmatterwhichofthethree
variablesischosenastheonewhosedifferentlevelsarerepresentedasdiffer-
entgraphs—ifthethree-wayinteractionisstatisticallysignificant,therewillbe
significantdifferencesinthesimpleinteractioneffectsineachcase.
Varieties of Three-way Interactions
Just as there are many patterns of cell means that lead to two-way interac-
tions (e.g., one line is flat while the other goes up or down, the two lines go
in opposite directions, or the lines go in the same direction but with differ-
ent slopes), there are even more distinct patterns in a three-way design. Per-
haps the simplest is when all of the means are about the same, except for
one, which is distinctly different. For instance, in our present example the
results might have shown no effect for the men (all cell means about 40), no
difference for the control women (both means about 40), and a mean of 40
for at-risk women exposed to the nice experimenter. Then, if the mean for at-
risk women with the exploitive experimenter were well above 40, there
would be a strong three-way interaction. This is a situation in which all three
variables must be at the “right” level simultaneously to see the effect—in this
variation of our example the subject must be female andraised by an alco-
hol-dependent parent andexposed to the exploitive experimenter to attain a
high score. Not only might the three-way interaction be significant, but one
cell mean might be significantly different from all of the other cell means,
making an even stronger case that all three variables must be combined
properly to see any effect (if you were sure that this pattern were going to
occur, you could test a contrast comparing the average of seven cell means
to the one you expect to be different and not bother with the ANOVA at all).
More often the results are not so clear-cut, but there is one cell mean
that is considerably higher than the others (as in Figure 22.1). This kind of
pattern is analogous to the ordinal interaction in the two-way case and tends
to cause all of the effects to be significant. On the other hand, a three-way
interaction could arise because the two-way interaction reverses its pattern
when changing levels of the third variable (e.g., imagine that in Figure 22.1
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the labels of the two lines were reversed for the graph of men but not for the
women). This is analogous to the disordinal interaction in the two-way case.
Or, the two-way interaction could be strong at one level of the third variable
and much weaker (or nonexistent) at another level. Of course, there are
many other possible variations. And consider how much more complicated
the three-way interaction can get when each factor has more than two levels
(we will deal with a greater number of levels in Section B).
Fortunately, three-way (between-subjects) ANOVAs with many levels for
each factor are not common. One reason is a practical one: the number of
subjects required. Even a design as simple as a 2 ×3 ×4 has 24 cells (to find
the number of cells, you just multiply the numbers of levels). If you want to
have at least 5 subjects per cell, 120 subjects are required. This is not an
impractical study, but you can see how quickly the addition of more levels
would result in a required sample size that could be prohibitive.
Main Effects
Inadditiontothethree-wayinteractiontherearethreemaineffectstolookat,
oneforeachfactor.Tolookatthegendermaineffect,forinstance,justtakethe
averageofthescoresforallofthemenandcompareittotheaverageofallof
thewomen.Ifyouhavethecellmeanshandyandthedesignisbalanced,you
canaverageallofthecellmeansinvolvingmenandthenallofthecellmeans
involvingwomen.InTable22.1,youcanaveragethefourcellmeansforthe
men(40,28,36,48)toget38(alternatively,youcouldusetherowmeansinthe
extremerightcolumnandaverage34and42togetthesameresult).Theaver-
ageforthewomen(30,22,40,88)is45.Themeansfortheothermaineffects
havealreadybeenincludedinTable22.1.Lookingatthebottomrowyoucan
seethatthemeanforthenurturingexperimenteris36.5ascomparedto46.5
fortheexploitiveone.Intheextremerightcolumnyou’llfindthatthemean
forthecontrolsubjectsis30,ascomparedto53fortheat-risksubjects.
Two-Way Interactions in Three-Way ANOVAs
Further complicating the three-way ANOVA is that, in addition to the three-
way interaction and the three main effects, there are three two-way inter-
actions to consider. In terms of our example there are the gender by
experimenter, gender by alcohol, and experimenter by alcohol interactions.
We will look at the last of these first. Before graphing a two-way interaction
in a three-factor design, you have to “collapse” (i.e., average) your scores over
the variable that is not involved in the two-way interaction. To graph the alco-
hol by experimenter (A ×B) interaction you need to average the men with the
women for each combination of alcohol and experimenter levels (i.e., each
cell of the A ×Bmatrix). These means have also been included in Table 22.1.
The graph of these cell means is shown in Figure 22.2. If you compare
this overall two-way interaction with the two-way interactions for the men
and women separately (see Figure 22.1), you will see that the overall inter-
action looks like an average of the two separate interactions; the amount of
interaction seen in Figure 22.2 is midway between the amount of interaction
for the men and that amount for the women. Does it make sense to average
the interactions for the two genders into one overall interaction? It does if
they are not very different. How different is too different? The size of the
three-way interaction tells us how different these two two-way interactions
are. A statistically significant three-way interaction suggests that we should
be cautious in interpreting any of the two-way interactions. Just as a signif-
icant two-way interaction tells us to look carefully at, and possible test, the
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Figure 22.2 Average of Men and Women
Graph of Cell Means in
70
Table 22.1 after Averaging
At risk
Across Gender
60
50
40
30
Control
20
0 Nurturing Exploitive
simple main effects (rather than the overall main effects), a significant three-
way interaction suggests that we focus on the simple interaction effects—the
two-way interactions at each level of the third variable (which of the three
independent variables is treated as the “third” variable is a matter of con-
venience). Even if the three-way interaction falls somewhat short of signifi-
cance, I would recommend caution in interpreting the two-way interactions
and the main effects, as well, whenever the simple interaction effects look
completely different and, perhaps, show opposite patterns.
So far I have been focusing on the two-way interaction of alcohol and
experimenter in our example, but this choice is somewhat arbitrary. The two
genders are populations that we are likely to have theories about, so it is
often meaningful to compare them. However, I can just as easily graph the
three-way interaction using “alcohol” as the third factor, as I have done in
Figure 22.3a. To graph the overall two-way interaction of gender and exper-
imenter, you can go back to Table 22.1 and average across the alcohol factor.
For instance, the mean for men in the nurturing condition is found by aver-
aging the mean for control group men in the nurturing condition (40) with
Figure 22.3a Control At Risk
Women
Graph of Cell Means in
Table 22.1 Using the 80 80
“Alcohol” Factor to
Distinguish the Panels 70 70
60 60
50 50
Men
40 40
30 Men 30
Women
20 20
0 Nurturing Exploitive 0 Nurturing Exploitive
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Average of Control and at Risk Figure 22.3b
Graph of Cell Means in
70 Table 22.1 after Averaging
Across the “Alcohol”
60 Factor
Women
50
40
Men
30
20
0 Nurturing Exploitive
the mean for at-risk men in the nurturing condition (36), which is 38. The
overall two-way interaction of gender and experimenter is shown in Figure
22.3b. Note that once again the two-way interaction is a compromise. (Actu-
ally, the two two-way interactions are not as different as they look; in both
cases the slope of the line for the women is more positive—or at least less
negative). For completeness, I have graphed the three-way interaction using
experimenter as the third variable, and the overall two-way interaction of
gender and alcohol in Figures 22.4a and 22.4b.
An Example of a Disordinal Three-Way Interaction
In the three-factor example I have been describing, it looks like all three
main effects and all three two-way interactions, as well as the three-way
interaction, could easily be statistically significant. However, it is important
to note that in a balanced design all seven of these effects are independent;
the seven Fratios do share the same error term (i.e., denominator), but the
sizes of the numerators are entirely independent. It is quite possible to have
Nurturing Exploitive Figure 22.4a
Women Graph of Cell Means in
80 80 Table 22.1 Using the
“Experimenter” Factor to
70 70 Distinguish the Panels
60 60
50 50
Men
Women
40 40
Men
30 30
20 20
0 Control At risk 0 Control At risk
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Figure 22.4b Average of Nurturing and Exploitive
Graph of Cell Means in
Table 22.1 after Averaging 70
Across the “Experimenter” Women
Factor 60
50
Men
40
30
20
0 Control At risk
a large three-way interaction while all of the other effects are quite small. By
changing the means only for the men in our example, I will illustrate a large,
disordinal interaction that obliterates two of the two-way interactions and
two of the main effects. You can see in Figure 22.5a that this new three-way
interaction is caused by a reversal of the alcohol by experimenter interaction
from one gender to the other. In Figure 22.5b, you can see that the overall
interaction of alcohol by gender is now zero (the lines are parallel); the gen-
der by experimenter interaction is also zero (not shown). On the other hand,
the large gender by alcohol interaction very nearly obliterates the main
effects of both gender and alcohol (see Figure 22.5c). The main effect of
experimenter is, however, large, as can be seen in Figure 22.5b.
An Example in which the Three-Way
Interaction Equals Zero
Finally, I will change the means for the men once more to create an example
in which the three-way interaction is zero, even though the graphs for the
Figure 22.5a Women Men
At risk
Rearranging the Cell
Control
Means of Table 22.1 to 80 80
Depict a Disordinal
3-Way Interaction 70 70
60 60
50 50
40 40
30 30
Control
20 20 At risk
0 Nurturing Expoitive 0 Nurturing Expoitive
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Average of men and women Figure 22.5b
Regraphing Figure 22.5a
70 after Averaging Across
Gender
60 At risk
50 Control
40
30
20
0 Nurturing Exploitive
Average of Nurturing and Exploitive Figure 22.5c
Regraphing Figure 22.5a
70 after Averaging Across
the “Experimenter”
Women
60 Factor
Men
50
40
30
20
0 Control At risk
two genders do not look the same. In Figure 22.6, I created the means for the
men by starting out with the women’s means and subtracting 10 from each
(this creates a main effect of gender); then I added 30 only to the men’s
means that involved the nurturing condition. The latter change creates a
two-way interaction between experimenter and gender, but because it
affects both the men/nurturing means equally, it does not produce any three-
way interaction. One way to see that the three-way interaction is zero in Fig-
ure 22.6 is to subtract the slopes of the two lines for each gender. For the
women the slope of the at-risk line is positive: 88− 40 =48. The slope of the
control line is negative: 22 −30 =−8. The difference of the slopes is 48 −(−8) =56.
If we do the same for the men, we get slopes of 18 and −38, whose difference
is also 56. You may recall that a 2 ×2 interaction has only one df, and can be
summarized by a single number, L, that forms the basis of a simple linear
contrast. The same is true for a 2 × 2 × 2 interaction or any higher-order
interaction in which all of the factors have two levels. Of course, quantifying
a three-way interaction gets considerably more complicated when the fac-
tors have more than two levels, but it is safe to say that if the two (or more)
graphs are exactly the same, there will be no three-way interaction (they will
continue to be identical, even if a different factor is chosen to distinguish the
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Figure 22.6 Women Men
At risk
Rearranging the Cell
Means of Table 22.1 to 80 80
At risk
Depict a Zero Amount of
Three-Way Interaction 70 70
60 60
50 50
40 40
30 30
Control
20 20
Control
0 Nurturing Expoitive 0 Nurturing Expoitive
graphs). Bear in mind, however, that even if the graphs do not lookthe same,
the three-way interaction will be zero if the amount of two-way interaction
is the same for every graph.
Calculating the Three-Way ANOVA
Calculatingathree-wayindependent-groupsANOVAisasimpleextensionof
themethodforatwo-wayindependent-groupsANOVA,usingthesamebasic
formulas.Inparticular,thereisreallynothingnewaboutcalculatingMS (the
W
errortermforalltheFratios);itisjusttheordinaryaverageofthecellvari-
anceswhenthedesignisbalanced.(Itishardtoimaginethatanyonewould
calculate an unbalanced three-way ANOVA with a calculator rather than a
computer, so I will not consider that possibility. The analysis of unbalanced
designsisdescribedingeneralinChapter18,SectionA).Ratherthangiveyou
allofthecellstandarddeviationsorvariancesfortheexampleinTable22.1,I’ll
justtellyouthatSS equals6,400;laterI’lldividethisbydf toobtainMS .(If
W W W
youhadalloftherawscores,youwouldalsohavetheoptionofobtainingSS
W
bycalculatingSS andsubtractingSS asdefinedinthefollowing.)
total between-cells
Main Effects
The calculation of the main effects is also the same as in the two-way
ANOVA; the SS for a main effect is just the biased variance of the relevant
group means multiplied by the total N.Let us say that each of the eight cells
in our example contains five subjects, so N equals 40. Then the SSfor the
T
experimenter factor (SS ) is 40 times the biased variance of 36.5 and 46.5
exper
(the nurturing and exploitive means from Table 22.1), which equals 40(25) =
1000 (the shortcut for finding the biased variance of two numbers is to take
the square of the difference between them and then divide by 4). Similarly,
SS =40(132.25) =5290, and SS =40(12.25) =490.
alcohol gender
The Two-Way Interactions
When calculating the two-way ANOVA, the SSfor the two-way interaction is
found by subtraction; it is the amount of the SS that is left after sub-
between-cells
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tracting the SSs for the main effects. Similarly, the three-way interaction SS
is the amount left over after subtracting the SSs for the main effects and the
SSs for all the two-way interactions from the overall SS . However,
between-cells
finding the SSs for the two-way interactions in a three-way design gets a lit-
tle tricky. In addition to the overall SS , we must also calculate some
between-cells
intermediate “two-way” SS terms.
between
To keep track of these I will have to introduce some new subscripts. The
overall SS is based on the variance of all the cell means, so no factors
between-cells
are “collapsed,” or averaged over. Representing gender as G,alcohol as A,and
experimenter as E, the overall SS will be written as SS . We will
between-cells GAE
also need to calculate an SS after averaging over gender. This is based on
between
the four means (included in Table 22.1) I used to graph the alcohol by exper-
imenter interaction and will be represented by SS . Because the design is
AE
balanced, you can take the simple average of the appropriate male cell mean
and female cell mean in each case. Note that SS is not the SSfor the alco-
AE
hol by experimenter interaction because it also includes the main effects of
those two factors. In similar fashion, we need to find SS from the means
GA
you get after averaging over the experimenter factor and SS by averaging
GE
over the alcohol factor. Once we have calculated these four SS terms, all
between
of the SSs we need for the three-way ANOVA can be found by subtraction.
Let’s begin with the calculation of SS ;the biased variance of the eight
GAE
cell means is 366.75, so SS =40(366.75) =14,670. The means for SS are
GAE AE
35, 25, 38, 68, and their biased variance equals 257.75, so SS =10,290. SS
AE GA
is based on the following means: 34, 26, 42, 64, so SS =40(200.75) =8,030.
GA
Finally, SS ,based on means of 38, 38, 35, 55, equals 2,490.
GE
Next we find the SSs for each two-way interaction:
SSA×E=SSAE−SSalcohol−SSexper=10,290 −5,290 −1,000 =4,000
SSG×A=SSGA−SSgender−SSalcohol=8,030 −490 −5,290 =2,250
SSG×E=SSGE−SSgender−SSexper=2,490 −490 −1,000 =1,000
Finally, the SSfor the three-way interaction (SSG×A×E) equals
SSGAE−SSA×E−SSG×A−SSG×E−SSgender−SSalcohol−SSexper
=14,670 −4,000 −2,250 −1,000 −490 −5,290 −1,000 =640
Formulas for the General Case
It is traditional to assign the letters A, B, and, C to the three independent
variables in the general case; variables D, E,and so forth, can then be added
to represent a four-way, five-way, or higher ANOVA. I’ll assume that the fol-
lowing components have already been calculated using Formula 14.3
applied to the appropriate means: SS , SS , SS , SS , SS , SS , SS . In
A B C AB AC BC ABC
addition, I’ll assume that SS has also been calculated, either by averaging
W
the cell variances and multiplying by df or by subtracting SS from SS .
W ABC total
The remaining SScomponents are found by Formula 22.1:
a. SSA×B=SSAB−SSA−SSB Formula 22.1
b. SSA×C=SSAC−SSA−SSC
c. SSB×C=SSBC−SSB−SSC
d. SSA×B×C=SSABC−SSA×B−SSB×C−SSA×C−SSA−SSB−SSC
At the end of the analysis, SS (whether or not it has been calculated
total
separately) has been divided into eight components: SS , SS , SS ,the four
A B C
interactions listed in Formula 22.1, and SS .Each of these is divided by its
W
corresponding df to form a variance estimate, MS.Using ato represent the