Three Way ANOVA

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688 Chapter 22 Three-Way ANOVA

ACONCEPTUALFOUNDATION

22Chap t e r

Three-Way ANOVA

You will need to use the following from previous chapters:

Symbolsk: Number of independent groups in a one-way ANOVA

c: Number of levels (i.e., conditions) of an RM factorn: Number of subjects in each cell of a factorial ANOVA

NT: Total number of observations in an experiment

FormulasFormula 16.2: SSinter (by subtraction) also Formulas 16.3, 16.4, 16.5

Formula 14.3: SSbet or one of its components

ConceptsAdvantages and disadvantages of the RM ANOVA

SS components of the one-way RM ANOVASS components of the two-way ANOVA

Interaction of factors in a two-way ANOVA

So far I have covered two types of two-way factorial ANOVAs: two-way inde-pendent (Chapter 14)andthe mixed designANOVA (Chapter 16). There is onlyone more simple two-way ANOVA to describe: the two-way repeated measuresdesign. [There are other two-way designs, such as those including random-effectsor nested factors, but they are not commonlyusedsee Hays (1994) fora description of some of these.] Just as the one-way RM ANOVA can bedescribed in terms of a two-way independent-groupsANOVA, the two-way RMANOVA can be described in terms of a three-way independent-groups ANOVA.This gives me a reason to describe the latter design next. Of course, the three-way factorial ANOVA is interesting in its own right, and its frequent use in the

psychological literature makes it an important topic to cover, anyway. I willdealwith the three-way independent-groups ANOVA and the two-way RM ANOVAin this section and the two types of three-way mixed designs in Section B.

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 greaternumber of times to extract a greater number ofSS components from SStotal(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, andthat is something that can get quite complicated, as you will see. To give you amanageable view of the complexities that may arise when dealing with threefactors, Ill 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 publishedstudy, 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 didnot. Ill call this the alcohol factor and its two levels, at risk (of codepen-dency) andcontrol. The other factor (theexperimenterfactor) also had twolevels; in one level subjects were told that the experimenter was an exploitiveperson, and in the other level the experimenter was described as a nurturingperson. All of the subjects were women. If we imagine that the experimentwas replicated using equal-sized groups of men and women, the original688

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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 samenumber 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 displayedin published articles in the form of a table, such as Table 22.1.

Section A Conceptual Foundation 689

Nurturing Exploitive Row Mean

Control: Men 40 28 34Women 30 22 26Mean 35 25 30

At risk: Men 36 48 42Women 40 88 64Mean 38 68 53Column mean 36.5 46.5 41.5

Table 22.1

Figure 22.1

Graphing Three Factors

The easiest way to see the effects of this experiment is to graph the cellmeans. However, putting all of the cell means on a single graph would not bean easy way to look at the three-way interaction. It is better to use twographs side by side, as shown in Figure 22.1. With a two-way design one hasto decide which factor is to be placed along the horizontal axis, leaving theother to be represented by different lines on the graph. With a three-waydesign one chooses both the factor to be placed along the horizontal axisandthe factor to be represented by different lines, leaving the third factor to berepresented by different graphs. These decisions result in six different waysthat 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 considerablylesser extent (the lines for the men are closer to being parallel). This difference

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Graph of Cell Means for

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in amount of two-way interaction for men and women constitutes a three-wayinteraction. If the two graphs had looked exactly the same, the Fratio for thethree-way interaction would have been zero. However, that is not a necessarycondition. A main effect of gender could raise the lines on one graph relativeto the other without contributing to a three-way interaction. Moreover, an

interaction of gender with the experimenter factor could rotate the lines onone graph relative to the other, again without contributing to the three-wayinteraction. As long as the difference in slopes (i.e., the amount of two-wayinteraction) is the same in both graphs, the three-way interaction will be zero.

Simple Interaction Effects

A three-way interaction can be defined in terms of simple effects in a way thatis analogous to the definition of a two-way interaction. A two-way interactionis a difference in the simple main effects of one of the variables as you changelevels of the other variable (if you look at just the graph of the women in Fig-ure 22.1, each line is a simple main effect). In Figure 22.1 each of the twographscanbe considered a simpleeffect of thethree-waydesignmore specif-ically, a simple interaction effect. Each graph depicts the two-way interaction

of alcohol and experimenter at one level of the gender factor. The three-wayinteraction can be defined as the difference between these two simple interac-tion effects. If the simple interaction effects differ significantly, the three-wayinteraction will be significant. Of course, it doesnt matter which of the three

variables is chosen as the one whose different levels are represented as differ-entgraphsif thethree-way interaction is statistically significant, there will besignificant differences in the simple interaction effects in each case.

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 goin 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 forone, which is distinctly different. For instance, in our present example theresults might have shown no effect for the men (all cell means about 40), nodifference for the control women (both means about 40), and a mean of 40for at-risk women exposed to the nice experimenter. Then, if the mean for at-risk women with the exploitive experimenter were well above 40, therewould 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 effectin thisvariation of our example the subject must be femaleand raised by an alco-hol-dependent parentand exposed to the exploitive experimenter to attain ahigh score. Not only might the three-way interaction be significant, but onecell mean might be significantly different from all of the other cell means,making an even stronger case that all three variables must be combinedproperly to see any effect (if you were sure that this pattern were going tooccur, you could test a contrast comparing the average of seven cell meansto 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 meanthat is considerably higher than the others (as in Figure 22.1). This kind ofpattern is analogous to the ordinal interaction in the two-way case and tendsto cause all of the effects to be significant. On the other hand, a three-wayinteraction could arise because the two-way interaction reverses its patternwhen 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 thewomen). 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 variableand much weaker (or nonexistent) at another level. Of course, there aremany 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 foreach factor are not common. One reason is a practical one: the number ofsubjects required. Even a design as simple as a 2 3 4 has 24 cells (to findthe number of cells, you just multiply the numbers of levels). If you want tohave at least 5 subjects per cell, 120 subjects are required. This is not animpractical study, but you can see how quickly the addition of more levelswould result in a required sample size that could be prohibitive.

Main Effects

In addition to the three-way interaction there are three main effects to look at,one for eachfactor. To lookat the gender main effect, for instance, just take the

average of the scores for all of the men and compare it to the average of all ofthe women. If you have the cell means handy and the design is balanced, youcan average all of the cell means involving men and then all of the cell meansinvolving women. In Table 22.1, you can average the four cell means for themen (40, 28, 36, 48) to get 38 (alternatively, you could use the row means in theextreme right column and average 34 and 42 to get the same result). The aver-age for the women (30, 22, 40, 88) is 45. The means for the other main effectshave already been included in Table 22.1. Looking at the bottom row you cansee that the mean for the nurturing experimenter is 36.5 as compared to 46.5for the exploitive one. In the extreme right column youll find that the meanfor the control subjects is 30, as compared to 53 for the at-risk subjects.

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 byexperimenter, gender by alcohol, and experimenter by alcohol interactions.We will look at the last of these first. Before graphing a two-way interactionin a three-factor design, you have to collapse (i.e., average) your scores overthe 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 thewomen for each combination of alcohol and experimenter levels (i.e., eachcell of theA B matrix). These means have also been included in Table 22.1.

The graph of these cell means is shown in Figure 22.2. If you comparethis overall two-way interaction with the two-way interactions for the menand 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 ofinteraction seen in Figure 22.2 is midway between the amount of interactionfor the men and that amount for the women. Does it make sense to averagethe interactions for the two genders into one overall interaction? It does ifthey are not very different. How different is too different? The size of thethree-way interaction tells us how different these two two-way interactionsare. A statistically significant three-way interaction suggests that we shouldbe 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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Average of Men and Women

Graph of Cell Means in

Table 22.1 after Averaging

Across Gender

Figure 22.2

simple main effects (rather than the overall main effects), a significant three-way interaction suggests that we focus on the simple interaction effectsthetwo-way interactions at each level of the third variable (which of the threeindependent 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 interactionsand the main effects, as well, whenever the simple interaction effects lookcompletely different and, perhaps, show opposite patterns.

So far I have been focusing on the two-way interaction of alcohol andexperimenter in our example, but this choice is somewhat arbitrary. The twogenders are populations that we are likely to have theories about, so it isoften meaningful to compare them. However, I can just as easily graph thethree-way interaction using alcohol as the third factor, as I have done inFigure 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

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Table 22.1 Using the

Alcohol Factor to

Distinguish the Panels

Figure 22.3a

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Figure 22.3b

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the mean for at-risk men in the nurturing condition (36), which is 38. Theoverall two-way interaction of gender and experimenter is shown in Figure22.3b. 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 bothcases the slope of the line for the women is more positiveor at least lessnegative). For completeness, I have graphed the three-way interaction usingexperimenter as the third variable, and the overall two-way interaction ofgender 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 threemain effects and all three two-way interactions, as well as the three-wayinteraction, could easily be statistically significant. However, it is importantto 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 thesizes of the numerators are entirely independent. It is quite possible to have

Figure 22.4a

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Table 22.1 Using the

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a large three-way interaction while all of the other effects are quite small. Bychanging the means only for the men in our example, I will illustrate a large,disordinal interaction that obliterates two of the two-way interactions andtwo of the main effects. You can see in Figure 22.5a that this new three-wayinteraction is caused by a reversal of the alcohol by experimenter interactionfrom one gender to the other. In Figure 22.5b, you can see that the overallinteraction 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 maineffects of both gender and alcohol (see Figure 22.5c). The main effect ofexperimenter is, however, large, as can be seen in Figure 22.5b.

An Example in which the Three-Way

Interaction Equals ZeroFinally, I will change the means for the men once more to create an examplein which the three-way interaction is zero, even though the graphs for the


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Figure 22.4b

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Rearranging the Cell

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3-Way Interaction

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two genders do not look the same. In Figure 22.6, I created the means for themen by starting out with the womens means and subtracting 10 from each(this creates a main effect of gender); then I added 30 only to the mensmeans that involved the nurturing condition. The latter change creates atwo-way interaction between experimenter and gender, but because itaffects 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 thewomen the slope of the at-risk line is positive: 88 40 = 48. The slope of thecontrol 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 differenceis also 56. You may recall that a 2 2 interaction has only one df, and can besummarized by a single number, L, that forms the basis of a simple linearcontrast. The same is true for a 2 2 2 interaction or any higher-orderinteraction in which all of the factors have two levels. Of course, quantifyinga 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 willcontinue to be identical, even if a different factor is chosen to distinguish the


Figure 22.5b

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Regraphing Figure 22.5a

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Figure 22.5c

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graphs). Bear in mind, however, that even if the graphs do not look the same,the three-way interaction will be zero if the amount of two-way interactionis the same for every graph.

Calculating the Three-Way ANOVA

Calculating a three-way independent-groups ANOVA is a simple extension ofthe method for a two-way independent-groups ANOVA, using the same basicformulas. In particular, there is really nothing new about calculating MSW(theerror term for all the Fratios); it is just the ordinary average of the cell vari-ances when the design is balanced. (It is hard to imagine that anyone wouldcalculate an unbalanced three-way ANOVA with a calculator rather than acomputer, so I will not consider that possibility. The analysis of unbalanced

designs is described in general in Chapter 18, Section A). Rather than give youallof thecell standard deviations or variancesfortheexample in Table 22.1, Ill

just tell you that SSWequals 6,400; later Ill divide this by dfWto obtain MSW. (Ifyou had all of the raw scores, you would also have the option of obtaining SSWby calculating SStotal and subtracting SSbetween-cells as defined in the following.)

Main Effects

The calculation of the main effects is also the same as in the two-wayANOVA; the SS for a main effect is just the biased variance of the relevantgroup means multiplied by the totalN. Let us say that each of the eight cellsin our example contains five subjects, soNT equals 40. Then the SS for theexperimenter factor (SSexper) is 40 times the biased variance of 36.5 and 46.5(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 takethe square of the difference between them and then divide by 4). Similarly,SSalcohol = 40(132.25) = 5290, and SSgender = 40(12.25) = 490.

The Two-Way Interactions

When calculating the two-way ANOVA, the SS for the two-way interaction isfound by subtraction; it is the amount of the SSbetween-cells that is left after sub-


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Figure 22.6

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tracting the SSs for the main effects. Similarly, the three-way interaction SSis the amount left over after subtracting the SSs for the main effects and theSSs for all the two-way interactions from the overall SSbetween-cells. However,finding the SSs for the two-way interactions in a three-way design gets a lit-tle tricky. In addition to the overall SSbetween-cells, we must also calculate some

intermediate two-way SSbetween terms.To keep track of these I will have to introduce some new subscripts. The

overall SSbetween-cells is based on the variance of all the cell means, so no factorsare collapsed, or averaged over. Representing gender as G, alcohol asA, andexperimenter as E, the overall SSbetween-cells will be written as SSGAE. We willalso need to calculate an SSbetween after averaging over gender. This is based onthe four means (included in Table 22.1) I used to graph the alcohol by exper-imenter interaction and will be represented by SSAE. Because the design isbalanced, you can take the simple average of the appropriate male cell meanand female cell mean in each case. Note that SSAE is not the SS for the alco-hol by experimenter interaction because it also includes the main effects ofthose two factors. In similar fashion, we need to find SSGA from the meansyou get after averaging over the experimenter factor and SSGE by averagingover the alcohol factor. Once we have calculated these four SSbetween terms, allof the SSs we need for the three-way ANOVA can be found by subtraction.

Lets begin with the calculation ofSSGAE; the biased variance of the eightcell means is 366.75, so SSGAE = 40(366.75) = 14,670. The means for SSAE are35, 25, 38, 68, and their biased variance equals 257.75, so SSAE = 10,290. SSGAis based on the following means: 34, 26, 42, 64, so SSGA = 40(200.75) = 8,030.Finally, SSGE, based on means of 38, 38, 35, 55, equals 2,490.

Next we find the SSs for each two-way interaction:

SSA E = SSAE SSalcohol SSexper = 10,290 5,290 1,000 = 4,000SSG A = SSGA SSgender SSalcohol = 8,030 490 5,290 = 2,250SSG E = SSGE SSgender SSexper = 2,490 490 1,000 = 1,000

Finally, the SS for 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 independentvariables in the general case; variablesD, E, and so forth, can then be addedto represent a four-way, five-way, or higher ANOVA. Ill assume that the fol-lowing components have already been calculated using Formula 14.3applied to the appropriate means: SSA, SSB, SSC, SSAB, SSAC, SSBC, SSABC. Inaddition, Ill assume that SSW has also been calculated, either by averagingthe cell variances and multiplying by dfWor by subtracting SSABC from SStotal.The remaining SS components are found by Formula 22.1:

a. SSA B = SSAB SSA SSB Formula 22.1b. SSA C = SSAC SSA SSCc. SSB C = SSBC SSB SSCd. SSA B C = SSABC SSA B SSB C SSA C SSA SSB SSC

At the end of the analysis, SStotal (whether or not it has been calculatedseparately) has been divided into eight components: SSA, SSB, SSC, the fourinteractions listed in Formula 22.1, and SSW. Each of these is divided by itscorresponding df to form a variance estimate, MS. Usinga to represent the


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number of levels of theA factor, b for theB factor,c for the C factor, and nfor the number of subjects in each cell, the formulas for the df componentsare as follows:

a. df A = a 1 Formula 22.2

b. df B = b 1c. df C = c 1d. df A B = (a 1)(b 1)e. df A C = (a 1)(c 1)f. df B C = (b 1)(c 1)g. df A B C = (a 1)(b 1)(c 1)h. df W= abc (n 1)

Completing the Analysis for the Example

Because each factor in the example has only two levels, all of the numeratordfs are equal to 1, which means that all of the MS terms are equal to theircorresponding SS termsexcept, of course, for the error term. The df for theerror term (i.e., dfW) equals the number of cells (abc) times one less than the

number of subjects per cell (this gives the same value asNTminus the num-ber of cells); in this case dfW = 8(4) = 32. MSW = SSW/dfW; therefore, MSW =6400/32 = 200. (Reminder: I gave the value of SSW to you to reduce theamount of calculation.)

Now we can complete the three-way ANOVA by calculating all of thepossible Fratios and testing each for statistical significance:

Fgender = = = 2.45

Falcohol = = = 26.45

Fexper = = = 5

FA E = = = 20

FG A = = = 11.35

FG E = = = 5

FG A E = = = 3.2

Because the df happens to be 1 for all of the numerator terms, the critical Ffor all seven tests is F.05 (1,32), which is equal (approximately) to 4.15. Exceptfor the main effect of gender, and the three-way interaction, all of the Fratiosexceed the critical value (4.15) and are therefore significant at the .05 level.

Follow-Up Tests for the Three-Way ANOVA

Decisions concerning follow-up comparisons for a factorial ANOVA aremade in a top-down fashion. First, one checks the highest-order interaction

640

200

MSG A E

MSW

1000

200

MSG E

MSW

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200

MSG A

MSW

4,000

200

MSA E

MSW

1,000

200

MSexper

MSW

5,290

200

MSalcohol

MSW

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200

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MSW


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for significance; in a three-way ANOVA it is the three-way interaction. (Two-way interactions are the simplest possible interactions and are called first-order interactions; three-way interactions are known as second-orderinteractions, etc.) If the highest interaction is significant, the post hoc testsfocus on the various simple effects or interaction contrasts, followed by

appropriate cell-to-cell comparisons. In a three-way ANOVA in which thethree-way interaction is not significant, as in the present example, attentionturns to the three two-way interactions. Although all of the two-way interac-tions are significant in our example, the alcohol by experimenter interactionis the easiest to interpret because it replicates previous results.

It would be appropriate to follow up the significant alcohol by experi-menter interaction with four t tests (e.g., one of the relevant t tests woulddetermine whether at-risk subjects differ significantly from controls in theexploitive condition). Given the disordinal nature of the interaction (see Fig-ure 22.2), it is likely that the main effects would simply be ignored. A similarapproach would be taken to the two other significant two-way interactions.Thus, all three main effects would be regarded with caution. Note thatbecause all of the factors are dichotomous, there would be no follow-up teststo perform on significant main effects, even if none of the interactions weresignificant. With more than two levels for some or all of the factors, itbecomes possible to test partial interactions, and significant main effects forfactors not involved in significant interactions can be followed by pairwiseor complex comparisons, as described in Chapter 14, Section C. I will illus-trate some of the complex planned and post hoc comparisons for the three-way design in Section B.

Types of Three-Way Designs

Cases involving significant three-way interactions and factors with morethan two levels will be considered in the context of mixed designs in SectionB. However, before we turn to mixed designs, let us look at some of the typ-ical situations in which three-way designs with no repeated measures arise.One situation involves three experimental manipulations for which repeatedmeasures are not feasible. For instance, subjects perform a repetitive task inone of two conditions: They are told that their performance is being meas-ured or that it is not. In each condition half of the subjects are told that per-formance on the task is related to intelligence, and the other half are toldthat it is not. Finally, within each of the four groups just described, half thesubjects are treated respectfully and half are treated rudely. The work outputof each subject can then be analyzed by a 2 2 2 ANOVA.

Another possibility involves three grouping variables, each of whichinvolves selecting subjects whose group is already determined. For instance, agroup of people who exercise regularly and an equal-sized group of those whodont are divided into those high and those relatively low on self-esteem (by amedian split). If there are equal numbers of men and women in each of thefour cells, we have a balanced 2 2 2 design. More commonly one or two ofthe variables involve experimental manipulations and two or one involvegrouping variables. The example calculated earlier in this section involved twogrouping variables (gender and having an alcohol-dependent parent or not)and one experimental variable (nurturing vs. exploitive experimenter).

To devise an interesting example with two experimental manipulationsand one grouping variable, start with two experimental factors that areexpected to interact (e.g., one factor is whether or not the subjects are toldthat performance on the experimental task is related to intelligence, and theother factor is whether or not the group of subjects run together will know


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each others final scores). Then, add a grouping variable by comparing subjects who are either high or low on self-esteem, need for achievement, orsome other relevant aspect of personality. If the two-way interaction differssignificantly between the two groups of subjects, the three-way interactionwill be significant.

The Two-Way RM ANOVA

One added benefit of learning how to calculate a three-way ANOVA is thatyou now know how to calculate a two-way ANOVA in which both factorsinvolve repeated measures. In Chapter 15, I showed you that the SS compo-nents of a one-way RM design are calculated as though the design were atwo-way independent-groups ANOVA with no within-cell variability. Simi-larly, a two-way RM ANOVA is calculated just as shown in the preceding forthe three-way independent-groups ANOVA, with the following modifica-tions: (1) One of the three factors is the subjects factoreach subject repre-sents a different level of the subjects factor, (2) the main effect of subjects isnot tested, and there is no MSW error term, (3) each of the two main effectsthat is tested uses the interaction of that factor with the subjects factor as the

error term, and (4) the interaction of the two factors of interest is tested byusing as the error term the interaction of all three factors (i.e., including thesubjects factor). If one RM factor is labeled Q and the other factor,R, and weuse S to represent the subjects factor, the equations for the three Fratios canbe written as follows:

FQ = , FR = FQ R =

Higher-Order ANOVA

This text will not cover factorial designs of higher order than the three-wayANOVA. Although higher-order ANOVAs can be difficult to interpret, no newprinciples are introduced. The four-way ANOVA produces 15 different F

ratios to test: four main effects, 6 two-way interactions, 4 three-way interac-tions, and 1 four-way interaction. Testing each of these 15 effects at the .05level raises serious concerns about the increased risk of Type I errors. Usu-ally, all of the Fratios are not tested; specific hypotheses should guide theselection of particular effects to test. Of course, the potential for an inflatedrate of Type I errors only increases as factors are added. In general, anN-wayANOVA produces 2N 1 Fratios that can be tested for significance.

In the next section I will delve into more complex varieties of the three-way ANOVAin particular those that include repeated measures on one ortwo of the factors.

1. To display the cell means of a three-way factorial design, it is convenientto create two-way graphs for each level of the third variable and placethese graphs side by side (you have to decide which of the three vari-ables will distinguish the graphs and which of the two remaining vari-ables will be placed along theXaxis of each graph). Each two-way graphdepicts a simple interaction effect; if the simple interaction effects aresignificantly different from each other, the three-way interaction will besignificant.

2. Three-way interactions can occur in a variety of ways. The interaction oftwo of the factors can be strong at one level of the third factor and close

MSQ R

MXQ R S

MSR

MSR S

MSQ

MSQ S


ASUMMARY

_ . . :


14/59

to zero at a different level (or even stronger at a different level). Thedirection of the two-way interaction can reverse from one level of thethird variable to another. Also, a three-way interaction can arise whenall of the cell means are similar except for one.

3. The main effects of the three-way ANOVA are based on the means at

each level of one of the factors, averaging across the other two. A two-way interaction is the average of the separate two-way interactions(simple interaction effects) at each level of the third factor. A two-wayinteraction is based on a two-way table of means created by averagingacross the third factor.

4. The error term for the three-way ANOVA, MSW, is a simple extension ofthe error term for a two-way ANOVA; in a balanced design, it is the sim-ple average of all of the cell variances. All of the SSbetween components arefound by Formula 14.3, or by subtraction using Formula 22.1. There areseven Fratios that can be tested for significance: the three main effects,three two-way interactions, and the three-way interaction.

5. Averaging simple interaction effects together to create a two-way inter-action is reasonable only if these effects do not differ significantly. Ifthey do differ, follow-up tests usually focus on the simple interactioneffects themselves or particular 2 2 interaction contrasts. If the three-way interaction is not significant, but a two-way interaction is, the sig-nificant two-way interaction is explored as in a two-way ANOVAwithsimple main effects or interaction contrasts. Also, when the three-wayinteraction is not significant, any significant main effect can be followedup in the usual way if that variable is not involved in a significant two-way interaction.

6. All three factors in a three-way ANOVA can be grouping variables (i.e.,based on intact groups), but this is rare. It is more common to have justone grouping variable and compare the interaction of two experimentalfactors among various subgroups of the population. Of course, all threefactors can involve experimental manipulations.

7. The two-way ANOVA in which both factors involve repeated measures isanalyzed as a three-way ANOVA, with the different subjects serving asthe levels of the third factor. The error term for each RM factor is theinteraction of that factor with the subject factor; the error term for theinteraction of the two RM factors is the three-way interaction.

8. In anN-way factorial ANOVA, there are 2N 1 Fratios that can be tested.The two-way interaction is called a first-order interaction, the three-wayis a second-order interaction, and so forth.


1. Imagine an experiment in which each sub-ject is required to use his or her memories tocreate one emotion: either happiness, sad-ness, anger, or fear. Within each emotiongroup, half of the subjects participate in arelaxation exercise just before the emotioncondition, and half do not. Finally, half thesubjects in each emotion/relaxation condi-tion are run in a dark, sound-proof chamber,and the other half are run in a normally litroom. The dependent variable is the subjects

systolic blood pressure when the subject sig-nals that the emotion is fully present. Thedesign is balanced, with a total of 128 sub-

jects. The results of the three-way ANOVA forthis hypothetical experiment are as follows:SSemotion = 223.1, SSrelax = 64.4, SSdark = 31.6,SSemo rel = 167.3, SSemo dark = 51.5; SSrel dark =127.3, and SSemo rel dark = 77.2. The total sumof squares is 2,344.a. Calculate the seven Fratios, and test each

for significance.

EXERCISES

_ . . :


15/59

b. Calculate partial eta squared for each ofthe three main effects (use Formula 14.9).Are any of these effects at least moderatein size?

2. In this exercise there are 20 subjects in each

cell of a 3 3 2 design. The levels of the firstfactor (location) are urban, suburban, andrural. The levels of the second factor are nosiblings, one or two siblings, and more thantwo siblings. The third factor has only twolevels: presently married and not presentlymarried. The dependent variable is the num-ber of close friends that each subject reportshaving. The cell means are as follows:

a. Given that SSWequals 1,094, complete thethree-way ANOVA, and present yourresults in a summary table.

b. Draw a graph of the means for Location Number of Siblings (averaging across mar-ital status). Describe the nature of theinteraction.

c. Using the means from part b, test the sim-ple effect of number of siblings at eachlocation.

3. Seventy-two patients with agoraphobia arerandomly assigned to one of four drug condi-tions: SSRI (e.g., Prozac), tricyclic antidepres-sant (e.g., Elavil), antianxiety (e.g., Xanax), ora placebo (offered as a new drug for agora-phobia). Within each drug condition, a thirdof the patients are randomly assigned to eachof three types of psychotherapy: psychody-namic, cognitive/behavioral, and group. Thesubjects are assigned so that half the subjects

in each drug/therapy group are also de-pressed, and half are not. After 6 months oftreatment, the severity of agoraphobia ismeasured for each subject (30 is the maxi-mum possible phobia score); the cell means(n = 3) are as follows:a. Given that SSW equals 131, complete the

three-way ANOVA, and present your resultsin a summary table.

b. Draw a graph of the cell means, with sep-arate panels for depressed and notdepressed. Describe the nature of thetherapy drug interaction in each panel.Does there appear to be a three-way inter-action? Explain.

c. Given your results in part a, describe a setof follow-up tests that would be justifi-able.

d. Optional: Test the 2 2 2 interactioncontrast that results from deleting Grouptherapy and the SSRI and placebo condi-tions from the analysis (extend the tech-niques of Chapter 13, Section B, andChapter 14, Section C).

4. An industrial psychologist is studying therelation between motivation and productiv-ity. Subjects are told to perform as manyrepetitions of a given clerical task as theycan in a 1-hour period. The dependent vari-able is the number of tasks correctly per-

formed. Sixteen subjects participated in theexperiment for credit toward a requirementof their introductory psychology course(credit group). Another 16 subjects wererecruited from other classes and paid $10for the hour (money group). All subjectsperformed a small set of similar clericaltasks as practice before the main study; ineach group (credit or money) half the sub-

jects (selected randomly) were told they hadperformed unusually well on the practicetrials (positive feedback), and half were toldthey had performed poorly (negative feed-back). Finally, within each of the four

groups created by the manipulations justdescribed, half of the subjects (at random)were told that performing the tasks quicklyand accurately was correlated with otherimportant job skills (self motivation),whereas the other half were told that goodperformance would help the experiment(other motivation). The data appear in thefollowing table:


Urban Suburban Rural

No Siblings

Married 1.9 3.1 2.0

Not Married 4.7 5.7 3.5

1 or 2 Siblings

Married 2.3 3.0 3.3


2 or more Siblings

Married 3.2 4.5 2.9


SSRI Tricyclic Antianxiety Placebo

PsychodynamicNot Depressed 10 11.5 19.0 22.0Depressed 8.7 8.7 14.5 19.0

Cog/Behav

Not Depressed 9.5 11.0 12.0 17.0Depressed 10.3 14.0 10.0 16.5

GroupNot Depressed 11.6 12.6 19.3 13.0Depressed 9.7 12.0 17.0 11.0

_ . . :


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a. Perform a three-way ANOVA on the data.Test all seven Fratios for significance, andpresent your results in a summary table.

b. Use graphs of the cell means to help youdescribe the pattern underlying eacheffect that was significant in part a.

c. Based on the results in part a, what posthoc tests would be justified?

5. Imagine that subjects are matched in blocksof three based on height, weight, and otherphysical characteristics; six blocks areformed in this way. Then the subjects in eachblock are randomly assigned to three differ-

ent weight-loss programs. Subjects are meas-ured before the diet, at the end of the dietprogram, 3 months later, and 6 monthslater. The results of the two-way RM ANOVAfor this hypothetical experiment are given in

terms of the SS components, as follows:SSdiet = 403.1, SStime = 316.8, SSdiet time = 52,SSdiet S = 295.7, SStime S = 174.1, and SSdiet time S= 230.a. Calculate the three Fratios, and test each

for significance.b. Find the conservatively adjusted critical F

for each test. Will any of your conclusionsbe affected if you do not assume thatsphericity exists in the population?

6. A psychologist wants to know how both theaffective valence (happy vs. sad vs. neutral)and the imageability (low, medium, high) ofwords affect their recall. A list of 90 words isprepared with 10 words from each combina-tion of factors (e.g., happy, low imagery: pro-motion; sad, high imagery: cemetery)randomly mixed together. The number ofwords recalled in each category by each ofthe six subjects in the study is given in thefollowing table:

a. Perform a two-way RM ANOVA on the data.Test the three Fratios for significance, andpresent your results in a summary table.

b. Find the conservatively adjusted critical Ffor each test. Will any of your conclusionsbe affected if you do not assume thatsphericity exists in the population?

c. Draw a graph of the cell means, anddescribe any trend toward an interactionthat you can see.

d. Based on the variables in this exercise,and the results in part a, what posthoc tests would be justified and mean-ingful?

Section B Basic Statistical Procedures 703

CREDIT SUBJECTS PAID SUBJECTSPositive Negative Positive Negative

Feedback Feedback Feedback Feedback

Self 22 12 21 2525 15 17 23

26 12 15 3030 10 21 26

Other 11 20 33 2118 23 29 2212 21 35 1914 26 29 17

SAD NEUTRAL HAPPYSubject No. Low Medium High Low Medium High Low Medium High

1 5 6 9 2 5 6 3 4 82 2 5 7 3 6 6 5 5 63 5 7 5 2 4 5 4 3 7

4 3 6 5 3 5 6 4 4 55 4 9 8 4 7 7 4 5 96 3 5 7 4 5 6 6 4 4

An important way in which one three-factor design can differ from anotheris the number of factors that involve repeated measures (or matching). Thedesign in which none of the factors involve repeated measures was coveredin Section A. The design in which all three factors are RM factors will not becovered in this text; however, the three-way RM design is a straightforwardextension of the two-way RM design described at the end of Section A. Thissection will focus on three-way designs with either one or two RM factors(i.e., mixed designs), and it will also elaborate on the general principles ofdealing with three-way ANOVAs, as introduced in Section A, and consider

BBASICSTATISTICALPROCEDIRES

_ . . :


17/59

the complexities of interactions and post hoc tests when the factors havemore than two levels each.

One RM Factor

I will begin with a three-factor design in which there are repeated measureson only one of the factors. The ANOVA for this design is not much morecomplicated than the two-way mixed ANOVA described in the previouschapterfor instance, there are only two different error terms. Such designsarise frequently in psychological research. One simple way to arrive at sucha design is to start with a two-way ANOVA with no repeated measures. Forinstance, patients with two different types of anxiety disorders (generalizedanxiety vs. specific phobias) are treated with two different forms of psy-chotherapy (psychodynamic vs. behavioral). The third factor is added bymeasuring the patients anxiety at several points in time (e.g., beginning oftherapy, end of therapy, several months after therapy has stopped); I willrefer to this factor simply as time.

To illustrate the analysis of this type of design I will take the two-wayANOVA from Section B of Chapter 14 and add time as an RM factor. You may

recall that that example involved four levels of sleep deprivation and threelevels of stimulation. Performance was measured only onceafter 4 days inthe sleep lab. Now imagine that performance on the simulated truck drivingtask is measured three times: after 2, 4, and 6 days in the sleep lab. The rawdata for the three-factor study are given in Table 22.2, along with the variousmeans we will need to graph and analyze the results; note that the datafor Day 4 are identical to the data for the corresponding two-way ANOVAin Chapter 14. To see what we may expect from the results of a three-way ANOVA on these data, the cell means have been graphed so that we canlook at the sleep by stimulation interaction at each time period (see Figure22.7).

You can see from Figure 22.7 that the sleep stimulation interaction,which was not quite significant for Day 4 alone (see Chapter 14, section B),increases over time, perhaps enough so as to produce a three-way interac-tion. We can also see that the main effects of stimulation and sleep, signifi-cant at Day 4, are likely to be significant in the three-way analysis. Thegeneral decrease in scores from Day 2 to Day 4 to Day 6 is also likely to yielda significant main effect for time. Without regraphing the data, it is hard tosee whether the interactions of time with either sleep or stimulation arelarge or small. However, because these interactions are less interesting in thecontext of this experiment, I wont bother to present the two other possiblesets of graphs.

To present general formulas for analyzing the kind of experiment shownin Table 22.2, I will adopt the following notation. The two between-subjectfactors will be labeled A and B. Of course, it is arbitrary which factor iscalledA and whichB; in this example the sleep deprivation factor will beA,and the stimulation factor will beB. The lowercase lettersa and b will standfor the number of levels of their corresponding factorsin this case, 4 and3, respectively. The within-subject factor will be labeledR, and its number oflevels,c, to be consistent with previous chapters.

Let us begin with the simplest SS components: SStotal, and the SSs forthe numerators of each main effect. SStotal is based on the total number ofobservations, NT, which for any balanced three-way factorial ANOVA isequal toabcn, where n is the number of different subjects in each cell of theA B table. So,NT= 4 3 3 5 = 180. The biased variance obtained by enter-ing all 180 scores is 43.1569, so SStotal = 43.1569 180 = 7,768.24. SSA is based


_ . . :


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PLACEBO

MOTIVATION

CAFFEINE

Subject

Subjec

t

Su

bject

Row

Day

2

Day4

Day6

Means

Day2

Day4

Day6

Means

Day2

Day4

Day6

M

eans

Means

26

24

24

24.6

7

29

28

26

27.6

7

29

26

26

27.0

30

29

25

28.0

26

23

23

24.0

24

22

23

23.0

None

29

28

27

28.0

23

24

25

24.0

23

20

17

20.0

23

20

20

21.0

29

30

27

28.6

7

31

30

30

30.3

3

21

20

20

20.3

3

35

33

22

30.0

29

27

25

27.0

AB

means

25.8

24.2

23.2

24.4

28.4

27.6

24.6

26.8

7

27.2

25.0

24.2

25.4

7

25.5

8

24

22

17

21

27

26

33

28.6

7

24

25

20

23

20

18

15

17.6

7

29

30

17

25.3

3

30

27

24

27.0

JetLag

15

16

13

14.6

7

34

32

25

30.3

3

30

31

25

28.6

7

27

25

19

23.6

7

23

20

18

20.3

3

25

24

17

22.0

28

27

22

25.6

7

25

23

20

22.6

7

23

21

22

22.0

AB

means

22.8

21.6

17.2

20.5

3

27.6

26.2

22.6

25.4

6

26.4

25.6

21.6

24.5

3

23.5

1

17

16

9

14.0

25

16

10

17.0

23

23

20

22.0

19

19

6

14.6

7

21

13

9

14.3

3

29

28

23

26.6

7

Interrupt

22

20

11

17.6

7

19

12

8

13.0

28

26

23

25.6

7

11

11

7

9.6

7

25

18

12

18.3

3

20

17

12

16.3

3

15

14

10

13.0

24

19

14

19.0

21

19

17

19.0

AB

means

16.8

16.0

8.6

13.8

22.8

15.6

10.6

16.3

3

24.2

22.6

19.0

21.9

3

17.3

5

16

14

5

11.6

7

24

15

14

17.6

7

25

23

18

22.0

18

17

6

13.6

7

19

11

8

12.6

7

16

16

14

15.3

3

Total

20

18

10

16.0

20

11

15

15.3

3

19

18

12

16.3

3

14

12

7

11.0

27

19

17

21.0

27

26

21

24.6

7

11

10

7

9.3

3

26

17

10

17.6

7

26

24

21

23.6

7

AB

means

15.8

14.2

7.0

12.3

3

23.2

14.6

12.8

16.8

7

22.6

21.4

17.2

20.4

16.5

3

Columnmeans

20.3

19.0

14.0

17.7

7

25.5

21.0

17.6

5

21.3

8

25.1

23.6

5

20.5

23.0

8

Table22.2

_ . . :


19/59

on the means for the four sleep deprivation levels, which can be found in therightmost column of the table, labeled row means. SSB is based on themeans for the three stimulation levels, which are found where the bottomrow of the table (Column Means), intersects the columns labeled SubjectMeans (these are averaged over the three days, as well as the sleep levels).The means for the three different days are not in the table but can be foundby averaging the three Column Means for Day 2, the three for Day 4, andsimilarly for Day 6. The SSs for the main effects are as follows:

SSA = 2

(25.58, 23.51, 17.35, 16.53)

180 = 15.08

180 = 2,714.4.SSB =

2(17.77, 21.38, 23.08) 180 = 4.902 180 = 882.36.SSR =

2(23.63, 21.22, 17.38) = 6.622 180 = 1,192.0

As in Section A, we will need the SS based on the cell means, SSABR, andthe SSs for each two-way table of means: SSAB, SSAR, and SSBR. In addition,because one factor has repeated measures we will also need to find themeans for each subject (averaging their scores for Day 2, Day 4, and Day 6)and the SS based on those means, SSbetween-subjects.


30

25

20

15

10

7

None Jet-Lag0 Interrupt Total

Placebo

Caffeine

Day 2

Motivation

30

25

20

15

10

7


Placebo

Caffeine

Day 4Motivation

30

25

20

15

10

7


Placebo

Caffeine Day 6

Motivation

Graph of the Cell Means

in Table 22.2

Figure 22.7

_ . . :


20/59

The cell means we need for SSABR are given in Table 22.2, under Day 2,Day 4, and Day 6, in each of the rows labeledAB Means; there are 36 of them(a b c). The biased variance of these cell means is 30.746, so SSABR =30.746 180 = 5,534.28. The means for SSAB are found by averaging acrossthe 3 days for each combination of sleep and stimulation levels and are

found in the rows for AB Means under Subject Means. The biased varianceof these 12 (i.e.,a b) means equals 22.078, so SSAB = 3,974. The nine meansfor SSBR are the column means of Table 22.2, except for the columns labeledSubject Means. SSBR =

2(20.3, 19.0, 14.0, 25.5, 21.0, 17.65, 25.1, 23.65,20.5) 180 = 2,169.14. Unfortunately, there was no convenient place in Table22.2 to put the means for SSAR. They are found by averaging the (AB) meansfor each day and level of sleep deprivation over the three stimulation levels.SSAR =

2(27.13, 25.6, 24, 25.6, 24.47, 20.47, 21.27, 18.07, 12.73, 20.53, 16.73,12.33) 180 = 4,066.6. Finally, we need to calculate SSbetween-subjects for the 60(a b n) subject means found in Table 22.2 under Subject Means (ignor-ing the entries in the rows labeledAB Means and Column Means, of course).

SSbetween-subjects = 32.22 180 = 5,799.6.

Now we can get the rest of the SS components we need by subtraction.The SSs for the two-way interactions are found just as in Section A fromFormula 22.1a, b, and c (except that factor C has been changed toR):

SSA B = SSAB SSA SSBSSA R = SSAR SSA SSRSSB R = SSBR SSB SSR

Plugging in the SSs for the present example, we get

SSA B = 3,974 2,714.4 882.4 = 377.2SSA R = 4,066.6 2,714.4 1,192 = 160.2SSB R = 2,169.14 882.4 1,192 = 94.74

The three-way interaction is found by subtracting from SSABR the SSs forthree two-way interactions and the three main effects (Formula 22.1d).

SSA B R = SSABR SSA B SSA R SSB R SSA SSB SSRSSA B R = 5,534.28 377.2 160.2 94.74 2,714.4 882.4 1192 = 113.34

As in the two-way mixed design there are two different error terms. Oneof the error terms involves subject-to-subject variability within each groupor, in the case of the present design, within each cell formed by the twobetween-group factors. This is the error component you have come to knowas SSW, and I will continue to call it that. The total variability from one sub-

ject to another (averaging across the RM factor) is represented by a term wehave already calculated: SSbetween-subjects, or SSbet-s, for short. In the one-wayRM ANOVA this source of variability was called the subjects factor (SSsub),or the main effect of subjects, and because it did not play a useful role, weignored it. In the mixed design of the previous chapter it was simply dividedbetween SSgroups and SSW. Now that we have two between-group factors, that

source of variability can be divided into four components, as follows:SSbet-s = SSA + SSB + SSA B + SSW

This relation can be expressed more simply as

SSbet-s = SSAB + SSW

The error portion, SSW, is found most easily by subtraction:

SSW= SSbet-S SSAB Formula 22.3


_ . . :


21/59

This SS is the basis of the error term that is used for all three of the between-group effects. The other error term involves the variability within subjects.The total variability within subjects, represented by SSwithin-subjects, or SSW-S,for short, can be found by taking the total SS and subtracting the between-subject variability:

SSW-S = SStotal SSbet-S Formula 22.4

The within-subject variability can be divided into five components,which include the main effect of the RM factor and all of its interactions:

SSW-S = SSR + SSA R + SSB R + SSA B R + SSS R

The last term is the basis for the error term that is used for all of theeffects involving the RM factor (it was called SSS RM in Chapter 16). It isfound conveniently by subtraction:

SSS R = SSW-S SSR SSA R SSB R SSA B R Formula 22.5

We are now ready to get the remaining SS components for our example.

SSW= SSbet-S SSAB = 5,799.6 3,974 = 1,825.6

SSW-S = SStotal SSbet-S = 7,768.24 5,799.6 = 1,968.64SSS R = SSW-S SSR SSA R SSB R SSA B R

= 1,968.64 1,192 160.2 94.74 113.34 = 408.36

A more tedious but more instructive way to find SSS R would be to findthe subject by RM interaction separately for each of the eight cells of thebetween-groups (AB) matrix and then add these eight components together.This overall error term is justified only if you can assume that all eight inter-actions would be the same in the entire population. As mentioned in the pre-

vious chapter, there is a statistical test (Boxs M criterion) that can be used togive some indication of whether this assumption is reasonable.

Now that we have divided SStotal into all of its components, we need todo the same for the degrees of freedom. This division, along with all of thedf formulas, is shown in the degrees of freedom tree in Figure 22.8.

The dfs we will need to complete the ANOVA are based on the followingformula:

a. df A = a 1 Formula 22.6b. df B = b 1c. df A B = (a 1)(b 1)d. df R = c 1e. df A R = (a 1)(c 1)f. df B R = (b 1)(c 1)g. df A B R = (a 1)(b 1)(c 1)h. df W= ab(n 1)i. df S R = dfW dfR = ab(n 1)(c 1)

For the present example,

dfA = 4 1 = 3

dfB = 3 1 = 2dfA B = 3 2 = 6dfR = 3 1 = 2dfA R = 3 2 = 6dfB R = 2 2 = 4dfA B R = 3 2 2 = 12dfW= 4 3 (5 1) = 48dfS R = dfW dfR = 48 2 = 96


_ . . :


22/59

Note that the sum of all the dfs is 179, which equals dftotal (NT 1 = abcn 1 =180 1).

The next step is to divide each SS by its df to obtain the correspondingMS. The results of this step are shown in Table 22.3 along with the Fratiosand their p values. The seven Fratios were formed according to Formula22.7:


Figure 22.8dftotal[abcn1]

dfgroups[ab1]

dfW[ab(n1)]

dfbetween-subjects[abn1]

dfR[c1]

dfS R[ab(n1)(c1)]

dfA[a1]

dfB[b1]

dfA B[(a1)(b1)]

dfA R[(a1)(c1)]

dfB R[(b1)(c1)]

dfA B R[(a1)(b1)(c1)]

dfwithin-subjects[abn(c1)]

Degrees of Freedom Tree

for Three-Way ANOVA

with Repeated Measures

on One Factor

Source SS df MS F p

Between-subjects 5,799.6 59Sleep deprivation 2714.4 3 904.8 23.8


23/59

a. FA = Formula 22.7

b. FB =

c. FA B =

d. FR =

e. FA R =

f. FB R =

g. FA B R =

Interpreting the Results

Although the three-way interaction is significant, the ordering of most of theeffects is consistent enough that the main effects are interpretable. The sig-nificant main effect of sleep is due to a general decline in performanceacross the four levels, with no deprivation producing the least deficit andtotal deprivation the most, as would be expected. It is also no surprise thatoverall performance significantly declines with increased time in the sleep

lab. The significant stimulation main effect seems to be due mainly to theconsistently lower performance of the placebo group rather than the fairlysmall difference between caffeine and reward.

In Figure 22.9, I have graphed the sleep by stimulation interaction, byaveraging the three panels of Figure 22.7. Although the interaction looks likeit might be significant, we know from Table 22.3 that it is not. Rememberthat the error term for testing this interaction is based on subject-to-subject

variability within each cell and does not benefit from the added power ofrepeated measures. The other two interactions use MSS RM as their errorterm and therefore do gain the extra power usually conferred by repeatedmeasures. Of course, even if the sleep by stimulation interaction were sig-nificant, its interpretation would be qualified by the significance of thethree-way interaction. The significant three-way interaction tells us to becautious in our interpretation of the other six Fratios and suggests that welook at simple interaction effects.

There are three ways to look at simple interaction effects in a three-wayANOVA (depending on which factor is looked at one level at a time), but themost interesting two-way interaction for the present example is sleep depri-

vation by stimulation, so we will look at that interaction at each level of thetime factor. The results have already been graphed this way in Figure 22.7. Itis easy to see that the three-way interaction in this study is due to the pro-gressive increase in the sleep by stimulation interaction over time.

MSA B R

MSS R

MSB R

MSS R

MSA R

MSS R

MSR

MSS R

MSA B

MSW

MSB

MSW

MSA

MSW


_ . . :


24/59

Assumptions

The sphericity tests and adjustments you learned in Chapters 15 and 16 areeasily extended to apply to this design as well. Boxs M criterion can be used

to test that the covariances for each pair of RM levels are the same (in thepopulation) for every combination of the two between-group factors. If M isnot significant, the interactions can be pooled across all the cells of the two-way between-groups part of the design and then tested for sphericity withMauchleys W. If you cannot perform these tests (or do not trust them), youcan use the modified univariate approach as described in Chapter 15. A fac-torial MANOVA is also an option (see section C). The dfs and p levels forthe within-subjects effects in Table 22.3 were based on the assumption ofsphericity. Fortunately, the effects are so large that even using the most con-servative adjustment of the dfs (i.e., lower-bound epsilon), all of the effectsremain significant at the .05 level (although the three-way interaction is justat the borderline withp = .05).

Follow-up Comparisons: Simple Interaction EffectsTo test the significance of the simple interaction effects just discussed, theappropriate error term is MSwithin-cell, as defined in section C of Chapter 16,rather than MSW from the overall analysis. This entails adding SSW to SSS Rand dividing by the sum of dfW and dfS R. Thus, MSwithin-cell equals (1,827 +407)/(48 + 96) = 2,234/144 = 15.5. However, given the small sample sizes inour example, it would be even safer (with respect to controlling Type Ierrors) to test the two-way interaction in each simple interaction effect asthough it were a separate two-way ANOVA. There is little difference betweenthe two approaches in this case because MSwithin-cell is just the ordinary aver-age of the MSW terms for the three simple interaction effects, and these donot differ much. The middle graph in Figure 22.7 represents the results ofthe two-way experiment of Chapter 14 (Section B), so if we dont pool errorterms, we know from the Chapter 14 analysis that the two-way interactionafter 4 days is not statistically significant (F= 1.97). Because the interactionafter 2 days is clearly less than it is after 4 days (and the error term is simi-lar), it is a good guess that the two-way interaction after 2 days is not statis-tically significant, either (in fact, F< 1). However, the sleep stimulationinteraction becomes quite strong after 6 days; indeed, the Ffor that simpleinteraction effect is statistically significant (F= 2.73,p < .05).

Although it may not have been predicted specifically that the sleep stim-ulation interaction would grow stronger over time, it is a perfectly reasonable


Figure 22.9

20

30

10

5


Placebo

Caffeine

Motivation

Graph of the Cell Means

in Table 22.2 After

Averaging Across the

Time Factor

_ . . :


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result, and it would make sense to focus our remaining follow-up analyses onDay 6 alone. We would then be dealing with an ordinary 4 3 ANOVA with norepeated measures, and post hoc analyses would proceed by testing simplemain effects or interaction contrasts exactly as described in Chapter 14, Sec-tion C. Alternatively, we could have explored the significant three-way inter-

action by testing the sleep by time interaction for each stimulation level or thestimulation by time interaction for each sleep deprivation level. In these twocases, the appropriate error term, if all of the assumptions of the overallanalysis are met, is MSS RM from the omnibus analysis. However, as you knowby now, caution is recommended with respect to the sphericity assumption,which dictates that each simple interaction effect be analyzed as a separatetwo-way ANOVA in which only the interaction is analyzed.

Follow-up Comparisons: Partial Interactions

As in the case of the two-way ANOVA, a three-way ANOVA in which at leasttwo of the factors have three levels or more can be analyzed in terms of partialinteractions, either as planned comparisons or as a way to follow up a signif-icant three-way interaction. However, with three factors in the design, thereare two distinct options. The first type of partial interaction involves forminga pairwise or complex comparison for one of the factors and crosses that com-parison with all levels of the other two factors. For instance, you could reducethe stimulation factor to a comparison of caffeine and reward (pairwise) or toa comparison of placebo with the average of caffeine and reward (complex)but include all the levels of the other two factors. The second type of partialinteraction involves forming a comparison for two of the factors. For example,caffeine versus reward and jet lag versus interrupted crossed with the threetime periods. If a pairwise or complex comparison is created for all three fac-tors, the result is a 2 2 2 subset of the original design, which has only onenumerator df and therefore qualifies as an interaction contrast. A significantpartial interaction may be decomposed into a series of interaction contrasts,or one can plan to test several of these from the outset. Another alternative isthat a significant three-way interaction can be followed directly by post hocinteraction contrasts, skipping the analysis of partial interactions, even whenthey are possible. A significant three-way (i.e., 2 2 2) interaction contrastwould be followed by a test of simple interaction effects, and, if appropriate,simple main effects (i.e., t tests between two cells).

Follow-Up Comparisons: Three-WayInteraction Not Significant

When the three-way interaction is not significant, attention shifts to thethree two-way interactions. If none of the two-way interactions is signifi-cant, any significant main effect with more than two levels can be exploredfurther with pairwise or complex comparisons among its levels. If only oneof the two-way interactions is significant, the factor not involved in the inter-action can be explored in the usual way if its main effect is significant. Anysignificant two-way interaction can be followed up with an analysis of itssimple effects or with partial interactions and/or interaction contrasts, asdescribed in Chapter 14, Section C.

Planned Comparisons for the Three-Way ANOVA

Bear in mind that a three-way ANOVA with several levels of each factor cre-ates so many possibilities for post hoc testing that it is rare for a researcher


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to follow every significant omnibus Fratio (remember, there are seven ofthese) with post hoc tests and every significant post hoc test with more local-ized tests until all allowable cell-to-cell comparisons are made. It is morecommon when analyzing a three-way ANOVA to plan several comparisonsbased on ones research hypotheses.

Although a set of orthogonal contrasts is desirable, more often theplanned comparisons are a mixture of simple effects, two- and three-wayinteraction contrasts, and cell-to-cell comparisons. If there are not too manyof these, it is not unusual to test each planned comparison at the .05 level.However, if the planned comparisons are not orthogonal, and overlap in var-ious ways, the cautious researcher is likely to use the Bonferroni adjustmentto determine the alpha for each comparison. After the planned comparisonshave been tested, it is not unusual for a researcher to test the seven Fratiosof the overall analysis but to report and follow up only those effects that areboth significant and interesting (and whose patterns of means make sense).

When the RM Factor Has Only Two Levels

If you have only one RM factor in your three-way ANOVA, and that factor has

only two levels, you have the option of creating difference scores (i.e., the dif-ference between the two RM levels) and conducting a two-way ANOVA on thedifference scores. For this two-way ANOVA, the main effect of factor A isreally the interaction of the RM factor with factorA, and similarly for factorB. TheA B interaction is really the three-way interaction ofA, B, and theRM factor. The parts of the three-way ANOVA that you lose with this trick arethe three main effects and theA B interaction, but if you are only interestedin interactions involving the RM factor, this shortcut can be convenient. Themost likely case in which you would want to use difference scores is when thetwo levels of the RM factor are measurements taken before and after sometreatment. However, as I mentioned in Chapter 16, this type of design is agood candidate for ANCOVA (you would use factorial ANCOVA if you hadtwo between-group factors).

Published Results of a Three-way ANOVA (One RM Factor)

It is not hard to find published examples of the three-way ANOVA with oneRM factor; the 2 2 2 design is probably the most common and is illus-trated in a study entitled Outcome of Cognitive-Behavioral Therapy forDepression: Relation to Hemispheric Dominance for Verbal Processing(Bruder, et al., 1997). In this experiment, two dichotic listening tasks wereused to assess hemispheric dominance: a verbal (i.e., syllables) task forwhich most people show a right-ear advantage (indicating left-hemisphericcerebral dominance for speech) and a nonverbal (i.e., complex tones) taskfor which most subjects exhibit a left-ear advantage. These two tasks are thelevels of the RM factor. The dependent variable was a measure of perceptualasymmetry (PA), based on how much more material is reported from theright ear as compared to the left ear. Obviously, a strong main effect of theRM factor is to be expected.

All of the subjects were patients with depression. The two between-groups factors were treatment group (cognitive therapy or placebo) andtherapy response or outcome (significant clinical improvement or not). Theexperiment tested whether people who have greater left-hemisphere domi-nance are more likely to respond to cognitive therapy; this effect is notexpected for those responding to a placebo. The results exhibited a clearpattern, as I have shown in Figure 22.10 (I redrew their figure to make the


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presentation consistent with similar figures in this chapter). The authorsstate:

There was a striking difference in PA between cognitive-therapy responders and

nonresponders on the syllables test but not on the complex tones test. In contrast,

there was no significant difference in PA between placebo responders and nonre-

sponders on either test. The dependence of PA differences between responders

and nonresponders on treatment and test was reflected in a significant Out-

come Treatment Test interaction in an overall ANOVA of these data, F(1, 72) =

5.81,p = .018. Further analyses indicated that this three-way interaction was dueto the presence of a significant Outcome Test interaction for cognitive therapy,

F(1, 29) = 5.67,p = .025, but not for placebo, F(1, 43) = 0.96,p = .332. Cognitive-

therapy responders had a significantly larger right-ear (left-hemisphere) advantage

for syllables when compared with nonresponders, t (29) = 2.58,p = .015, but no sig-

nificant group difference was found for the tones test, t (29) = 1.12,p = .270.

Notice that the significant three-way interaction is followed by tests ofthe simple interaction effects, and the significant simple interaction is, inturn, followed by t tests on the simple main effects of that two-way interac-tion (of course, the t tests could have been reported as Fs, but it is commonto report tvalues for cell-to-cell comparisons when no factors are being col-lapsed). Until recently, Fvalues less than 1.0 were usually shown as F< 1,p > .05 (or ns), but there is a growing trend to report Fs andps as given byones statistical software output (note the reporting ofF= 0.96 above).

Two RM Factors

There are many ways that a three-way factorial design with two RM factorscan arise in psychological research. In one case you begin with a two-wayRM design and then add a grouping factor. For instance, tension in the browand cheek, as measured electrically (EMG), can reveal facial expressions

that are hard to observe visually. While watching a happy scene from amovie, cheek tension generally rises in a subject (due to smiling), whereasbrow tension declines (due to a decrease in frowning). The opposite patternoccurs while watching a sad scene. If tension is analyzed with a 2 (brow vs.cheek) 2 (happy vs. sad) ANOVA, a significant interaction is likely toemerge. This is not an impressive result in itself, but the degree of the two-way (RM) interaction can be used as an index of the intensity of a subjects(appropriate) emotional reactions. For example, in one (as yet unpublished)experiment, half the subjects were told to get involved in the movie scenes


20

25

10

15

5

0

5

Syllables Tones

Nonresponders

Placebo

Responders

20

25

10

15

5

0

5

Syllables Tones

NonrespondersNonresponders

Cognitive Therapy

RespondersGraph of Cell Means for

the Bruder, et al.

(1997) Study

Figure 22.10

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29/59

BELOW

AVERAGE

ABOVE

Row

Female

Male

Mean

Female

Male

Mean

Fem

ale

Male

Mean

Means

5.2

5.2

5.2

5.8

6.0

5.9

7.4

7.6

7.5

6.2

Low

5.8

6.0

5.9

6.4

5.2

5.8

7.6

8.0

7.8

6.5

5.6

5.6

5.6

6.0

6.2

6.1

6.6

7.8

7.2

6.3

4.4

5.8

5.1

7.0

6.8

6.9

7.8

6.4

7.1

6.3

7

CellMean

5.2

5

5.6

5

5.4

5

6.3

6.0

5

6.1

75

7.3

5

7.4

5

7.4

6.3

417

4.8

5.4

5.1

5.6

6.0

5.8

6.4

7.0

6.7

5.8

7

Moderate

5.4

4.8

5.1

5.4

6.6

6.0

5.8

7.6

6.7

5.9

3

4.2

5.2

4.7

5.0

5.8

5.4

7.6

6.8

7.2

5.7

7

4.6

6.0

5.3

6.2

5.4

5.8

7.2

6.4

6.8

5.9

7

CellMean

4.7

5

5.3

5

5.0

5

5.5

5

5.9

5

5.7

5

6.7

5

6.9

5

6.8

5

5.8

833

4.4

5.8

5.1

6.0

7.0

6.5

7.0

5.6

6.3

5.9

7

Hig

h

5.2

6.6

5.9

5.6

6.2

5.9

6.6

4.8

5.7

5.8

3

3.6

6.4

5.0

6.2

7.8

7.0

5.2

6.4

5.8

5.9

3

4.0

5.0

4.5

5.2

6.8

6.0

6.8

5.8

6.3

5.6

0

CellMean

4.3

0

5.9

5

5.1

25

5.7

5

6.9

5

6.3

5

6.4

0

5.6

5

6.0

25

5.8

333

ColMean

4.7

7

5.6

5

5.2

1

5.8

7

6.3

2

6.0

95

6.8

3

6.6

8

6.7

55

6.0

2

Table22.4

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SSA = 72 2(6.3417, 5.8833, 5.8333) = 3.77

SSQ = 72 2(5.8233, 6.2167) = 2.785

SSR = 72 2(5.21, 6.095, 6.755) = 28.85

As in the previous analysis we will need SSbetween-subjects based on the 12

overall subject means (across all six categories of applicants). These are therow means (ignoring the rows labeled Cell Means and Column Means, ofcourse) in Table 22.4.

SSbet-S = 72 2(6.2, 6.5, 6.3, 6.37, 5.87, 5.93, 5.77, 5.97, 5.97, 5.83, 5.93, 5.6)

= 4.694 = SSS

Because this is the same SS you would get if you were going to calculate amain effect of subjects, I will call this SS component SSS.

Before we get enmeshed in the complexities of dealing with two RM fac-tors, we can complete the between-groups part of the analysis. I will use For-mula 16.2 from the two-way mixed design with an appropriate change in thesubscripts:

SSW= SSS SSA Formula 22.8

For this example, SSW= 4.694 3.77 = .924dfA = a 1 = 3 1 = 2dfW= a(n 1) = an a = 12 3 = 9

Therefore,

MSA = = = 1.885 and

MSW= = = .103

Finally,

FA = = 18.4

The appropriate critical Fis F.05(2,9) = 4.26, so FA is easily significant. Alook at the means for the three groups of subjects shows us that managerswith greater experience are, in general, more cautious with their hirabilityratings (perhaps they have been burned more times), especially when com-paring low to moderate experience. However, there is no point in trying tointerpret this finding before testing the various interactions, which maymake this finding irrelevant or even misleading. I have completed thebetween-groups part of the analysis at this point just to show you that atleast part of the analysis is easy and to get it out of the way before the morecomplicated within-subject part of the analysis begins.

With only one RM factor there is only one error term that involves aninteraction with the subject factor, and that error term is found easily by

subtraction. However, with two RM factors the subject factor interacts witheach RM factor separately, and with the interaction of the two of them, yield-ing three different error terms. The extraction of these extra error termsrequires the collapsing of more intermediate tables, and the calculation ofmore intermediate SS terms. Of course, the calculations are performed thesame way as alwaysthere are just more of them. Lets begin, however, withthe numerators of the various interaction terms, which involve the sameprocedures as the three-way analysis with only one RM factor. First, we can

1.885

.103

.924

9

SSW

9

3.77

2

SSA

2


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get SSAQR from the 18 cell means in Table 22.4 (all of the female and malecolumns of the three Cell Means rows).

SSAQR = 72 2(cell means) = 72 .694 = 49.96

The means needed to find SSAR are averaged across the Q factor (i.e.,

gender); they are found in the three Cell Mean rows, in the columns labeledMeans.

SSAR = 72 2(5.45, 6.175, 7.4, 5.05, 5.75, 6.85, 5.125, 6.35, 6.025)

= 72 .5407 = 38.93

The means for SSQR are the ColumnMeans in Table 22.4 for females and males(but not Means) and are averaged across all subjects, regardless of group.

SSQR = 72 2(4.77, 5.65, 5.87, 6.32, 6.83, 6.68) = 72 .4839 = 34.84

The means needed for SSAQ do not have a convenient place in Table 22.4;those means would fit easily in a table in which the female columns are alladjacent (for Below, Average, and Above), followed by the three malecolumns. Using Table 22.4, you can average together for each group all of thefemale cell means and then all of the male cell means, thus producing the sixAQ means.

SSAQ = 72 2(6.3, 6.383, 5.683, 6.083, 5.483, 6.185) = 72 .1072 = 7.72

Now we can get the SSs for all of the interactions by subtraction, using For-mula 22.1 (except thatB and C have been changed to Q andR):

SSA Q = SSAQ SSA SSQ = 7.72 3.77 2.785 = 1.16SSA R = SSAR SSA SSR = 38.93 3.77 28.85 = 6.31SSQ R = SSQR SSQ SSR = 34.84 2.785 28.85 = 3.21

SSA Q R = SSAQR SSA Q SSA R SSQ R SSA SSQ SSR= 49.96 1.16 6.31 3.21 3.77 2.785 28.85 = 3.875

The next (and trickiest) step is to calculate the SSs for the three RMerror terms. These are the same error terms I described at the end of SectionA in the context of the two-way RM ANOVA. For each RM factor the appro-priate error term is based on the interaction of the subject factor with thatRM factor. The more that subjects move in parallel from one level of the RMfactor to another, the smaller the error term. The error term for each RM fac-tor is based on averaging over the other factor. However, the third RM errorterm, the error term for the interaction of the two RM factors, is based onthe three-way interaction of the subject factor and the two RM factors, withno averaging of scores. To the extent that each subject exhibits the same two-way interaction for the RM factors, this error term will be small.

Two more intermediate SSs are required: SSQS, and SSRS. These SSscome from two additional two-way means tables, each one averaging scoresover one of the RM factors but not the other. (Note: TheA factor isnt men-tioned for these components because it is simply being ignored. Some of thesubject means are from subjects in the same group, and some are from sub-

jects in different groups, but this distinction plays no role for these SS com-ponents.) You can get SSRS from the entries in the columns labeled Means(ignoring the rows labeled Cell Means and Column Means, of course) inTable 22.4; in all there are 36 male/female averages, or RS means:

SSRS = 72 2(RS means) = 72 .6543 = 47.11

To find the QS means, you need to create, in addition to the row means inTable 22.4, two additional means for each row: one for the females in thatrow, and one for the males, for a total of 24 gender row means.


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SSQS = 72 2(6.13, 6.27, 6.6, 6.4, 6.07, 6.53, 6.4, 6.33, 5.6, 6.13, 5.53,

6.33, 5.6, 5.93, 6, 5.93, 5.8, 6.13, 5.8, 5.87, 5.0, 6.87, 5.33, 5.87)= 72 .1737 = 12.51

(Note: the means are in the order female, male for each subject (i.e.,row)

top to bottomof Table 22.4.)Now we are ready to get the error terms by subtraction, using Formula

22.9A.

SSQ S = SSQS SSQ SSS SSA Q Formula 22.9A

So,

SSQ S = 12.51 2.785 4.694 1.16 = 3.87

[Note: I subtracted the group by gender interaction at the end of the pre-ceding calculation because what we really want (and what I mean bySSQ S) is the gender by subject interaction within each group (i.e., level oftheA factor), added across all the groups. This is n

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