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Factorial Designs

CHAPTER OVERViEW

Step 6 of the research process involves selecting a research design. In

this chapter we discuss in detail research designs thatinclude more

than one independent variable—«factorial designs. The unique infor-

mation they provide is considered, as well as different types and vari-

ous applications of factorial designs.

292 Chapter 12 Factorial Designs

In most research situations the goal is to examine the relationship between twovariables by isolating those variables within the research study. The idea is toeliminate or reduce the influence of any outside variabler; that may disguise orobscure the specific relationship under investigation. For example, experimen—tal research (discussed in Chapter 8) typically focuseson one independent vari—able (Which is expected to influence behavior) and one dependent variable(which is a measure of the behafior). In real life, however, variables rarely existin isolation. That is, behavior usually is influenced by a variety of different vari-ables acting and interacting simultaneously. To examine these more complex,real-life situations, researchers often design research studies that include morethan one independent variable. Consider the following example:

It is common practice for a host to serve coffee at the end of a party

Where the guests have been drinking alcohol. Presumably, the caffeineWill counteract someof the effects of the alcohol so that the guests willbe more mentally alert when they head out for the trip home. Most of

us believe that we have a good understanding of the effects of caffeineand alcohol on mental alertness. For many people, the first cup of coffeeeach morning is necessary to get started; on the other hand, many peoplehave a glass of Wine in the evening to help them relax and unwind at theend of a busy day, But do we really know how these substances influenceour ability to react in an emergency situation?Does that cup of coffee atthe end of the party really improve reaction time? These questions wereaddressed in a study by Lignori and Robinson (2 001). They designed anexperiment in which both alcohol and caffeine consumption were ina—nipulated within the same study. They observed how quickly partici-pants With different levels of alcohol and (laft-eine could apply the brakes

in a simulateddriving test. Figure 12.] shows the general structure ofthis experiment. Notice that the study involves two independent vari—ables: Alcohol consumption is varied from no alcohol to enough to cre—ete dizziness or a “high”; caffeine consumptionis varied from no caffeineto 200 mg to 400 mg. The two independent variables create a matrix,with the different values of caffeine defining the columns and the differ-ent levels ofalcohol defining the rows.The resulting 2 >< 3 matrix showssix different combinations of the variables, producing six treatment con—ditions to be examined. The dependent variable for the study is a mea—sure of reaction time in a simulated driving emergency situation forpeople observed in each of the six conditions.

To simplify further discussion of this kind of research study: some basic ter-minology and definitions are in order. When two or more independent vari—ables are combined in a single study, the independent variables are commonlycalled factors. For the study in our example, the two factors are alcohol con—sumption and caffeine consumption.A researchstudy involving two or morefactors is called a factorial design. This kind of design is often referred to bythe number of its factors, as a two—factor design or a three—factor design.

No Caffeine

Reaction time scores

tor a group of

participants who

200 MgCaffeine

Reaction time scoresfor a group ofparticipants who

29312.1 Introduction to Factorial Designs

400 MgCaffeine

Reaction time scoresfora group ofparticipants who

Naomi received alcohol received alcohol received alcohol

and a 0-mg dose and a 200—mg dose and a 400mg doseof caffeine of caffeine of caifeine

Reaction time scores Reaction time scores Reaction time scoresfor a group of for a group of for a group of

No Alcohol participants who participants who participants whoreceived no alcohol received no alcohol received no alcoholand a 0-mg dose and a ZOO-mg dose and a 400—mg doseof caffeine of caffeine of caffeine

Figure 12.1 The Structure of aTwo-Factor Experiment WhereAlcohol Consumption (Factor 1) and Caffeine Consumption

(Factor 2) Are Manipulated in the Same StudyThe purpose of the experiment is to examine how different

combinations of alcohol and caffeine affect reaction timein a simulated emergency driving situation.

Our example is a two—factor design. A research study with only one indepen—dent Variable is often called a single—factor design.

Generically, each factor is denoted by a letter (A, B, C, etc.). In addition, fac—torial designs use a notation systemthat identifies both the number of iåctors andthe number of values or levels thatexist for each factor (see Ch. 8, page 194).The previous example has two leVels for the alcohol factor (factor A) and three forthe caffeine factor (factor B) and can be denoted as a 2 ><3(read as “two by three”)factorial design, with 2 indicating two levels of the first factor (alcohol) and 3 sym—bolizing three levels of the second factor (caffeine). The number of treatmentconditions can be determined by multiplying the levels for each factor. Additionalexamples: a 2 >< Z factorial design (the simplest factorial design) would represent atwo—factor design With two levels of the first factor and two levelsof the second,with a total of four treatment conditions; and a 2 >< 3 >< 2 design would representa three—factor design With two, three, and two levels of each of the factors, respec—

tively, for a total of twelve conditions. Factorialdesigns including more than twoindependent variables are discussed in Section 12 .4.

In an experiment, an independent variable is often called a factor, especially inexperiments that include two or more independent variables.

A research design that includes two or more factors is called a factorial design.

As we have noted, one advantage of a factorial design is that it creates amore “realistic” situation than can be obtained by examining a single factor inisolation. Because behavior is influenced by a variety of factors usually actingtogether, it is sensible to examine two or more factors simultaneously in & single

it is possible to haveä _factor that is not ama 'njpuiated variable—rand ; 'therefore is not a'tr'u'éiindependent variable hunrather a quasE—indepehe"dent variable; For exam-ple, a researcher'co'uidf ;examine work proficiency:for males and females '"

w{factor jl‘mnd er different

temperature conditions ',(factor 2). in this case, '

gender is a factor, but

it is not manipulated.This kind of factorialstudy is discussed inSection 1Z4.

Definititms’ffs

”J.W.—..

294 Chapter 1 2

ln addition to a separategroup for each cell (abetween-subjects ole-—sign), the same groupcan participate in all ofthe different cells [a

within—subjects design).

These two designs can. also be mixed, for ex—ample, using separate

, groups for one factor

, ferent conditions defined

but having each groupparticipate in all the clif-

. by the second factor.

' These different versions” of atwofiactor study are, discussed in Section 12.41.

Facrorial Designs

study. At first glance it may appear that this kindof research is unnecessarilycomplicated. Why not do two separate, simplestudies looking at each factor byitself? The answer to this question is thatcombining of factorsWithin one studyprovides researchers with a unique opportunity to examine not only how thefactors influence behaviorbut also how the factors influence or interact witheach other. Returning to the alcohol/caffeine example, a researcher who manip-ulated only alcohol consumption would observe how alcohol affects behavior.Similarly, manipulating only caffeine consumptionwould demonstrate how caf—feine affectsbehavior. However, combining the two variables permits research—ers to examine how changes in caffeineconsumptioncan influence the effects ofalcohol on behavior. The idea that two factors can act together, creating uniqueconditions that are different from either factor acting alone, underlies the valueof a factorial design.

Suppose a researcher is interested in examining the effects of mood and fooddeprivation on eating. Females listen to one of two types of music to induceeither a happy or a sad mood, following either 19 hours of food deprivation(breakfast and lunch are skipped) or no deprivation. In a feeding laboratorysetting, the amount of food consumed is measured in all participants.

a. How many independent variables or factors does this study have? Whatare they?

b. Describe thisstudy using the notation system that indicates factors andnumbers of levels of each factor.

7:r“(_7\'§1,‘>

12.2 MAIN EFFECTS AND INTERACTIONS

The primary advantage of a facmria} design is that it allows researchers toexamine how unique combinations of factors, acting together, influence behav-ior. To explore this feature in more detail, we focus on designs involving onlymo factors; that is, the simplest possible example of a factorial design. In Sec-tion 114, we look briefly at more complex situations involving three or morefactors.

The structure of a two—factor design can be represented by a matrix in whichthe levels of one factor determine the columns and the levels of the second factordetermine the rows (see Figur—e 12.1). Each cell in the matrix corresponds to aspecific combination of the factors; that is, a separate treatment condition. Theresearch study would involve observing and measuring a group of individualsunder the conditions described by each cell.

With a matrix, diagram the example study examining the effects of mood andfood deprivation on eating.

The data from a two—factor Study provide three separate and distinct sets ofinformation describing how the two factors independently and jointly affectbe-

12 .2 Main Effects and Interactions 295

havior. To demonstrate the three kinds of information, the general structureofthe alcohol/caffeine study is repeated in Table 12. l, with hypothetical dataadded showing the mean reaction time (in milliseconds) for participants in eachof the cells. The data provide the following information:

1.

2.

Each column of the matrix corresponds to a specific level of caffeine con—sumption. For example, all of the participants tested in the first column(both sets of scores) were measured with no caffeine. By computing themean score for each column, we obtain an overall mean for each of the threedifferent caffeine conditions. The resulting three column means provide anindication of how caffeine consumption (factor 1) affects behavior. The dif—ferences among the three column means are called the main eficctfnr mf—feine. In more general terms, the mean differences among the columnsdetermine the main effect for one factor. Notice that the calculation of themean for each column involvesaveraging both levels of alcohol consump—tion. (Half the scores were obtained With alcohol and half were obtainedwith no alcohol.) Thus, the alcohol consumption is balanced or matchedacross all three caffeine levels, which means that any differences obtainedbetween the columns cannot be explained by differences in alcohol.

For the data in Table 12.1, the participants in the no-caffeinc conditionhave an average score of 250. This column mean Was obtained by averagingthe two groups (Mean = 290 and Mean = 210) in the no-caffeine column. Ina similar way, the other column means are computed as 225 and 200. Thesethree means show a general tendency for faster reaction times as caffeineconsumption increases. This relationship beUNeen caffeine consumption andreaction time is the main effect for caffeine. Finally, note that the mean dif—ferences among columns simply describe the main effect for caffeine. A sta—

tistical test is necessary to determine whether or not the mean differencesare significant.

Just as we determine the overall main effect for caffeine by calculating thecolumn means for the data in Table 12.1, we can determine the overall effectof alcohol consumption by examining the rows of the data matrix. For ex-ample, all of the participants in the top row were tested With alcohol. Themean score for these participants (all three sets of scores) provides a measureof reaction rime under that level of alcohol. Similarly, the overall mean forthe bottom row describes reaction time when no alcohol is given. The dif—ference between these two means is called the min (33%t alcohol. As be—fore, notice that the process of obtaining the row means involves averagingall three levels of caffeine. Thus, each row mean includes exactly the samecalfeine conditions such that caffeine is matched across rows and cannot ex—plain the mean differences between rows. In general terms, the differencesbetween the column means define the main effect for one factor, and the dif—ferences between the row means denne the main effect for the second factor.

For the date shown in Table 12.1, the overall mean for the first row (al—cohol) is 250. This mean is obtained by averaging the three treatmentmeans in the top row (290, 250, and 210). Similarly, the overall mean reac—

tion time for participants in the no—alcohol condition is 200. The SO—point

We use hypotheticaldata throughout the »chapter to demonstrate fdifferent patterns of re—suits. If you are curious !about the real results.see the Liguori and ,Robinson (2001) article.

5

ll.

For reaction time, ; _

smaller numbers indicate—faster reaction times. '


3.

difference between the two row means (250 and 200) deserihes the main ef—fect for alcohol. ln this study, alcohol consumption decreases (slows) reac-tion time by 50 points.

A factorial design allowsresearchers to examine how combinations of fac—tors, working together, affect behavior. Although the effects of each inde—pendent variable, and hence, each main effect, can be examined in its ownseparate study, a fectorial design provides an opportunity to explore poten—tial interaction between the two variablesas well. For the data in Table 12.1,this means looking at the effects of caffeine and alcohol that are differentfrom the effect of caffeine alone or the effect of alcohol alone. In the datamatrix, each cell corresponds to a specific combination of the two factors,and the mean differences between cells indicate differential treatment ef—fects. The question is whether the differences between cells can be explainedby the OVerall main effects of the two factors, or whether the cell differences

show effects that are separate and distinct from the overall main effects.For example, the data in Table 12.l Show a 50—point main effect for alco—

hol. However, in the first column, Where the caffeine level is I), the change inalcohol produces an 80—point effect on reaction time (means of 290 versus210). Also, in the third column (where the caffeine level is 400 mg), the twaalcohol conditions Show only a lil—point mean difference in reaction time,from a mean of 210 to a mean of 190. ln each case, the mean difference he—

tween cells is not explained by the overall 5 O—point effect of alcohol. For thesedata, individual combinations of caffeine end alcohol appear to have effectsthat are different from the overall effects of caffeine or alcohol acting alone.The "extra” mean differences between cells that are not explained by the maineffects are called an interaction between factors. In this example, the alco-hol has a large effect on reaction time (more than 50 points) when there is no

TABLE *! 21

, Hypothetical Data Showing the Treatment Means for a Two—FactorStudy Examining How Different Combinations of Alcohol andCaffeine Affect Reaction Time (in Milliseconds) in a SimulatedEmergency Driving Situation

The data are souctured to create main effects for both factors and an interaction.

200 Mg 400 M9N0 Caffeine Caffeine Caffeine

Alcohol M=290 M=250 M=210 OverallM: 250

No Alcohol M: 210 M = 200 M = 190 Overall M = 200

Overall Overall Overall

M=250 M=225 M.:ZOO

1243 More About Interactions 297

caffeine but a small effect (less than S (1 points) when combined with 400 mgof mffeine, The special advantage of combining two factors Within the samestudy is the ability to examine the unique effects caused by an interaction be—tween factors.

Thus, the results from a uro—factor design reveal how each factor indepen—dently affects behavior (the main effects) and how the two factors operating to-gether (the interaction) can affect behavior. For the data in Table 12.1, the main

effect for alcohol consumption deseribes how reaction time in a simulated driv-ing testisinfluenced by alcohol independent of caffeine consumption. The maineffect for caffeine consumption describes how reaction time is affected bychanges in caffeine independent of alcohol. Finally, the interaction describeshow combinations of alcohol and caffeine consumption influence reaction timein a simulated driving test in ways that are different from alcohol acting aloneor from caffeine acting alone.

In general, when the data from a two—factor 'study are organized in a matrixas in Table 12.1, the mean differences between the columns describe the main ef—fect for one factor and the mean differences between rows describe the maineffect for the second factor. The main effects reflect the results that would be ob—tained if each factor were examined in its own separate experiment. The extramean differences that exist between cells in the matrix (differences that are not

explained by the overall main effects) describe the interaction and represent theunique information that is obtained by combining the two factors in a single study.

The mean differences among the levels of one factor are called the main effect ofthat factor. When the research study is represented as a matrix With one factordefining the rows and the second factor defining the columns, then the mean dif—ferences among the rows define the main effect for one factor, and the meandifferences among the columns define the main effect for the second factor.

An interaction between factors occurs Whenever the mean differences be—tween individual treatment conditions, or cells, are different from what is pre—dicted from the overall main effects of the factors.

Use the data presented in Table 12.1 to determine what numbers are comparedi assess:

a. the main effecm for caffeineb. the main effects for alcoholc. the interaction between caffeine and alcohol

må 'MORE ABOUT INTERACTIONSThe previous section introduced the concept of interaction as the unique effectsproduced by two factors working together. Now we define interaction more

Defin‘stionsi

LearningCheck


formally and look in detail at this unique component of a factorial design. Forsimplicity, we continue to examine two—factor designs and postpone discussionof more complex designs (and more complex interactions).

As noted, the data in Table 12.1 Show an 80—point difference between alco—hol consumption and no alcohol consumption when there is no caffeine. Thisvalue is different from the SO—point difference shown as the main effect for al—cohol and therefore indicates an interaction. In a similar fashion, the main ef—

fects for caffeine show thatreaction times speed up from a mean 225 to a meanof 200 when caffeine level is increased from 200 mg to 400 mg. However, this25—point main effect does not explain the mean differences between individualcells. For example, When the alcohol is high, increasing caffeine from 200 mg to400 mg results in a 40—point change in performance, not the 25—point changepredicted from the main effect. Again, the main effects do not explain the celldifferences, Which indicates an interaction.

As a counterexample, consider the data in Table 12.12. To cons—met this newset of data, the same column means and row means that appeared in Table 12.1are used. Thus, the new (lata show exactly the Same main effects as the original setof data. However, the individual cell means have been adjusted to eliminate theinteraction. Notice, for example, that the SiO—point main effect for alcohol (rowmeans of 250 versus 200) now explains the individual cell differences for everycaffeine condition. When the caffeine level is 0, there is a 50-point difference be—tWeen the alcohol and no—alcohol conditions. This 50—point difference (the maineffect) exists for all three caffeine levels. Similarly, the main effect for caffeineshows a 25 «point change in reaction time when the caffeine level is increased from200 mg to 400 mg (overall reaction times speed up from & mean of 225 to a meanof 200). This 25-point main effect explains the individual cell differences for 110

alcohol (a ZS—point drop from 200 to 175) and for alcohol (a 25—point drop from

TABLE 12.2

Hypothetical Data Showing the Treatment Means for a TWO—FactorStudy Examining How Different Combinations of Alcohol andCaffeine Affect Reaction Time (in Milliseconds) in 3 Simulated

Emergency Driving Situation

The data are Structured to create the same main effecm as in Table 12.1 but withoutan interaction.

ZOO'MQ 400 M9N0 Caffeine Caffeine Caffeine

Alcohol M: 275 M=250 M=225 OverallM: 250

No Alcohol M: 225 M: 200 M: 175 Overall M: 200

Overall Overall 4 Overall

M=250 M=225 M=200

12 .3 More About Interactions 299

250 to 225). For these data, there are no "extra” unexplained mean differencesamong the cells; all the mean differences are accounted for by the main effects,and there is no interaction.

ALTERNATIVE DEFINITIONS OF INTERACTION

A slightly different perspective on the conceptof interaction focuses on the no—tion of interdependency between the factors, as Opposed to independence. Morespecifically, if the two factors are independent so that the effect of either one isnot influenced by the other, then there is no interaction. 011 the other hand, if the

two factors are interdependent so that one factor does influence the effectof theother, then there is an interaction. The notion of interdependence is consistentwith our earlier discussion of interactions; if one factor does influence the effectsof the other, then unique combinations of the factors produce unique effects.

When the effects of one factor depend on the different levels of a second factor,then there is an interaction between the factors.

This second definition of an interaction uses different terminology but is

equivalent to the first definition (page 297). When the effects of a factor varydepending on the levels of another factor, the two factors are combining to pro—

duce unique effects. Returning to the data in Table 12.l, notice that the size ofthe alcohol effect (top row versus bottom row) dependr on the caffeine. At no caf—feine, for example, the alcohol effect is 80 points. However, the alcohol effectdecreases to 50 points at 200 mg and decreasesfurther to 20 points When thecaffeine level is 400 mg. Again, the effect of one factor (alcohol) depends on thelevels of the second factor (caffeine), which indicates an interaction. By con-trast, the data in Table 12.2 show that the effect of alcohol is independent ofcaffeine. For these data, the change in alcohol shows the same SD-poineeffeet—for all three levels of caffeine. Thus, the alcohol "effect does not depend on caf—feine, and there is no interaction.

When the results of a two—factor study are presented in a graph, the con—Cept of interaction can be defined in terms of the pattern displayed in the graph.Figure 12.2 shows the original data from Table 12.1 thatwere used to illustratethe existence of an interaction. To construct this figure, one of the factors was

selected as the independent variable to appear on the horizontal axis; in thiscase, the different levels of caffeine are displayed. The dependent variable, reac—_tion time, is shown on the vertical axis. Notice that the figure actually containstwo separate graphs; the top line shows the relationship between caffeine andreaction time When alcohol is given, and the bottom line shows the relationshipwhen no alcohol is given. In general, the graph matches the structure of thedata matrix; the columns of the matrix appear as values along the X-axis, andeach row of the matrix appears as a separate line in the graph.

For this particular set of data (Figure 12 ,2), notice that the lines in the graphare not parallel. The distance between the lines decreases from left to right. Forthese data, the distance between the lines corresponds to the alcohol effect, that

Definition,?


BOD

275

250

ReactionTime 225 Alcohol

(in Msec)

200 No Alcohol

175 A '

150

0 200 400

Amount of Caffeine (In Mg)

Figure 12.2 A Line Graph of the Data From Table 12.1The hypothetical data are structured to show main

effects for both factors and an interaction.

is, the mean difference in reaction timesfor alcohol and no alcohol. The factthat this difference depends on caffeine indicates an interaction between factors.

Dye fi ni U 0 n When the results of & two—factor study are graphed, The existence of nonparallel«

lines (lines that cross or converge) is an indication of an interaction betweenthe two factors. (Note thata statistical testisneeded to determine whether ornot the interaction is significant.)

To demonstrate the pattern that appears in a graph when there is no inter—action, che data from Table 12.2 are presented in Figure 12,3. Notice that thetwo lines in this figure are parallel; that is, the distance between lines is con—stant.In this case, the distance between lines reflects the 50—p0int difference inreaction time between alcohol and no alcohol, and this 50-point differfsnce isthe same for all three caffeine conditions.

Evaluate the means in the following matrix.

Treatment 1 Treatment 2

Males M = 10 M = 30

Females M: 20 M": 50

a. Is there evidence of a maineffect for the treatment factor?

b. Is there evidence of a main effect for the gender factor?

c. Is there evidence of an interaction? (Hint: Sketch a graph of the data.)

12.3 More About Interaction-15

300

275

250

ReactionTime 225 Alcohol

[in lec) '200

175 N0 Alcohol

150

O 200 400

Amount of Caffeine (in Mg)

Figure 12.3 A Line Graph of the Data From Table 12.2The hypothetical data are structured to Show

main effects for both factors but no interaction.

INTERPRETING MAIN EFFECTS AND INTERACTIONS

As we have noted, the mean differences between columns and between rows de—

scribe the main effects in a two-factor study, and the extra mean differences de—scribe the interaction. However, you shouldrealize that these mean differencesare simply derwiptiee and must be evaluated by a statistical hypothesis test he-fore they can be considered significant. That is, the obtained mean differencesmay not represent a real treatment effect but rather may be simply due to chanceor error. Until the data are evaluated by a hypothesis test, be cautious about in-terpreting any results from a two-factor study.

Wien a statistical analysis does indicate significant effects, you must still Abecareful about interpreting the outcome. In particular, if the analysis msultsinasignificant interaction, then the main effects, Whether significant: or not, maypresent a distorted View of the actual outcome. Remember, the main effect forone factor is obtained by averaging all the different levels of the second factor.Because each main effect is an average, it may not accurately represent any of theindividual effects that were used to computc the average. To illustrate this point,Figure 12.4 presents two hypotheticaloutcomes from a two-factor experiment.

Figure 12.4a shows a data matrixand a graph presenting one set of data froma two—factor study examining the effects of a drug intended to reduce arthritispain. The first factor in the study is drug versus no drug (placebo), and the sec—ond factor is administration of the drug on a full stomach versus an empty stom-ach. The dependent variable is a measure of arthritis pain. Looking first at thedate matrix, recognize that these results show a 10—point main effect for the drugfactor; the overall mean With the drug (second column) is 10 points lower thanthe overall mean for participants with no drug (first column). However, lookingat the graph, recognize that these data show an interaction. One interpretationof the interaction is that the drug effect depends cmstomachfullness; that is, theeffect of the drug is different for a full sbomach than for an empty stomach.Specifically, the drug seems to have no effect on an empty stomach, but it has a

301


(8} Data showing how an overall main effect can be distorted by an interaction

No Drug Drug

Em Overall Empty510mg Mean = 30 Mean = 30 Mean =30 so o———-—-o szomach

ø.få 20

, p a;Full Mean _ 30 Mean _ 10 Overall E

Stomach " — Mean = 20 10 FU"Stomach

Overall Overall No DrugMean = GU Mean = 20 Drug

(b) Data showing how an interaction can disguise main effects

No Drug Drug 40 EmptyStomach

Empty _ _ OverallStomach Mean _ 30 Mean _ 4-0 Mean = 35 % 30 \\

n.

': FullGl

% 20 StomachFull Overall

Stomach Mean ” 30 Mean — 20 Mean = 25 YO

Overall Overall No DrugMean = 30 Mean = 30 Drug

Figure 12.4 Hypothetical Results for a Two-Fatter StudyExamining the Effect of a Drug Intended to Reduce Arthritis PainOne factor ls drug dosage (drug versus no drug), and the second

factor is the state of the participant (empty stomach versus full stomach).

20—point effect on a full stomach. The 10—pointmain effect was obtained by av-eraging these two values (0 and 20), but in truththe 10-pointmain effect doesnot accurately describe the influence of the drug for anyone. In this example,concluding that the drug has a 10-point effect on behavior (based on the maineffect) would be misleading.

Now consider the second set of data shownin Figure 12.4b. Starting withthe data matrix, these data show no main effect for the drug factor; participantswith the drug and participants without the drug both show a mean of 30. How—ever, the data once again Show an interaction, suggesting thatthe effect of thedrug is different for empty stomach than it is for full stomach. Specifically, thedrug appears to increase pain scores by 10 points for empty stomach and to de—

crease scores by 10 points for full stomach. Averaging these two values (+10 and

—10) results in the zero difference for the main effect. However, the main effect

12 .4 Types of Fa ctori al Designs

does not accurately describe the results. In particular, it would be incorrect toconclude thatthe drug has no effect.

In general, the presence of an interaction can obscure or distort the maineffects of either factor. Wienever a statistical analysis produces a significant in—teraction, you should take a close—look at the date before giving any credibilityto the main effects.

INDEPENDENCE OF MAIN EFFECTS AND INTERACTIONS

The two—factor study allows researchers to evaluate three separate sets of meandifferences: (i) the mean differences from the main effect of factor 1, (2) the

mean differences from the main effect of factor 2, and (3) the mean differencesfrom the interaction between factors. The diree sets of mean differences are notonly separate, they are also completely independent. Thus, it is possible for theresults from a two-factor study to show any possible combination of main ef-fects and interaction. The dam sets in Figure 12.5 show several possibilities. Tosimplify discussion, the two factorsare labeled A and B, with factor A definingthe rows of the data matrix and factor B defining the columns.

Figure 12.53 shows data With mean differences between levels of factor Abut no mean differences for factor B and no interaction. To identify the maineffect for factor A, notice that the overall mean for the top row is 10 pointshigher than the overall mean for the bottom row. This 10—point difference isthe main effect for factor A, or simply the A effect To evaluate the mean effectfor factor B, notice that both columns have exactly the same overall mean, indi-

cating no difference between levels of factor B; hence, no B effect. Finally, theabscnce of an interaction is indicated by the fact that the overall A effect (theII)—point difference) is constant within each column; thatis, the A effect does notdepend on the levels of factor B. (Alternatively, the data indicate that the overallB effect is constant Within each row.) '

Figure 12.51) shows data with an A effectandeafBAeffect but no interaction.For these data, the A effect is indicated by the "IO—point mean difference be—tween rows, and the B effect is indicated by the ZO—point mean differencebetween columns. The fact that the 10—point A effect is constant within eachcolumn indicates no interaction.

Finally, Figure 12.5c shows data that display an interaction but no main ef—fect for factor A or for factor B. For these data, note that there is no mean dif—ference between rows (no A effect) and no mean difference between columns(no B effect). However, Within each row (or within each column) there are mean

differences. The “extra” mean differences within the rows and columns cannotbe explained by the overall main effects and therefore indicate an interaction.

12.4 NTYPES OF FACTORIAL DESIGNSThus far, we have examined only one version of all the many different types offactorial designs. In particular,

' All of the designs that we have considered use 3 separate group of partici—pants for each of the individual treatment combinations or cells. In researchterminology, we have looked exclusively at bemeen—mbjectr designs.

303


(a) Data showing a main effect for factor A but no main effect for factor B and no interaction

Factor B40

OverallM = 20 M _ 20 M: 20 so

Factor A o———-—-——-oM , 10 M _10 Overall 20 Two Levels

_ _ M=1O 10 I . ofFactorA

Overall Overall ! mf-M=15 M=15 FactorB

(b) Dala showing main effects for both factor A and factor B but no interaction

Factor B40 Two Levels

_ Overall of Factor AM : 10 M -— 30 M = 20 30

Factor A O H 20vera

M = 2° M = 40 M = 30 10

Overall OverallM: 15 M=35 FactorB

(c) Data showing no main effect for either factor, but an interaction

Factor B40

OverallM: 10 M: 20 M = 15 SD

Factor A 20 T elO all wu Lev s

M= 20 . M: 10 Mfr“; 10 >< of Factor A

Overall OverallM=15 M: 15 FactorB

Figure 12.5 Three Possible Combinations of Main Effectsand Interactions in a Two—Factor Experiment

' All of the previous examples use factors that are trueindependent variables.That is, the factors are manipulated by the researcher so thatthe researchstudy is an example of the expmmentai strategy.

Although it is possible to have a separate group for each of the individualcells (a between-subjects design), it is also possible to have the same group of in—dividuals participate in all of the different cells (a within—subjects design). In addi-tion, it is possible to constructa factorial design Where the factors are notmanipulated but rather are quasi—independent variables (see Chapter 9, page 236).Finally, a factorial design can use any combination of factors. As a result, a facto-rial Study can combine elements of experimental and nonexperimental designs,and it can combine elements of between—subjects and within—subjects designswithin a single research study. Atwo—factordesign, for example, may include onebetween—subjectsfactor(with a separate group for each level of the factor) and

12.4- Types of Factorial Designs

one within—subjects factor (with each group measured in several different treat—ment conditions).The same study could also include one experimental factor(with a manipulated independent variable) and one nonexperiinental factor (Witha pre—existing, nomnanipulated variable). The ability to mix designs within asingle research study provides researchers with the potential to blend several dif—ferent research strategies within one study. This potential allows researchers todevelop studies that address scientific questions that could not be answered byany single strategy. In the following sections we examine some of the possibilitiesfor mixed or blended factorial designs.

BETWEEN-SUBJECTS AND WITHIN-SUBJECTS DESIGNS

It is possible to constructa factorial study that is purely a between—«subjects de—sign; that isJ in Which there is a separate group of participants for each of the treat-ment conditions. As we noted in Chapter 10, this type of design has some definiteadvantages as well as some disadvantages. A particular disadvantage for a factorialstudy is that a between—subjects design can require a large number of participants.For example, a 2 >< 4 factorial design has eight different treatment conditions. Aseparate group of 30 participants in each condition requires a total of 240 (8 X30) participants. As noted in Chapter 10, another disadvantage of between—subject designs is that individual differences (characteristics that differ from oneparticipant to another) can become confounding variables and can increase thevariability of the scores. On the positive side, 3 between—subjects design com—pletely avoids any problem from order effects because each score is completelyindependent of every other score. In general, between—subjects designs are bestsuited to situations Where a lot of participants are available, Where individual dif—ferences are relatively small, and where order effects are likely.

At the other extreme, it is possible to construct a factorial study that is purely

a within—subjects design. In this case, a single group of individuals participates inall of the separate treatment conditions. As we noted in Chapter—H: tidstypenf"design has some definite advantages and disadvantages. A particular disadvantagefor a factorial study is the number of different treatment conditions that eachparticipant mustundergo. In a 2 >< 4 design, for example, each participant mustbe measured in eight different treatment conditions. The large number of differ»ent treatments can be very time consuming, which increases the chances that

participants Will quit and walk away before the study is ended (participant attri—tion). ln addition, having each participant undergo a long series of treatmentconditions can increase the potential for order effects (such as fatigue or practiceeffects) and can make it more difficult to counterbalanee the design to controlfor order effects. Two advantages of Within—subjects designs are; (l) they requireonly one group of participants and (Z) they eliminate ot greatly reduce the prob—lems associated with individual differences. In general, Within—subjects designsare best suited for situations where individual differences are relatively large andWhere there is little reason to expect order effects to be large and disruptive.

Mixed Designs: Within and Between SubjectsVery often a researcher encounters a situation in Which the advantages or con—venience of a between—subjects design apply to one factor but a within-subjectsdesign is preferable for a second factor. For example, a researcher may prefer to

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use a within—subjects design to take maximum advantage of a small group ofparticipants. However, if one factor is expected to produce large order effects,then a heten—subjects design should be used for that factor. In this situation itis possible to construct a mixed design with one between—subjects factor andone within—subjects factor. If the design is pictured as a matrix with one factordefining the rows and the second factor defining the columns, then the mixeddesign has a separate group for each row with each group participating in all ofthe different columns.

A factorial study that combines two different research designs is called a mixeddesign. A common example of a mixed design is a factorial study with one be-tween-subjects factor and one Within-subjects factor.

Figure 12.6 shows a mixed factorial design in a study examining the relation—ship between mood and memory. The qxpical result in this research area is thatpeople tend to recall information that is consistent with their current mood. T hus,people remember happy things when they are happy and remember sad thingswhen they are sad. In a study like the one shown in Figure 12.6, Teasdale and

Fogarty (1979) manipulated mood by instructing one group of participants toread a series of increasingly depressing statements (such as “Looking back on mylife, I wonder ifI have accomplished anything really worthwhile”) and anothergroup to read aseries of increasingly euphoric statements(such as “Life is so fulland interesting, it is great to be aliVe”). Thus, the researchers created a between—subjects factor consisting of a happy mood group and a sad mood group. In Fig—ure 126, the two groups correspond to the two rows in the matrix. All participantswere then presented a list of words that contained some words With positive/pleas—ant associations and some Words with negative/unpleasant associations. For eachparticipant, the reseurchers recorded how many pleasant words were recalled andhow many unpleasant words were recalled. Thus, the researchers created a within—subjects factor consisting of pleasant words and unpleasant words. The within—subjects factor corresponds to the Columns in Figure 12.6.

For a two-factor research study With three levels for factor A and two levels forfactor B, how many participants are needed to obtain five scores in eachtreatmentcondition for each of the following situations?

:1. Both factorsare between subjects.

1). Both factors are Within subjects.

c. Factor A is a between—subjects factor, and factorB is Within subjects.

A researcher would like to use a factorial study to compare two different strata»gies for solving problems. Participants will be trained to use one of the strate—gies and then tested on three different types of problems. Thus, the twostrategies make up one factor, the three types of problems make up the sec-ond factor, and problem-solving scores are the dependent variable. For thisstudy, which Factor-(s) should be between—subjects and which should be withinsubjects? Explain your answer.

12 .4 Types of Factorial Designs 307

Two separate groups of participants

Positive Negative 80

Words Words

Moan 60Happy Mean recall Mean recallMood M: 70 M: 23 “Baa" 40 Sad

. HaSad Mean rec-eii Mean reoail 20 ppy

Mood M = 48 M = 35

Posin've NegativeWords Words

Figure 12.6 Hypothetical Results From a Mixed Two—Factor Study ThatCombines Ono Between—Suhiects Factor and One Within-Subjects Factor

The researchers induced a feeling of happiness in one group ofparticipants and afeeling of sadness in another group to create

the between—subjects factor (happy/sad). For each group, memorywas tested for a set of words with positive connotations and a

set of words with negative connotations to create awithin—subjectsfactor (positive/negative).

EXPERIMENTAL AND NONEXPERIMENTAL DESIGNS

As we demonstrated with the alcohol~and~caffeine example at the beginning ofthis chapter, it is possible to constructa factorial study that is a purely experimen-tal research design. In this case, both factors are true independent variables thatare manipulated by the researcher. It alsois possible to construct a factorial studythat is purely nonexperimental. In a two—factor study, for example, a purely non—experimental design simply means that both factors are nonmanipulated, quasi—independent variables. For example, Bahr-ich and Hall (1991) examined thepermanence of memmy by testing recall for high school algebra 21nd geometry.The study compared two groups of participants, those who had taken non?”level math courses andthosewho had no advanced math courses in college. Notethat these groups were not created by manipulating an independent variable; in—stead, they are pre—existing, nouequivulent groups, and therefore they form a non—experimental factor. The second factorin the study was time. The researcherstested recall at different time intervals ranging from 3 years up to 55 years after

high school. Again, note that time is a nonmanipulated variable and hence an—other nonexperimenral factor. Thus, the study contains no manipulated variablesand is a purely nonexperimenral design. Incidentally, the group With no advanced ,college math showed a systematic decline in mathematics knowledge over time,but the group With college math showed excellent recall of mathematics evendecades after their high school courses.

Mixed Designs: Experimental and Nonexperimentalln the behavioral sciences it is very common for a factorial design to mix anexperimental design for one fuctor and a nonexperimental design for anotherfactor. This type of study is another example of a mixed design. Typically, thiskind of study involves onefactor thatis atrueindependent variable consisting ofa set of manipulated treatment conditions, and a second factor that is a quasi—independent variable falling into one of the following categories:

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1. The second factor is a pre-existing" participant characteristic such as age or genider. In this case, the researcher wants to examine whether the treannent condi—

tions have the same impact on males as on females, or the question is whetherthe treatment effects change as a function of age. Note that pre—existing char-

acteristics create noncquivalent groups; thus this is a quasi—experimental factor.

2. The second factor is lime. In this case, the research question concerns howthe different treatment effects persist over time. For example, two differenttherapy techniques may be equally effective immediately after the therapyis concluded, but one may continue to have an effect over time, While theother loses effectiveness as time passes. Note that time is not controlled ormanipulated by the researcher, so this is a quasi—cxpérimenral factor.

A factorial study that combines two different research strategies is called amixed design. A commonexample of a mixed design is a factorial study withone experimental factor and one nonexp erimental factor.

For example, Shrauger (1972) examined how the presence or absence of anaudience can influence people’s performance. Half of the research participantsworked alone (no audience) on a concept—formation task, and half of the partic—ipants worked with an audience of people Who claimed to he interested in 01)—serving the experiment. Note that the audience-versus—noaudience variable ismanipulated by the researcher, so this is an experimental factor. The secondfactor in Shrauger’s study was self—esceem. In each of the audience groups, par—ticipants Were divided into high self—esteem and low self—esteem groups. Notethat the second factor, self—esteem, is a pre-existing participant variable andtherefore a nonexperimental factor. The structure of this study, including re—sults similar to Shrauger’s actualdata, is shown in Figure 12.7. Notice that theresults show an interaction between the two factors. Specifically, the presence ofan audience had a large effect on participants With low self—esteem, but the au-dience has essentially no effect on those with high self—esteem.

A researcher would like to compare CW0 therapy techniques for treating deprES—sion. One technique is suspected to have only temporary effects, and theother is expected to produce permanent or long-lasting effects. Describe afactorial research studythat would compare the effectiveness of the twotechniques and answer the question about the duration of their effective-ness. Identify which factor is experimental and which is nonexperirnental,

A researcher would like to compare'two therapy techniques for treating depres—sion. One technique is expected to be very effective for patients with rela—tively mild depression, and the other is expected to be more effective fortreating moderate to severe depression. Describe a factorial research studythat would compare the effectiveness of the two techniques and answer thequestion about which is better for different lcls of depression. Identifywhich factor is experimental and which is nonexperimental.

12 .4 Types of Factorial Designs

Experimental! Factor (Manipulator!) Low

10 Self-EsteemNo 8

Audience Audience

High Mean errors Mean errors yeah 6Self—Esteem M = 2.1 M = 2.2 "ors 4

Low Mean errors Mean errors 2 o————————o High

Self-Esteem M = 6.1 M: 9.5 SSH-Este”

No AudienceAudience

Figure 12.7 Hypothetical Results From a Mixed Two-Factor Study ThatCombines One Experimental Factor and One Quasi-Experimental Factor.

To create the experimental factor (audience versus no audience), theresearchers manipulated whether or not the participants performed infront of an audience. Within each experimental condition, to create a

quasi-experimental factor, two nonequivalent groups of participants wereobserved (high versus low self—esteem). The dependent variable is the

number of errors committed by each participant.

PRETEST—POSTTEST CONTROL GROUP DESIGNS

In Chapter 9 we introduced a quasi-experimental design known as thepretest—posttest nonequivslent control group design (page 223). This design in—volves two separate groups of participants. One group, called the treatmentgroup, is measured before and after receiving a treatment. A second group, thecontrol group, also is measured twice (pretest and posttest) but does not receiveany treatment between the two measurements. Using the notation introducedin Chapter 9, this design can be represented ss follows:

0 X O (treatment group)

O 0 (control group)

Each 0 represents an observation or measurement, and the X indicates a treat—ment. Each row corresponds to the series of events for one group.

You should recognize this design as an example of a two-factor mixed de—sign. One factor, treatment/control, is a between-subjects factor. The other fac-

tor, pre/post, is a within—subjects factor. Figure 12.8shows the design using thematrix notation customary for factorial designs.

Finally, the design introduced in Chapter 9 was classified as quasi-experimen—tal because it used nonequivalent groups (for example, students from two differenthigh schools or patients from two different clinics). On the other hand, if a re-searcher has one sample of participants and can randomly assign themto the twogroups, then the design is classified as a true experiment. The experimental ver—sion of the pretest—posttest control group design can he represented as follows:

R O X 0 (treatment group)

R O O (conan! group)

309


Protest Posttest

Protest scores Posttest scoresTreatment for participants for participants

Group who receive who receivethe treatment the treatment

Protest scores Posttest scoresControl for participants for participantsGroup who do not receive who do not receiva

the treatment the treatment

Figure 12.8 The Structure of a Pretest—Posttest Control GroupStudy Organized as aTwo—Factor Research Design

Notice that the treatment/control factor is abetween-subjectsfactor and the pre/post factor isawithin-subjects factor.

The letter R symbolizes random assignment, which means that the researcherhas control over assignment of participants to groups and therefore can createequivalentgroups.

HIGHER-ORDER FACTORIAL DESIGNS

The basic concepts of a two—factor research design can be extended to morecomplex designs involving three or moro factors; suchdesigns are referred to ashigher—order factorial designs. A three—factor design, for example, might lookat academic performance scores for We different teaching methods (factor A),for boys versus girls (factor B), and for first—grade versus second—grade classes(factor C). In the three—factor design, the researcher would evaluate main ef-fects for each of the three factors, as well as a set of two—way interactions: A >< B,B >< C, and A >< C. ln addition, the extra factor introduces the potential for athree-way interaction, A >< B >< C.

The logic for defining and interpreting higher—order interactions followsthe pattern set by two—way interactions. For example, a two-way interaction,A >< B, indicates that the effect of factor A depends on the levels of factor B. Ex—tending this definition, a three—way interaction, A >< B >< C, indicates that thetwoeway interaction between A and B depends on the levels of factor C. For ex—ample, two teaching methods may be equally effective for boys and girls in thefirst grade (no two—way interaction between method and gender), but in thesecond grade one of the methods works better for boys and the other methodworks better for girls (an interaction between method and gender). Because themethod by gender pattern of results is different for the first—graders and thesecond—graders, there is a three—way interaction. Although the general idea of athree—way interaction is easily grasped, mostpeople have great difficulty com—prehending or interpreting a four—way (or higher) interaction. Although it ispossible to add factors to a research study Without limit, studies that involvethree or more factors can produce very complex resultsthatare difficult to un—derstand and thus often have limited practical value.

12.5 Applications of Factorial Designs

»:v fr

12.5 APPLICATEONS OF FACTORIAL DESEGNS

Factorial designs provide researchers with a tremendous degree of flexibilityand freedom for constructing research smdies. As noted earlier, the primary ad—vantage of factorial studies is that they allow researchers to observe the influ—ence of two (or more) variables acting and interacting simultaneously. Thus,factorial designs have an almost unlimited range of potential applications. Inthis section, however, we focus on three specific situations where adding a sec—ond factor to an existing study answers a specific research question or solves aspecific research problem.

ADDING A SECOND FACTOR TO A PREVIOUS STUDY

Often factorial designs are developed when researchers plan studies thatare in—

tended to build on previous research results. For example, & published reportmay compare a set of treatment conditions or demonstrate the effectiveness of aparticular treatment by comparing the treatment condition with a control con-dition. The critical reader asks questions suchas

Would the same treatment effects be obtained if the treatments were ad—ministered under different conditions?

Would the treatment outcomes he changed if individuals with differentcharacteristics had participated?

Developing a research study to answer these questions. would involve a fac—torial design. Answering the first question, for example, requires administeringthe treatments (one factor) under a variety of different conditions (a second fae—tor).The primary prediction for this research is to obtain an interaction he—tween factors-, that is, the researcher predicts that the effect of the treatmentsdepends on the conditions under which they are administered. (See Figure 12.9as an example.) Similarly, the second question calls for a factorial design involv—ing the treatments (factor one) and different types of participants (factor two).Again, the primary prediction is for an interaction.

Because current research tends to build on past research, factorial designs arefairly common and very useful. In 3 single study, a researcher can replicate and ex»pand previous research. The replication involves repeatin g the previous study byusing the same factor or independent variable exactly as it was used in the earlierstudy. The expansion involvm adding a second factor in the form of new eondi«tions or new participant characteristics to determine Whether or not the previouslyreported effects can be generalized to new situations or new populations.

One example of adding a new factor to an existing study comes from the areaof human memory. Psychologists have demonstrated repeatedly that the ability toremember a list of items is determined in large part by the method used to studythe items. If studying consists of shallow and superficial activity, then memorywill be poor. On the other hand, deeper, more substantial processing producesbetter memory. A common demonstration of this phenomenon involves present—'ing the same list of words to two groups of participants. One group is instructedto perform a rhyming task requiring relatively superficial attention to the soundof each word. The second group is assigned a more substantial task requiring that

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31 2 Chapter 12 Factorial Designs

(a) The original study simply compares two treatments (A and B) using participants who arereasonably wen rested,

Scores for a groupof participants in

Treatment A treatment A whoare measured whenwell rested

Scoresfor a group .of participants fn

Treatment B treatment B whoare measured whenwell rested

(b) The new study adds a second factor by manipulating the participants” state of rest. Thenew question is whether the difference between the two treatments depends on the participants’degree of rest.

Degree of Rest(the New Factor}

We” Rested Fatigued— '— '— — '"— — ":

Scores for a group Scores for a group Iof participants in of participants In l

Treatment A treatment A who treatment A who Iare measured when are measured whenwal'l rested fatigued l

— -—- -—— —- —- — »!

Scores fora group Scores for a group Iof participants in of participants in

Treatment B treatment B who treatment B whoare measured When are measured when lwe” rested fatigued

——————— JFigure 12.9 Creating a New Research Study by

Adding aSecond Factor to an Existing Study

they analyze the meaning of each Word, As expected, the task of analyzing mean—ing results in better memory performance. This outcome should not be surpris—ing, but it does not necessarily mean that the bestway to study a list of words is toanalyze their meaning. For example, if you were preparing for a spelling test, youWould probably do better by studying the sounds and letter patterns of each wordthan you would by analyzing its meaning. T0 test the notion thatthe “best” wayto study depends on the type of test, a two-factor experiment was created (Brans-ford, Franks, Morris, & Stein, 1979). The first factor simply repeated the stan—dard experiment comparing a rhyming task and a meaning task. The second factorconsisted of two different types of memory test. F or one test, participants weresimply asked to recall as many wordsas possible (replicating the previous re—search). The second memory test focused on rhymes. Participants were asked, forexample, “Was there a word in the hat that rhymes with boat?” When the memory

12. 5 Applications of Factorial Designs — 31 3

test was based on rhymes, the data show that the participants who studied byrhyming outperformed those who studied by analyzing meaning. Thus, the re-sults produced an interaction: The effectiveness of a study method depends onthe type of test.

A researcher has demonstrated that a new noncompetitive physical educationprogram significantly improves self—esteem for children in a kindergartenprogram.

a. What additional information can be obtained by introducing participantgender as a second factor to the original research study?

b. What additional information can be obtained by adding participant age(third grade, fifth grade, and so on) to the original study?

REDUCING VARIABIUTY IN BETWEEN-SUBJECTS DESIGNS

In Chapter 10 We noted that individual differences such as age or gender cancreate serious problems for between—subjects research designs. One such prob—lem is the simple fact that differences between participants can result in largevariability for the scores within a treatment condition. Recall that large variabil—ity can make it difficult to establish any significant differences between treatmentconditions (see page 251). Often a researcher has reason to suspect that a specificparticipant characteristic such as age is a major factor contributing to the vari—ability of the scores. In this situation, it often is tempting to eliminate or reducethe influence of the specific characteristic by holding it constant or by restrictingits range, For example, suppose that a researcher compares two treatment condi—tions using a separate group of children for each condition. VWthin each'group,

the children range in age from 5 to 10years—old. The researcher is concerned—that the older children may have higher scores than the younger children simplybecause they are more mature. If the scores really are related to age, then therewill be big differences and high variability within each group. In this situau'on,the researcher may be tempted to restrict the study by holding age constant (forexample, using only 10-year-old participants). This Will produce more homoge—neous groups With less variability, but: it will also limit the researchers ability togeneralize the results. Recall that limiting generalization reduces the external va—lidity of the study. Fortunately, there is a relatively simple solutionto thisdilemma that allows the researcher to reduce variability within groups Withoutsacrificing external validity. The solution involves using the specific variable as asecond factor, thereby creating a two—factorstudy.

Consider the between—subjects design shown in Figure 12.1021. The designinvolves a comparison of two treatment conditions (A and B) with a group of 12participants in each condition. Suppose that the dependent variable for this de—sign is performance on a memory task, and the researcher suspects thatthe par—ticipants” age is a variable that will influence memory performance. Because agevaries Within each group, this variable is likely to cause increased variability forthe memory scores within each group. To eliminate this problem, the researcher

Learning en åCheck


(a) A study comparing two treatments with large age differences among the participants in eachgroup

Treatment A Treatment B

A group of12 A group of 12participants participantsranging in age ranging in agefrom 5 to from 5 to15 years old 15 years old

' a

(b) Using participant age as a second factor, the participants have been separated into amalier,more homogeneous groupe, The smaller age differences within each group should reduce thevariability of the scores.


A group of 4 A group 01‘4Younger participants participants

_ ranging in age ranging in age(5 8 Years Old) from 5 to frem 5 to

8 years old 8 years old

Agroup 0f4 Agroup of4Middle participants participants

ranging in age ranging in age(941 Years Old) from B to from 9 to

11 years old 11 years old

A group of 4 A group of 4Older participants participants

— ranging in age ranging in age(12 15 Years CMJ from 12 to from 12 to

15 years Did 15 years eid

Figure 12.10 A Participant Characteristic (Agel used as a Second Factorto Reduce the Variabiiity of Scores in aResearch Study

(a) Each treatment condition contains a wide range of ages, which

probably produces large variability among the scores, (b) Theparticipants have been separated into more homogeneous agegroups, which should reduce the variability within each group.

simply regroups the participants Within each treatment. Instead of one group ineach treatment, the researcher uses age, as a second factor, to divide the partici—

pants inte three groups Within eachtreatment:a younger age group (SMB years),a middle age group (9—11 years), and-an older group (12—15 years). The result isthe tive—factor experiment shown in Figure 12.1013, with one factor consistingof the two treatments (A and B) and the second factorconsisting of the threeage groups (younger, middle, older).

By creating six groups of participants instead of only two, the researcher hasgreatly reduced the individual differences (age differences) within each groupwhile still keeping the full range of ages from the original study. In the new, two-factor design, age differences still exist, but now they are differences hertugen

125 Applications of Factorial Designs

gwupr rather than variability within groups. Thc variability has been reduced with—out sacrificing external Validity. Furthermore, die researcher has gained all of theother advantages that go with a two—factor design: In addition to examining howthe different treatment conditions affectmemory, the researcher can now exam—ine how age (the new factor) is related to memory and can determine whetherthere is any interaction between age and the treatment conditions.

Under What circumstances would a researcher reduce the variability of thescores by adding gender as a second factor in a between—subjects study com—paring two treatments?

EVALUATING ORDER EFFECTS IN WlTHIN-SUBJECTS DESIGNS

In Chapter 11 we noted that order effects can he a serious problem for Within—subjects research studies. Specifically, in a within—subjects design each partici—pant goes through a series of treatment conditions in a particular order. In thissituation, it is possible that treatments that occur early in the order may influ—ence a participant’s scoresfor treatments that occur later in the order. Becauseorder effects can alter and distort the true effects of a treatment condition, theyare generally considered a confounding reliable that should be eliminated fromthe study. In some circumstances, however, a researcher may want to investigatethe order effects (where and how big they are). In some situations, a researchermay be specifically interested in how the order of treatments influences the ef-fectiveness of treatments, (Is treatment A more effective if it comes before treat-ment B or after it?) Or a researcher may simply Want to remove the order effectsto obtain a clearer view of the date. In any of these situations, it is possible tocreate a research design that actually measures the order effects and separatesthem from the rest of the data.

Using Order of Treatments as a Second FactdrTo measure and evaluate order effects, it is necessary to use counterhalancing(as discussed in Chapter 11). Remember that cotmterhalancing requires sepa—rate groups of participants, with each group going through the set of treatmentsin a different order. The simplest example of this procedure is a within-subjectsdesign comparing two treatments, A and B. The design is counterbalanced sothat half of the participants begin With treatment A and then move to treatmentB. The other half of the participants start with treatment B and then receive ,treatment A. The structure of this counterbalanced design can be presented as amatrix, with the two treatment conditions defining the columns and the orderof treatments defining the tows (see Figure 12.11).

You should recognize the matrix structure in Figure 12.11 as a two—factorresearch design and that this is an example of a mixed design. In particular, thetreatment factor (A and B) is a Within—subjects factor, and the order factor (ABand BA) is a between-subjects factor. By using the order of treatments as a sec—ond factor, it is possible to evaluate any order effects that may exist in the data.There are three possible outcomes that can occur, and each produces its ownpattern of results.

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fi—

316 Chapter 12 Factotitd Designs


X -—————> X Group 1X ——-—> X Halt the participantsX———————+ X get treatment A first andX -———+ X treatment B second.

X 4—-——- X Group 2X 4—————— X Half the participantsX <———«—— X get treatment B first andX +—-———— X treatment A second.

Treatment A Treatment 8

Mean score in Mean score inGroup 1 treatment A for treatment B forA —> B participants who had participants who had

treatment A first treatment B second

Order ofTreatments

Mean score In Mean score inGroup 2 treatment A for treatment B torB —> A participants who had participants who had

treatment A second treatment B first

Figure 12.11 Order of Treatments Added as aSecond Factor to aWithin-Subjects Study

The original study uses a counterbalanced design to compare twotreatment conditions (A and B). Thus, hahc of the participants have

treatment A first, and half have treatment B first. Similarly, half of theparticipants have treatment A second, and half have treatment B second.

1.

2.

No order fists; When there are no order effects, it does not matter whethera treatment: is presented firs: or second. An example of this type of result isshown in Figure 12.12. For these data, when treatment A is presented first(group l), the mean is 29, and when tree tment Ais presented second (group2), the mean is still 20. Similarly, the order of presentation has no effect onthe mean for treatment B. As a result1 the difference between treatments is10 points for both groups of participants. Thus, the treatment effect (fac-tor 1) does not depend on the order—of—treatments (factor 2). You should rec—ognize this pattern as an example of data with no interaction. When there areno order effects, the data show a pattern with no interaction. It makes no dif—ference Whether a treatment is presented first or second; the mean is the samein either case.

Symmetri-tal order afiémr When order effects exist, the scores in the secondtreatment are influenced by participationin the firSt treatment. An exampleof this outcome is shown in Figure 12.13. To create these data a 5—pointorder effect was added to the data in F igure 12 . 12. Thus, when treatment B


XH—————>X+OX—-———>X+0X——-——>X+0X—————-—>X+O

X+0+———XX+0<——-———-—XX+04-————'XX+0<—-——r——-X

Treatment A

12.5 Applications of Factorial Designs 31 7

Group 1Half the participantsget treatment A first andtreatment B second.

_ Group 2

Half the participantsget treatment B first andtreatment A second.

Treatment B

. Mean score inMean score mGroup 1 treatment B plus a Overall M ; 25

A ——-> B "same—”åg G-poim order effect‘ M = 30

Order ofTreatments

Mean score in Mean score in

Group 2 treatment A plus aB —+ A O—point order effect ingame—mag Overall M: 25

M = 20 —

Overall M : 20 Overall M : 30

Figure 12.12 Treatment Effects and Order Effects Revealed in a Two-Factor Design Using Order of Treatments as a Second Factor

A ii)-point difference between the two treatment conditions isassumed, with the mean score for treatment A equal to M = 20 andthe mean score for treatment B equal to M = 30. No order efiectsis

also assumed. Thus, participating in one treatment has no effect (0 points)on en individual's score in the following treatment. In the two—factor

analysis, the treatment effect shows up as a10—point main effect for thetreatment factor, and the absence of any order effects is indicated by theabsence of an interaction between treatments and order of treatments.

occurs second (after A), the mean score is raised by 5 points. This occursfor the participants in group 1. Also, When treatment A is second (after B),the mean is raised by 5 points. This occurs for the individuals in group 2.

In this situation, thetreatmenteffect: (A versus B), depends on the orderof treatments. Thus, the effect of one factor depends on the other factor.

You should recognize thisas an example of interaction. When order effectsexist, they Show up in the two—factor analysis as an interaction betweentreatments and the order of treatments.

For these data, a symmetrical order effect was created; that is, the sec—ond treatment always gets an extra 5 points, whether it is treatmentA ortreatment B. This symmetry appears in the data as a symmetrical interac-tion. In the graph of the data, for example, the two lines cross exactly” at the



X -———-b X + 5 Group 1X ——-——-> X + 5 Hatt the participantsX —-—-——> X + 5 get treatment A first andX ————-—> X + 5 treatment B second.

X + 5 <——— X Group 2X + 5 <-—-———- X Half the participantsX + 5<——-——— X get treatment B first andX + 5 4—————— x treatment A second.


. Mean score inMean score InGroup 'I treatment B plus e Overall = 275

A -—+ B ”Gaming? 5—point order effect M_ M: 35

Order ofTreatments

Mean score in .Mean score rn

Group 2 treatment A plus a _B —> A 5-point order effect treatgrfe—nåä Overall M * 27'5

M = 25 *

Overall M = 22.5 Overall M: 32.5

40

35 Group i (A — B)

SD Group2 (B—A)

MeanScore 25 ”

20

15

1O

5

A B

TreatmentsFigure 12.13 Symmetrical Order Effects Revealed in a Two—Factor Design

Using Order of Treatments as a Second FactorA it)-point difference between the» two treatment conditions is assumed,

with the mean score for treatment A equal to M = 20 and the meanscore for treatment B equal to M = 30. A symmetrical 5-point order

effect is added. After participating in one treatment, the order effectadds 5points to each participant’s score in the second treatment. In

this situation, the order effect appears as an interaction betweentreatments and the order of treatments.

Chapter Summary 31 9

center. Also, the 5 —point difference between the two groups in treatment A(leit—hand side of the graph) is exactly equal to the 5—point difference be—tween the groups in treatment B (right—hand side of the graph). This syin—metry only exists in situations Where the order effects are symmetrical.

. 3. Asymmetriml order agam: Figure 12.14 shows data Where the order effectsare not symmetrical. To create these data we started with the original meansin Figure 12.12, and then added order effects as follows:

a. The participants in group l received the treatments in the order AB. For

these participants, we added a 10-point order effect. Thus, the mean fortreatment B is increased by 10 points.

b. The participants in group 2 receive the treatments in the order BA. Forthese participants, we added a 5—point order effect. Thus, the mean fortreatment A is increased by 5 points.

Notice that the graph in Figure 12.14 shows an interaction, just as withsymmetrical order effects. Again, the existence of an interaction in thisanalysis is an indication thatorder effects exist. For these data, however, theinteraction is not symmetrical; in the graph, the two lines do intersect attheir midpoints. Also, the difference between groups in treatment A is muchsmaller than the difference in treatment B. In general, asymmetrical ordereffects produce a lopsided or asymmetrical interaction between treatmentsand ordets as seen in Figure 12.14.

In the preceding examples, we created order effects and added them into thedata. In this artificial situation, we knew that order effects existed and how big

they were. In an actual experiment, however, a researcher cannot see the order ef—fects. However, as we have demonstrated in the three examples, using order oftreatments as a second factor makes it possible to examine any order effects thatexist in a set of data; their magnitude and nature are revealed in the interaction.

Thus, researchers can observe the order effectsin their data and can separatethem from the effects of the different treatments.

LearningWhat does it mean to say that order effects are “symmetrical” or “asymmetrical”? C h l<e C S

If order effects exist in a research study, how do they show up in a two—factoranalysis where order of treatments has been added as a second factor?

CHAPTER SUMMARY

To examine more complex, real-life situations, re—

searchers often design research studies that includemore than one independent variable. These designsare called factor-lai designs. Factorial designs are com—monly described With a notation system that identi—fies not only the number of factors in the design but

also the number of values or levels that exist for eachfactor. For example, a 2 X 3 factorial design is a de—sign With two independent variables, two levels of thefirst factor, and three levels of the second factor.

The results from a factorial design provide in—formation ahout how each factor individually effects

320 Chaptm‘ 12 Factorial Designs


X——‘——+X+1UX-—-—-——>X+1OX—————*X+1OX—--——->X+1O

Group 1Half the participantsget treatment A first andtreatment B second.

X+5<*—'——XX+5<--——-—XX+5+——-——XX+5<————=—=—-———X

&

Group 2Half the participantsget treatment B first andtreatment A second.

Treatment A

Mean score in

Treatment B

Mean score in

Group 1 treatment B plus 3 Overall M: 30A -—> B Neame—må 10-polnt order effect

* M = 40Order of

Treatments

Mean score in Mean score inGroup 2 treatment A plus a _B ", A 5-point order effect treatment B Overall M — 27,5

M = 25 M = 30

Overall Mm 225 Overall M = 35

40 Group 1 (A — B)

35

30 Group 2 [B — A)

MeanScore 25

20

15

10

5

A B

Treatments

Figure 12.14 Asymmetrical Order Effects Revealed in aTwo-Factor Design Using Orderotc Treatments as a Second Factor

A 10—point difference betwaen the tw0 treatment conditions is assumed,with the mean score for treatment A equal to M = 20 and the mean scorefor treatment B equal to M = 30. An asymmetrical order effect is added.

After participating in treatment A, the order effect adds 10 points to eachparticipant's score, and after participating in treatment B the order effect

adds 5 points to each participant's score. in this situation, the ordereffects appear as an interaction between treatments and order of

treatments. Because the order effects are not symmetrical, thestructure of the interaction is aiso not symmetrical.

behavior {main effects) and how the factors j ointlyaffect behavior (interaction). The value of a factorialdesign is that it allows a researcher to examine howuniquecombinations of factors, acting together, h1—flueuce behavior. When the effects of a factor vary

depending on the levels of another factor, it meansthat the two factors are combining to produceunique effects and thatthere is an interaction he—tween the factors.

ln factorial designs, it is possible to have a sep—arate group for each of the conditions (a between—subjccts design) and to have the same group ofindividuals participate in all of the different condi-tions (a Within—subjects design). ln addition, it is

possible to constructa factorial design Where thefactors are not manipulated but rather are quasi—independent variables. Finally, a factorial designcan use any combination of factors to create a vari—

K E Y W O R D S

factor main effectfactorial design

J E X E R C l 5 E 5G

1. In addition to the key words, you should alsobe able to denne the following terms:two—factor design single-factor designthree—factor design levels

higher—order factorial design2. Describe the major advantage of using a facto-

rial design.3. Explain why it is better to use &factorialde—

sign in research than to conduct two separatestudies.

4. How many independent variables are there ina 4- X 2 >< 2 factorial design?

. What is a main (fired?6. HOW many main effects are there in a study ex—

amining the effects of treatment (behavioralversus psychoanalytic versus cognitive) and ex—perience of the therapist (experienced versusnot experienced) on depression?

7. Use the values in the following matrix to an—swer questions a, b, and c:

U1

Exercises 32’]

ety of mixed designs. As a result, a factorial studycan combine elements of experimental and quasi—experimental designs, and it can combine elementsof between—subjects and Within—subjects designswithin a single research study.

Although factorial designs can be used in a va—riety of situations, three specific applications werediscussed: (1) Often a new study builds 011 existingresearch by adding another factor to an earlier re—search study; (2) using a participant variable such asage or gender as a second factor can separate par—ticipants into more homogeneous groups and

thereby reduce variability in a between—subjects de—sign; and (3) When the order of treatments is used

as a second factor in a counterbalanced within—subjectsdesign, it is possible to measure and evalu—ate the order effects.

mixed designinteraction between factors

Before _ Afterligaatmgnt Treat-m cnr—,., ..

Meles M: 20 M: 24- overall M: 22Females M: 22 M: 32 overall M = 27

overall overall

M = 21 M : 28

a. What numbers are compared to evaluerethe main effect for the treatment?

b. What numbers are compared to evaluatethe main effect for gender?

c. What numbers are compared to evaluatethe interaction?

8. The following matrix represents the results(the means) from a 2 >< 2 factorial study. Onemean is not given.a. What value for the missing mean would re—

sult in no main effect for factor A?b. What value for the missing mean would re—

sult in no main effect for factor B?


c. What value for the missing mean would re—sult in no interaction?

Al A2

B1 40 20

B2 30

9. Examine the data in the following graph.

20

M Gan Tremmenn

Sonia Treatment 2

5

5 Years 6 Years 7 Years

Aga

a. Is there a main effect for the treatment factor?

b. Is there a main effect for the age factor?c. Is there an interaction between age and the

treatment?

(E: InfoTrac COLLEGE EDITION

1. Use Info—Trac College Edition to locate a re—search study using a factorial design. Note:You can try a subject term based on en area ofinterest, but the most direct path to findingfactorial designs is to use a key word such as2 >< 2 ut 2 >< 3. Once you find a factorial study,

do the following:a. Identify each factor and the specific levels

that are used for each factor.b. Specify whether each factor is an experi-

mental or a quasi—Experimental factor.c. Specify whether each factor is a between—

subjects or a within—subjects factor.d. Describe the results of the study in terms of

main effects and interactions (Which main

10.

11.

12.

13.

14.

15.

Explain the issue of interdependence and in—

dependence of factors and how it is related tointeraction.

Under what circumstances Will the main ef—fects in a factorial study not provide an accu—rate description of the results?A researcher conducts a 2 >< 3 X 2 factorialstudy with 20 participants in each treatment

. condition.

a. If the researcher uses an exclusively be—

tween—subjects design, how many individu—alsparticipatein the entire study?

b. If the researcher uses an exclusively Within-subjects design, how many individuals par—ticipate in the entire study?

A researcher would like to compare two ther—apy techniques for treating depression in bothadolescents and adults. Identify which factor isexperimental and Which is quasi—experimental.Suppose a researcher has demonstrated that aparticular treatment is effective in reducingstress in adults. Descrihe some ways to add asecond factor to expand these results.Under what circumstances would adding gen—der as a second factor in a between—subjectsstudy not reduce variability?

EXERCiSES

effects were significant? Was the interactionsignificant?) Then describe the results of thestudy in terms of the variables studied.(How did the independent variables affectthe dependent variable? What pattern of re—sults was obtained?)

l 2. Retrieve the between—subjects study you used forthe InfoTrsc College Edition exercise in Chap—ter 10 and the Within—subjects study you used inChapter 11. In each case, explain how the studycould be expanded by adding a second factor.What new information might be obtained fromyour proposed two—factor studies?

factorial designs overview 6 that - bi.no or reduce the influence of any outside variabler; ... for...

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