Estimating and Testing Mediation and Moderation in Within ...subject treatment variables (e.g., Fiske, Kenny, & Taylor, 1982; Neuberg, 1989). In general, these ar-ticles have explored

Psychological Methods2001. Vol. 6, No. 2, 115-134

Copyright 2001 by the American Psychological Association, Inc.1082-989X/01/S5.00 DOI: 10.1037//1082-989X.6.2.115

Estimating and Testing Mediation and Moderation inWithin-Subject Designs

Charles M. JuddUniversity of Colorado at Boulder

David A. KennyUniversity of Connecticut

Gary H. McClellandUniversity of Colorado at Boulder

Analyses designed to detect mediation and moderation of treatment effects areincreasingly prevalent in research in psychology. The mediation question concernsthe processes that produce a treatment effect. The moderation question concernsfactors that affect the magnitude of that effect. Although analytic procedures havebeen reasonably well worked out in the case in which the treatment varies betweenparticipants, no systematic procedures for examining mediation and moderationhave been developed in the case in which the treatment varies within participants.The authors present an analytic approach to these issues using ordinary leastsquares estimation.

The issues of mediation and moderation have re-ceived considerable attention in recent years in bothbasic and applied research (Baron & Kenny, 1986;James & Brett, 1984; Judd & Kenny, 198Ib; Mac-Kinnon & Dwyer, 1993). In addition to knowingwhether a particular intervention has an effect, theresearcher typically wants to know about factors thataffect the magnitude of that effect (i.e., moderation)and mechanisms that produce the effect (i.e., media-tion). Such knowledge helps in both theory develop-ment and intervention application.

To illustrate the difference between mediation andmoderation, consider a design in which a researcher isinterested in whether students who are taught with anew curriculum (the treatment condition) show higherperformance on a subsequent standardized test thanstudents taught under the old curriculum (the control

Charles M. Judd and Gary H. McClelland, Department ofPsychology, University of Colorado at Boulder; David A.Kenny, Department of Psychology, University of Connecti-cut.

Preparation of this article was partially supported byNational Institute of Mental Health Grants R01 MH45049and R01 MH51964.

Correspondence concerning this article should be ad-dressed to Charles M. Judd, Department of Psychology,University of Colorado, Boulder, Colorado 80309-0345.Electronic mail may be sent to [email protected].

condition). Assuming that a performance difference isfound, one might plausibly hypothesize different me-diating mechanisms for this effect. The new curricu-lum might increase students' interest in the subjectmatter; it might cause students to study harder outsideof class; or it might convey the material more clearly.These are alternative reasons why the performancedifference is found, that is, alternative mediators ofthe treatment effect. The researcher might also be in-terested in factors that affect the magnitude of thedifference between performance following the oldcurriculum and performance following the new one.That difference might be larger or smaller for differ-ent types of students or in different types of class-rooms or when taught by different kinds of teachers.All of these then are potential moderators of the treat-ment effect.

It is possible that the same variable may serve asboth a mediator and a moderator. For instance, studytime might serve both roles. First, as a mediator, thenew curriculum might lead to higher performance be-cause it causes students to study more. Second, as amoderator, the treatment might be especially effectivefor students who spend more time studying.

Procedures for assessing mediation and moderationhave been relatively well worked out through ordinaryleast squares regression and analysis of variance pro-cedures. Mediation is assessed through a four-stepprocedure (Baron & Kenny, 1986; Judd & Kenny,

115

116 JUDD, KENNY, AND McCLELLAND

1981b) that involves testing whether the treatmenteffect is reduced in magnitude when the mediator iscontrolled. The test of moderation involves a test ofthe statistical interaction between the treatment vari-able and the moderator, examining whether a variablethat is a product of the two predicts the outcome vari-able over and above the treatment and moderator(Aiken & West, 1991; McClelland & Judd, 1993).

These analytic procedures, however, apply only tothe situation in which the treatment variable variesbetween experimental units, with some units or par-ticipants receiving one treatment and others receivinganother. However, treatment comparisons often in-volve within-subject comparisons, examining everyexperimental unit or participant in each treatment con-dition. For example, as a means of evaluating theeffects of mood differences on judgment, the sameparticipants might be observed both in a positivemood condition and in a negative one. Or to evaluatea new remedy for chronic migraine headaches, thesame participant might one day take a placebo pill andthe next day the new remedy pill (or vice versa). Adesign in which the independent variable varieswithin participants rather than between them is par-ticularly useful when the effects of the independentvariable are relatively transitory and the independentvariable is easily varied. Such a design can often re-sult in a dramatic increase in statistical power relativeto between-subjects designs, because variance due tostable individual differences contributes to the errorvariance in testing between-subjects effects but not tothe error variance in within-subject analyses. As aresult, within-subject designs often involve smallersamples than those typically used in between-subjectsdesigns. Surprisingly, there is very little work that hasformally considered issues of mediation and modera-tion in within-subject designs (for an exception, seeJudd, McClelland, & Smith, 1996).

In the analysis of variance literature, a variety ofauthors have dealt with the use of covariates inwithin-subject designs (for instance, Bock, 1975; Hu-itema, 1980; Khattree & Naik, 1995; Myers, 1979;Winer, Brown, & Michels, 1991), and some of thisliterature is relevant.1 But this literature has not ex-plicitly addressed the conditions under which covari-ates or, more generally, concomitant variables may beconsidered to be either mediators or moderators of awithin-subject treatment effect. In addition, this lit-erature has largely confined itself to concomitant vari-ables that are measured only once (i.e., between-subject variables in within-subject designs) rather

than dealing with concomitant variables that varywithin participants (but see Khattree & Naik, 1995).Certainly, recent literature on multilevel modelingand hierarchical linear modeling (Bryk & Rauden-bush, 1992; Kenny, Kashy, & Bolger, 1998; Snijders& Bosker, 1999) has considered such time-vary ingconcomitant variables, but again this literature has notexplicitly addressed the conditions under which thesevariables serve mediating and moderating roles.

Interestingly, in spite of an absence of a formalliterature on within-subject mediation and modera-tion, there are several empirical articles that have at-tempted to evaluate mediational models with within-subject treatment variables (e.g., Fiske, Kenny, &Taylor, 1982; Neuberg, 1989). In general, these ar-ticles have explored mediation by analyzing differ-ence scores in ordinary least squares regression mod-els. But there has been no formal work that hasdeveloped the rationale for the use of such an ap-proach.

The purpose of this article is to provide a formalbasis for estimating and testing mediation and mod-eration effects in within-subject designs, wherein theobservations under one level of the treatment variableare not independent of observations under the otherlevel(s).2 Although much of what we present could bedeveloped within the context of multilevel modelingor hierarchical linear modeling procedures, we madethe decision to explore these issues in the context ofordinary least squares regression for a number of rea-sons. First, many researchers who are interested inquestions of mediation and moderation are likely to bemuch more conversant with analysis of variance and

1 Traditionally, in the analysis of variance literature, co-variates have been seen as variables assumed to be uncor-related with the between-subjects treatment variable. Theyare included in an analysis of covariance to increase statis-tical power. Because this assumption is not made for po-tential mediators and moderators, we refer to them as con-comitant variables rather than as covariates.

2 Although we generally refer to the designs we are con-sidering as within-subject designs, in fact the proceduresthat we develop are widely applicable whenever observa-tions are linked or nonindependent of each other becausethey come from the same participant, the same family, thesame classroom, the same setting, or whatever induces de-pendence. In general, the designs considered are designs inwhich the factor that causes the dependence, and underwhich observations are nested, is crossed with the indepen-dent or treatment variable of interest.

WITHIN-SUBJECT MEDIATION AND MODERATION 117

regression procedures that use ordinary least squaresestimation than with multilevel modeling procedures.Second, within-subject designs typically involve rela-tively few participants. As a result, the large-sampleassumptions underlying some estimation proceduresused in multilevel modeling may be untenable. Third,the detailed examination of the within-subject media-tion and moderation models we offer holds indepen-dent of what statistical methods are used to estimatethe parameters of those models. The analyses that weoutline involve the use of difference scores in regres-sion models, thus providing a more formal basis forthe procedures that researchers have occasionallyused on an ad hoc basis.

The within-subject case requires us to address a setof issues that are typically not addressed in the be-tween-subject case. Doing so will make clear certainassumptions underlying the definition and assessmentof mediation and moderation, both in within-subjectand between-subject designs.

The Basic Model

We start with a relatively simple research design inwhich there are two treatment conditions being com-pared. At a later point, we generalize our approach tomore complex designs involving multiple levels of awithin-subject treatment variable. In the two-condition case, each unit (e.g., participant, respon-dent, or case) is observed twice, once in each condi-tion, and these observations are denoted Yl and Y2.

3

The treatment effect is equal to the difference in thepopulation between the mean of K, and the mean ofK2- We assume, for now, that there are no confound-ing factors between the observations, so the differencebetween the means is an unbiased estimate of the truetreatment effect. Accordingly, following the model ofcausal inference developed by Rubin (1974, 1978)and Holland (1986, 1988; see also West, Biesanz, &Pitts, 2000), we assume both temporal stability (i.e.,the value of an observation does not depend on whenthe treatment is delivered) and causal transience (i.e.,the effect of a treatment or of a prior measurementdoes not persist over time). These are strong assump-tions that may not be met in many applications ofwithin-subject designs in psychological research.4

We occasionally illustrate our discussion by refer-ring to a hypothetical research design involving a drugversus placebo treatment comparison. In the design,participants who chronically suffer from migraineheadaches are given an experimental drug one day

and a placebo pill another. The variables Y{ and Y2 aremeasures of pain taken after the administration of theplacebo or drug. In a later section, we present hypo-thetical data from such a design and provide illustra-tive analyses of these data.

We start by considering the simple case without apotential mediator or moderator. We present more de-tail here than necessary to provide a basis for latergeneralization. Then we introduce a concomitant vari-able that may serve as a mediator or moderator. Wefirst consider the case of a stable concomitant vari-able, assumed to represent a stable individual-differences attribute or other variable that varies be-tween units but not across treatment levels. Finally,we consider the case of a varying concomitant vari-able, assumed to vary within units across treatmentlevels as well as between units.

Although we discuss estimation, in general we de-fine models in terms of population parameters. Theseare designated with Greek letters, whereas their esti-mates are in the Roman alphabet.

Case 1: No Concomitant Variable

Treatment effects in the case of between-subjectsexperimental designs are estimated by asking whetherthe mean value of Y differs between units exposed toone treatment and those exposed to the other. In thewithin-subject case, however, we have Yl and Y2 mea-sured on each unit, because each unit is observed ineach treatment condition. Formally, each response ismodeled as

(1)

In these equations, Yu and K2/ are the Y measurementsin each treatment condition for the fth participant. Theas are expected values of each Y, and the ss are errorsor residuals for each observed Y value. According tothese equations, we are assuming that all observationswithin a treatment condition are equal, except for er-

3 In the text, we exclude the subscripts that designate unitor participant. These are given in the equations, however.

4 In many within-subject designs in psychology, re-searchers may use a Latin square design, counterbalancingthe order of treatment exposure across participants. This iscommonly thought to yield unbiased estimates of the treat-ment effect when temporal stability cannot be assumed. Ata later point in this article, we examine the implications ofrelaxing this temporal stability assumption in greater detail.


ror, to a constant. In other words, the true scores areall equal within each condition. We assume that theerrors or residuals (es) share a multivariate normaldistribution, each with a mean of zero and a commonvariance, and, given dependence, a nonzero covari-ance between residuals from the same unit (typicallypositive).

If a,0 = a20, then there is no treatment differenceon average, although there may be differences be-tween the Ys for individual units as a result of eh notequaling e2/. Assuming that a lo + a20, the treatmentdifference for each unit equals the difference betweenthe preceding two equations:

YDi = (2)

It is assumed that the treatment difference, except forerror, is constant for all units. The mean difference

~ a io (3)

is the treatment effect. In a sample, it is estimated asthe difference between the sample mean of Yi: and thesample mean of Y2i, both computed across the partici-pants in the sample.

A test of whether this mean treatment effect differsfrom zero is provided equivalently by a paired- ordependent-samples t test, a one-factor repeated mea-sures analysis of variance, or a single-sample t testtesting whether the mean difference score differs fromzero. In our example, the mean pain difference be-tween the two experimental conditions estimates theoverall treatment effect, and each participant is as-sumed to manifest this same pain difference, exceptfor error.

Case 2: A Stable Concomitant Variable

Instead of assuming that the true scores are constantwithin each conditions, it may be more realistic toassume that they are related to one or more concomi-tant variables. We first consider the case of a singlestable concomitant variable, X, assumed to be an in-dividual-differences attribute or some other variablethat does not vary over the time frame of the experi-ment. In the context of our example, this variablemight be the participant's age or gender.

The Ys are now modeled as a function of this con-comitant variable:

(4)

In other words, the X variable contributes to some

extent to each individual's two Y scores. We considertwo cases depending on whether the two within-condition slopes are equal to each other, that is,whether or not (3U = (32i-

Parallel functions. If pu = (32i m Equation 4,

then the two linear functions for the Ys, one in each ofthe two conditions, are parallel. This situation is de-picted in Figure 1 with two participants, P and Q. Pl

and 2, are the data points for these participants inCondition 1, and P2 and Q2 are their data points inCondition 2. Accordingly, in this and subsequent fig-ures, both y, and Y2 are graphed on the commonordinate axis. Because each participant has only asingle X value, both data points for a given participanthave exactly the same location on the jc-axis. In ad-dition, with only two data points in each condition, weassume no error, so that all data points lie exactly onthe two linear functions. The bottom line, labeledy, :X, represents the linear function in Condition 1; theY2'.X line is the function in Condition 2. The treatmenteffect for each participant is then the vertical distancebetween the two data points for each participant, rep-resented by the two vertical dashed lines: between P,and P2 and between Ql and Q2. Importantly, becauseof the equal slopes in the two conditions, these twowithin-subject treatment effects are equal to eachother. With X centered in the figure so that the meanX score for the two participants equals zero, the solidvertical line represents the treatment effect on aver-

:X

0

Figure 1. Treatment effect (dashed vertical line) unrelatedto a stable concomitant variable. P and Q are two unitsmeasured on the concomitant X variable and on Yl and Y2 inTreatments 1 and 2. The treatment effect (solid vertical line)is the distance between the two lines relating the two Yscores, respectively, to the unvarying concomitant variableX. The parallel slopes indicate a constant treatment effectthat does not depend on X.


age. And note once again that this is equal to thevertical distance between the two linear functions.

Nonparallel functions. Now consider what hap-pens if the two within-condition slopes are not equalto each other, that is, (3n + (321. This situation, againwith two participants, P and Q, is depicted in Figure2. Here the slope of the Y2'.X linear function is morepositive than the slope for the X, :X linear function. Inother words, X relates to Y in Condition 2 morestrongly than it does in Condition 1. Again, the treat-ment difference for each participant is represented bythe dashed vertical line between the two Y:X func-tions, at that participant's particular X value. Giventhe slopes in the figure, participant P has a smaller Xvalue than participant Q, and so the treatment differ-ence for P is smaller than that for Q. Again, with Xcentered, the mean X equals zero, and the averagetreatment difference is represented by the solid verti-cal line at this point on the Jt-axis. It necessarily equalsthe average of the two within-subject treatment dif-ferences.

Figures 1 and 2 differ in that the within-conditionslopes are equal in the former and unequal in thelatter. The situation in Figure 2 represents moderationof the within-subject treatment effect. With unequalwithin-condition slopes, the relationship between Yand X depends on condition. Equivalently, the mag-nitude of the treatment effect—that is, the verticaldistance between the two functions—depends on thevalue of X.

Algebraically, the situation is captured by takingthe difference between the two within-condition func-

Figure 2. Treatment effect (dashed vertical line) moder-ated by a stable concomitant variable. P and Q and theregression lines are as in Figure 1. The nonparallel slopesindicate a varying treatment effect that depends on the con-comitant variable X.

tions in Equation 4. For each unit, the difference be-tween the two observations, one in Condition 2 andone in Condition 1, equals

YD, = = (P20 - 310) + (P21 -+ (e2/ - e,,-), (5)

which represents the treatment effect for each unit(plus error). There are two components to this treat-ment effect, one that is common to all units, (320 - (310,and one that varies between units, (f321 - (3, ,)X,. If thetwo within-condition slopes are equal to each other,PH = ^2i' men the second component of the treat-ment effect equals zero. Hence, the treatment differ-ence is the same for all units. On the other hand, if thetwo within-condition slopes are unequal, then thetreatment effect depends on the value of X,. This de-fines moderation. The magnitude of the treatment ef-fect depends on X,. Equivalently, the Y:X slopes aredifferent in the two conditions.

Estimation. Moderation due to a stable X is esti-mated by regressing the Y difference on X, thus esti-mating the model in Equation 5:

YDI - bo + b\X; + eYDi. (6)

The slope for X in this linear regression will exactlyequal the difference in the two slopes if separate re-gression models are estimated within each condition.Accordingly, a test of whether this slope differs fromzero provides a statistical test of moderation. In otherwords, moderation of the within-subject treatment dif-ference due to a stable X variable is indicated if Xpredicts the Y difference.

For ease of interpretation, it is helpful in estimationto center X by deviating each unit's X score from themean of X, as we did in Figures 1 and 2. If X has beencentered, then the estimated Y difference is given by

YD, = b0 + bj(X, - X) + eYDi,

and the estimated mean difference equals

YD = Y2- Yj = bo + b^X-

or

(7)

(8)

YD = Y2 - F, = bo, (9)

which equals the estimated treatment effect for a unitwith an average value of X. Thus, with centered X, theestimated intercept equals the average treatment ef-fect.5

5 At first, there may appear to be an inconsistency be-tween the -1,1 coding for the within-subject difference here


In the between-subjects case, moderation is testedby computing a product term between a dichotomoustreatment variable and the moderator and testingwhether that product variable predicts the outcomeonce both the treatment variable and the moderatorhave been partialed. In other words, in the between-subjects case, moderation is shown by the familiarpractice of testing whether an interaction, computedas a partialed product variable, predicts the outcome.Given this context, moderation in the within-subjectcase may seem a bit strange, because we are not form-ing a product variable and so one might wonderwhether an interaction is being tested or not. In fact,the model given in Equation 5 does imply an interac-tion, because X predicts the treatment difference foreach unit only if (32, ^ P in that is, only if the effectof A" on Y depends on whether the observation is inTreatment Condition 1 or Treatment Condition 2.

In the context of our example, suppose the con-comitant variable was the participant's age. If thewithin-condition relationships between pain judg-ments and age are different, then moderation of thewithin-subject treatment effect by age is indicated. Anassessment of whether such moderation is presentwould be provided by testing whether age is a pre-dictor of each participant's difference between his orher pain ratings in the two conditions. If age doespredict the pain difference, then the magnitude of thetreatment difference depends on the participant's age.Equivalently, the relationship between pain and age isdifferent in the two conditions.

Case 3: A Varying Concomitant Variable

We now allow there to be variation in X withinparticipants across the two treatment conditions. Justas we have two measures of Y for each participant (7,in one treatment condition and Y2 in the other), we

now have X} in the first treatment condition and X2 inthe other, and these X scores, measured after the ad-ministration of each treatment, may be influenced bythe treatment variable. As we did for Y, we make thestrong assumptions from Rubin and Holland of tem-poral stability and causal transience in these X mea-sures. Thus, any mean difference in X between thetwo treatment conditions is assumed to be a causaleffect of treatment. In addition, we assume that the Xvariables are causally prior to the Y variables in eachtreatment condition.

Just as we did in the case of the Y variables, we candevelop a model for the X variables:

X» = *io + ei,-. (1Q)

From these equations, we can derive the treatmentdifference for each unit:

XDi = %2i ~XU = <>20 - ^lo) + (S2i ~ eu)-

(11)And the mean treatment effect in X equals

In the context of our example, if the Y variables aremeasures of subjective pain on each day, followingthe administration of the experimental drug or theplacebo, X might be the degree to which a certainhormone, presumed to be released by the administra-tion of the experimental drug, is found in the bloodsystem on each of the 2 days.

Each of the Y variables can be modeled as a func-tion of the X variable measured in the same condi-tions:6

,,. = 810 + s, ,(13)

These two equations are very similar in form to thosewe presented for the case of the stable concomitant

and the -.5 and .5 contrast coding recommended by Judd,McClelland, and Culhane (1995) and West, Aiken, andKrull (1996) to estimate the treatment difference in the be-tween-subjects case. However, any apparent inconsistencyis entirely due to the difference between slope and interceptestimates. If we were to compute a regression equation foreach participant to estimate each participant's treatment ef-fect as a slope, it would equal (Z\,1})/Z\,2 = (-.5K, +.5y2)/(.25 + .25) = Y2 - Y,. This of course parallels thebetween-subjects regression for_which the slope, using the-.5 and .5 codes, equals Y2 - Y,. Note also that the inter-cept in a within-subject regression would equal the averageof the two observations.

6 As already stated, we are assuming that X in each con-dition is causally prior to Y in each condition. Thus, inmodeling the X effects, we did not include Y variables aspredictors. However, we do include X in modeling the Yvariables. In addition, because of our assumptions of tem-poral stability and causal transience, we can assume that inthe population the X in one condition exerts no causal effecton the Y in the other. In other words, each Y is a function ofthe X from that same condition alone and is uninfluenced bythe X measured in the other condition. At a later point atwhich we consider additional complexities, we examine de-signs in which this assumption cannot reasonably be made.


variable, the difference being that the X predictor vari-able varies between the two treatment conditions; thatis, there are two Xs, not one. Again, interpretationdepends on whether the within-condition slopes areequal, that is, whether 8n = 822. We consider eachcase in turn.

Parallel Junctions. If 8n = 822, then the two lin-ear functions in Conditions 1 and 2 are parallel. Fig-ure 3 presents a graph of these two parallel slopes.The top sloping line is the function in Condition 2relating Y2 to X2. The lower line graphs the function inCondition 1. As in the earlier two figures, both Ys areplotted on the common ordinate. Unlike the earlierfigures, where there was only one X, now both Xs areplotted on the common abscissa. Again, the importantassumption in this figure is that the two slopes areidentical, that is, 8n = 822.

We have plotted the two observations from a singleunit in Figure 3, participant Q, with observation <2ifrom the first condition and observation Q2 from thesecond. Both observations lie perfectly on the twofunctions; accordingly, for purposes of illustration, weare assuming that their Y scores are measured withouterror.7 Note that the two observations from this unitnot only are at two different locations on the v-axis orordinate axis (corresponding to their two different Yscores), but they also occupy two different locationson the x-axis, because we are not presuming that Xu

XQ

Figure 3. Treatment effect adjusted within units bychanges in the concomitant variable. Q is a unit measuredtwice on both the outcome variable Y and the concomitantvariable X after Treatments 1 and 2. The total treatmenteffect is the difference between Yl and Y2 (the dashed ver-tical line). The adjusted treatment effect (the solid verticalline) is the distance between the two lines relating each Yscore, respectively, to its concomitant X variable at themean value of X for that unit.

equals X2i. The total treatment effect for this unitequals the difference between Y2i and Ylt (the verticaldashed line). But note that these two different Y val-ues, one in Condition 1 and one in Condition 2, areassociated with different X values (and the differencebetween the two Xs is represented by the horizontaldashed line). Part of the difference in the two Y valuesfor this participant derives from the fact that higher Xvalues in both conditions lead to higher Y values, andthe X value for the observation in Condition 1 is lowerthan the X value for the observation from this unit inCondition 2. In other words, of the total treatmentdifference between Y2i and Yu for this unit, part of itis due to the X difference between the two conditions,and part of it is a treatment difference over and abovethat. The treatment difference that persists over andabove the X difference in the two conditions is indi-cated by the solid vertical line between the two lines,located above the participant's mean X value.

The Y difference for any unit (e.g., unit Q in Figure3) equals the difference between the two functions inEquation 13:

YDI = Y2i ~ Yu =z,- - £„•)• (14)

Assuming parallel slopes in the two conditions — thatis, 8U = 822 — this reduces to

($20 ~ ^10) ++ (e2l- - £„•),

2(- - Xu)(15)

where 8, l is the common slope in the two conditions.The total treatment effect is then equal to the sum oftwo different components: (820 - 810) and 8U (X2, -XH). The first of these is the vertical distance betweenthe two lines in the figure, at the mean level of X forthe unit. The second of these two components repre-sents the part of the treatment difference in the Ys thatis attributable to the X difference for that unit. Inessence, the vertical distance between the two lines, atthe value of the mean X for the unit, represents thetreatment effect in the Ys over and above or adjustingfor the X difference between the two conditions forthat unit. It represents an adjusted treatment differ-ence for the unit, adjusting for the within-unit differ-ences between the two Xs.

7 This is only an assumption of convenience to betterillustrate the situation for this particular unit in the figure. Ingeneral, there is assumed to be residual variation in allobservations.


With multiple units, interpretation depends onwhether or not there is a treatment effect for X onaverage. For simplicity, we first consider the case ofno X treatment effect. Figure 4 presents two differentparticipants, P and Q. As in Figure 3, unit Q has apositive X2i - X,,- difference. Given the positive rela-tionship between X and Y in the two conditions, thismeans that Qs adjusted treatment difference, over andabove his X difference, is smaller than his total orunadjusted treatment difference. That is, the verticaldistance between the two lines at his mean A" value issmaller than the total difference between his Y values.On the other hand, the adjustment for participant Pruns in exactly the opposite direction; her X2i - Xu

difference is negative, whereas her total treatment ef-fect, Y2i - Yu, is slightly positive. As a result, partici-pant P's adjusted treatment difference is larger thanthe unadjusted difference.

For there to be no average treatment effect in X, thedegree of the adjustments must be exactly the same,except for sign, for P and Q. That is, the X2i - Xu

differences are equal, but of opposite signs, for thetwo. Hence, the net adjustment collapsing across bothunits is exactly zero. Accordingly, the mean X-adjusted treatment effect, portrayed by the length ofthe solid vertical line between the two functions at thecentered value of zero on the X axis, exactly equalsthe mean of the two unadjusted treatment effects forthe two units. Although there is individual adjustment

Q2

Pi

X1X2

Figure 4. Treatment effect adjusted within units bychanges in the concomitant X without between-units media-tion. P and Q are units measured on both X and Y in Treat-ments 1 and 2. On average across the two units, the meansof X, and X2 are equal; they both can be centered at zerowithout loss of generality. The average treatment effect ad-justed within-unit (the solid vertical line) equals the averageof the two unadjusted treatment effects (the two dashedvertical lines).

Figure 5. Treatment effect adjusted within units bychanges in the concomitant X with between-units mediation.P and Q, the regression lines, and the within-unit adjust-ments are as in Figure 4. The difference in the means of X,and X2 implies that the treatment effect adjusting for theaverage change in the concomitant variable (the solid ver-tical line) is smaller than the average of the two unadjustedtreatment effects (the two dashed vertical lines).

of the Y treatment difference within each unit, there isno aggregate adjustment in the magnitude of the treat-ment difference when X is controlled.

Finally, consider Figure 5, again with two units, Pand Q. In this case, both units have a positive X2i - X,,difference. Hence, on average across the two units,there is a treatment difference in X as well as in Y.Accordingly, the two Xs are at different locations onthe joint abscissa (which has been centered at theirjoint mean). Given a positive relationship between Xand Y in each condition, the adjustment in the mag-nitude of the treatment effect is such that the totaltreatment effect for each unit, Y2i - Yu, is larger thanthe treatment effect adjusted for the X difference. Asa result, the average adjusted treatment difference,represented by the solid vertical line at the joint meanof the two Xs, is smaller than the average unadjustedtreatment effect.

The situation that is graphically displayed in Figure5 represents mediation. It is captured by the fact that(a) there are, on average, treatment differences in bothX and Y; (b) X and Y are related in each condition (weassume that X has been scaled to have a positive re-lationship with Y); and (c) the treatment effects in Yand X are in the same direction (assuming the scalingof X just specified).8 In terms of our example, treat-

8 It is possible that the mean X difference between the twoconditions is in the opposite direction from the mean Y


ment differences in hormone levels in the bloodwould be a mediator of treatment differences in painjudgments if (a) pain judgments are related to bloodhormone levels in each condition (scaled so that therelationship is positive) and (b) there are mean treat-ment differences in both variables in the same direc-tion. In this case, the adjusted treatment effect in painjudgments would be smaller than the unadjusted one,once those effects are estimated controlling for bloodhormone levels. The specifics of the estimation arediscussed later.

Nonparallel functions. Let us return to the twoequations relating Y and X in each condition (Equation13): Yu = 810 + 8UX, eu and Y2i = 520 + 522X2,+ e2,. Their difference tells us about the treatmentdifference in Y for each unit (Equation 14):

YDi = Y2i ~ Y\i = (°20 ~ 8lo) + 822*2/ ~ 8ll^li

+ (e2/ - EU).

In our discussion so far, we have been assumingthat the two slopes are homogenous, that is, 5U =822. Now, however, we want to examine the conse-quences if these two within-condition slopes are notequal to each other, 5n + 822. To examine this situ-ation, for pedagogical purposes we initially make theunrealistic assumption that there are no treatment dif-ferences in X, either on average or at the level ofindividual units, that is, XH = X2i. Then the treatmentdifference in Y specified in Equation 14 becomes

YD, = = (520 - 610) + (822 - SU)X,.+ (B2i - B,,.), (16)

where X, is either Xu or X2i, as they are assumed to beequal to each other. This model is identical to Equa-tion 5 when we were considering the case of the stableconcomitant variable. It captures what we earlier de-fined as moderation of the within-subject treatmenteffect and was graphically depicted in Figure 2. Ac-cordingly, when the within-condition slopes are un-

difference, even given the scaling of X to have a positiverelationship with Y in each condition. In such a case, wemight talk about inconsistent mediation, wherein the direc-tion of the indirect or mediated effect is opposite from thedirection of the unmediated effect. In this case, the averageadjusted treatment difference in Y, over and above the treat-ment difference in X, would actually be larger than the totaltreatment difference in Y. Although inconsistent mediationis possible, empirically it seems less likely than the case ofconsistent mediation described in the text.

equal, moderation of the treatment effect is once againindicated. Moderation, as shown earlier, has twoequivalent definitions: (a) The X:Y relationship de-pends on condition, and (b) the magnitude of the treat-ment effect depends on the value of X (in this case,assuming one common value).

We now relax the assumption that XH = X2;, con-tinuing to allow unequal within-condition slopes, sothat the model as specified in Equation 14 holds:

= Y2i - - (820 - 810)+ (e2i - E!,.

822X2, -

It is useful to reexpress this equation in terms of thecorresponding difference and the sum of the two Xvariables. This amounts to a 45° rotation of the twooriginal ;t-axes. The preceding difference equationcan be equivalently expressed as9

822-8n 8n +822

+ (e2l--e1/), (17)

where XSi = Xu + X2i and XDi = X2i - Xu. In this fullmodel, allowing unequal within-condition slopes aswell as X differences within participants, there arethree different components to each unit's treatmenteffect in Y, ignoring error.

First, there is a component that is common to allunits, 820 - 810. Its magnitude is unaffected by indi-vidual differences in the Xs.

Second, there is a component of each unit's treat-ment effect that varies as a function of the sum of eachunit's two X scores, [(822 - 8, !)/2]XS(. Its contributionto the unit's treatment effect will be large dependingon the magnitude of the condition differences in theY:X slopes. Assuming that X is scaled to have a posi-tive within-condition relationship with Y and assum-

9 Any orthogonal basis can be transformed by an orthogo-nal matrix to another orthogonal basis (Cliff, 1987, pp. 320-326). Solving

*\ n n/*,•J L-i uU2

Xs

X£

for X, and X2 and then substituting these values into theoriginal difference equations yields the reexpression. Suchtransformations have long been used for mathematical con-venience but often have substantive importance in factoranalysis and elsewhere. For a substantive example using the45° rotation to the sum and difference, see Coombs,Coombs, and McClelland (1975).


ing that 822 > 8,,, larger treatment differences will befound for units having larger average (or sum of) Xvalues across the two conditions. This second com-ponent reflects moderation: With different average Xvalues, we get different treatment effects. Equiva-lently, the relationships between Y and X are not ho-mogeneous in the two conditions.

The final component of each unit's treatment effectis that due to the difference between the two Xs, [(822

+ 8, ,)/2]XD,. It will be large whenever the sum of thetwo within-condition slopes is large and when thereare large differences in the unit's X values. This com-ponent reflects the individual adjustment due to dif-ferences between X2i and X,,- for each unit. Whenthere is a large X difference in the same direction asthe Y difference, and X and Y are highly related, thenpart of the individual's treatment effect in Y is due tothe individual's treatment effect in X. And if, on av-erage, across units, there is a mean treatment effect inX, then mediation is indicated.

Figure 6 captures the full complexity, again for twoperfectly measured participants, P and Q. First, thereis moderation of the treatment effect according toeach unit's average or typical X value in the two con-ditions. This is revealed by the unequal slopes be-tween the two conditions, with the Y:X slope beingsteeper in Condition 2 than in Condition 1. Thus, Xrelates to Y more strongly in the second condition than

X2

X, 0 X2

X1X2

Figure 6. Between-units mediation of treatment effectwith moderation of treatment effect due to concomitant X.All points and lines are as in Figure 5, except that thenonparallel slopes indicate a varying adjusted treatment ef-fect depending on each unit's average X score. Mediation isshown by the fact that the average adjusted treatment effect(the solid vertical line) is smaller than the average of the twounadjusted treatment effects (the two dashed vertical lines).Moderation is shown by the fact that the adjusted treatmenteffect depends on the average of X, and X2 for a given unit.

the first, and this means that the magnitude of thetreatment difference for each unit, represented by thedashed vertical line for each unit, increases as theunit's average X value increases. Thus, unit Q showsa larger treatment effect than does unit P, over andabove any within-unit adjustment due to differencesin the unit's two X values.

Second, each unit has different X values in the twodifferent conditions, and there are positive relation-ships between Y and X in both conditions. Hence, partof the Y difference for each unit is due to the X dif-ference between conditions. Adjusting for the X dif-ference within each unit leaves each unit's adjustedtreatment difference (i.e., the vertical distance be-tween the two lines at each unit's mean X). In addi-tion, as in Figure 5, averaging across the two units,there is an average treatment effect in X, such that themean X value in Condition 2 is higher than the meanX value in Condition 1. Given that the treatment effectin Y is in the same direction and that the within-condition relationships between Y and X are positive,then on average, across units, there is mediation dueto X: The overall treatment effect in Y is partly due tothe treatment difference in X and is reduced when theX difference is controlled.

As in previous figures, we have centered the jc-axisso that the joint mean of X,, and X2i equals zero. Thesolid vertical line at this point represents the averageadjusted treatment difference, over and above any Xdifference, averaged across the two units and allowingfor moderation. Note that it is smaller than the aver-age of the two unadjusted treatment effects.

Estimation. Having discussed this full model, wecan now turn to estimation. We need to estimate threedifferent models. First, Equations 2 and 11 should beestimated, testing for overall treatment effects in bothY and X. Second, the full model portrayed in Equation14 should be estimated. However, the equivalent for-mulation, given in Equation 17, is much more ame-nable to interpretation. Accordingly, one can estimatea single regression model in which the difference inthe two Y scores for each unit is regressed on twopredictors, the sum of that unit's X scores and thedifference in that unit's X scores (with the Y and Xdifferences taken in the same direction):

YDi = d2XD (18)

The regression coefficients in this model provide es-timates of the parameters in Equation 17. Thus, d0

estimates the component of the treatment differencethat is constant across all units, 820 - 810. For most


applications, we recommend centering the Xsi predic-tor in the model, as we did in Figure 6. If Xsi has beencentered, then the estimated intercept, dQ, estimatesthe mean treatment effect over and above any media-tion of that treatment effect due to X. In addition, thisestimate is at the mean value of XSi, allowing mod-eration as a function of the magnitude of Xsi. If thereis an overall treatment effect in both Y and X, but theintercept in Equation 18 is not different from zero(given XSi centered), then complete mediation is in-dicated.

The regression coefficient for the sum predictorvariable, dl, estimates the difference in the twowithin-condition slopes, (822 - 8n)/2. Accordingly, itestimates the degree to which the treatment effect in Yis moderated by a unit's average X score. Assumingthat X has been scaled to relate positively to Y in eachcondition, then a positive value for this regressioncoefficient indicates larger treatment effects for unitswith larger average X scores.

Finally, the regression coefficient for the differencepredictor variable, d2, estimates the degree to whicheach unit's treatment difference in Y is due to thatunit's X difference. It represents the within-unit ad-justment in the magnitude of the Y difference due tothe X difference. If this regression coefficient is sig-nificant (in a positive direction, given the scaling ofX), and if there is a mean difference in X between thetwo conditions, then mediation of the treatment effectin Y is indicated.

In summary, first, the slope d{ for the sum XSi (=X,, + X2,) estimates (822 - 8n)/2, half of the differ-ence of the within-condition slopes for the concomi-tant variable X, and indicates moderation of the treat-ment effect in Y depending on the level of the X sum.Second, the slope d2 for the difference XDi (= X2/ -Xu) estimates (822 + S,,)/2, the average of the twowithin-condition slopes for the concomitant variableX, and indicates mediation of the treatment effect in Y,assuming a mean difference between the two Xs.Third, the intercept d0 estimates 820 - 810, which is(assuming Xs, centered10) (a) the average treatmenteffect over and above mediation, if d2 differs fromzero, and (b) that treatment effect for a unit with anaverage sum of the X values, if d} differs from zero.

Illustration

Our example has been a hypothetical study involv-ing a drug treatment versus placebo comparison, ex-amining effects of these treatments on chronic pain

Table 1Data Used in Example

Y,

7357576780567281616774708362497473466093

Y26155614975607365594864556861557960515571

A

5960566967515660586158645860566361595462

#13730303137333843332043344135323536252646

H2

3328363035343529311741273930323738242539

judgments. We analyze hypothetical data to illustrateour proposed analysis strategy.

Data from this example are presented in Tables 1and 2. Variables Yl and Y2 are the pain judgmentscores for each participant in the two conditions (F, inthe placebo condition and Y2 in the drug condition,both measures taken on a hypothetical 100-point scalein which higher numbers indicate greater levels ofpain). Three other variables are measured. The first,A, is the participant's age, assumed to be unchangingfor all participants during the course of the study. Thesecond and third, Hl and H2, are measures of levels ofthe hypothetical hormone in the blood of each partici-pant, measured after the administration of either theplacebo or the drug. The hormone variables have beenscaled so that lower numbers indicate higher levels of

10 It is a useful default practice to mean-center predictorvariables in moderator models so that zero will be a mean-ingful value (Aiken & West, 1991). However, we specifi-cally do not do so for the difference predictor because itsuncentered zero value is meaningful; it represents no dif-ference in X between the two treatment conditions. Theintercept then equals the treatment effect in Y at the averagelevel of the Xs (assuming that the X sum is centered) andwhen the X difference equals zero (i.e., acting as if therewere no treatment difference in X).


Table 2Correlations, Means, and Standard Deviations Usedin Example

Variable

i . y ,2. Y2

3. A4.H,5.H2

MSD

1

—.64.42.74.48

67.7511.89

2

—.06.70.71

61.258.58

3

—.10

-.0559.60

4.21

4

—.78

34.256.41

5

—32.005.96

the hormone in the blood, and thus they are scaled tohave positive relationships with the pain variables:Higher levels of hormone in each condition coincidewith lower levels of pain. These variables are all mea-sured for 20 hypothetical participants.'l

The overall effect of treatment on pain judgments isestimated by the mean difference between Y2 and Yl,which equals -6.50. A test of the null hypothesis thatthe true difference is zero yields /(19) = -3.15, p <.01. Hence, in these data there is a statistically sig-nificant effect of the drug relative to the placebo, withlower levels of pain reported in the drug condition.Next, we estimate this treatment effect consideringage as the concomitant variable and then consideringhormone levels as the concomitant variables.

Regressing each pain score on age yields Yu =-3.39 +1.19 A,, and Y2i = 53.64 + 0.13 A,, In the firstequation, A is a marginally significant predictor of Yl,?(18) = 1.98, p < .10; in the second equation, theslope for A is not significantly different from zero,r(18) = 0.27, p> .50.

When the pain difference, YD (Y2 - 3j), is regressedon A, the following equation results: YDi = 57.03 -1.06 A. Here the slope for A is significantly differentfrom zero, f(18) = 2.36, p < .05. It equals the differ-ence in the two slopes from the separate Y2 and Yl

equations just described (i.e., -1.06 = .13 - 1.19).Thus, the test of whether this slope differs from zeroshows equivalently that the slope for A in the Y2 equa-tion differs from the slope for A in the y, equation. Asa result, we can conclude that there is a significantAge x Drug Condition interaction, meaning that age(A) relates more strongly to pain judgments (olderparticipants report greater pain) in the placebo condi-tion than in the drug condition. Equivalently, we cansay that age moderates the treatment difference, withlarger differences between the drug and placebo con-ditions for older participants; that is, older participants

have more negative pain differences between Condi-tion 2 (drug) and Condition 1 (placebo).

The interpretation of the parameter estimates in thismodel is made simpler if age is centered from itsmean (A' = A - A): YDi = -6.50 - 1.06 A'. In thisequation, the intercept equals the predicted treatmentdifference at the mean age (i.e., A' = 0), and it isidentical to the estimated treatment difference whenage was not included as a predictor. A test of whetherthis treatment difference differs from zero yields f(18)= 3.50, p < .01. Although this test of the mean treat-ment effect has one fewer degrees of freedom than thetest of the treatment difference when age was notincluded as a predictor, it provides a more powerfultest of the overall treatment effect because age isstrongly related to the treatment difference.

Turning to the two hormone variables, we want toknow whether the hormone levels in the two condi-tions both mediate and moderate the treatment differ-ence in pain judgments. Initially, we regress each painmeasure on the hormone measure from the same con-dition: YU = 20.51 + 1.38 //,, and Y2i = 28.54 + 1.02H2i. As expected in the present research context, hor-mone levels in each condition are significantly relatedto pain levels, with higher hormone levels associatedwith lower judgments of pain. (Remember that lowerscores on Y indicate less pain and lower scores on Hindicate higher hormone levels.)

When the pain difference is regressed on the hor-mone sum (Hs) and difference (HD), the followingestimate results:'2 YDi = 0.55 - 0.07//s, + \.22HDi. Inthis model, the hormone level difference is a signifi-cant predictor of the pain difference, t(ll) = 2.66,p < .05, but the hormone sum is not. To aid interpre-

1' Both types of concomitant variables (age and hormonelevel) could be included in one analysis. For simplicity ofexposition, we treat them separately, just as we have donethroughout.

12 Given Equation 16 presented earlier, it may come as asurprise that the intercept and slopes in this estimated modelare not exact functions of the intercepts and slopes in thetwo equations estimated in each treatment condition. Thealgebra underlying Equation 16 presumes that the twocrossed slopes (effect of H in one condition on P in theother) are exactly zero, which they will be in the populationif the assumptions that we made in the earlier section hold.Even though in the present data those crossed effects are notsignificant, they do not exactly equal zero. We discuss therole of these crossed slopes in a later section devoted tocomplications.


tation, as previously explained, we reestimate thismodel after centering the sum variable: YDi = -3.77- 0.07 H'Si + 1.22 HDi, where H's is the centered sum.

Differences in hormone levels can be said to me-diate the treatment difference in pain if two conditionsare met. First, there must be a condition difference inhormone levels that is in the same direction as thecondition difference in pain (assuming that hormonelevels have been scaled to have a positive relationshipwith pain judgments). The mean hormone level in thedrug condition is 32.00; in the placebo condition, it is34.25. The difference between these two is signifi-cant, f(19) = 2.45, p < .05. Second, the hormone leveldifference must be predictive of the pain difference,as we have shown it to be.

The intercept in the preceding model in which thesum predictor has been centered, -3.77, estimates themean difference in pain judgments between the twotreatment conditions over and above the hormone dif-ference, or acting as if the hormone difference werezero. Thus, it represents the residual treatment differ-ence over, and above mediation. In this case, that re-sidual difference remains marginally significantly dif-ferent from zero (p < .10). Nevertheless, it isnecessarily less than the total or unmediated treatmentdifference because of the fact that the hormone dif-ference is a significant predictor in this model andthere is a mean hormone difference between the twoconditions.

In conclusion, with these hypothetical data, wehave evidence of a significant treatment difference inpain judgments, with the age of the participant mod-erating that difference. Older participants show alarger treatment effect. In addition, there is a signifi-cant treatment effect on levels of the blood hormonethat serves as a mediator of the treatment difference inpain. Over and above the treatment difference in hor-mone levels, the residual pain difference between thetwo treatment conditions is marginally significant.There is no evidence that average hormone levels ofthe participant moderate the treatment effect.

Elaborations and Complications

Having laid out the underlying logic and estimationof mediation and moderation in within-subject de-signs, we now turn to two different complicating is-sues. The first is that many within-subject treatmentcomparisons involve independent variables havingmore than two levels. The second is that within-subject comparisons are frequently conducted in non-

experimental situations, wherein the assumptions un-derlying causal inference laid out by Rubin (1974,1978) and Holland (1986, 1988) are very unlikely tobe met.

Within-Subject Independent Variables WithMore Than Two Levels

The models developed earlier for the case of tworepeated measures readily generalize to studies withthree or more levels of a within-subject independentvariable. For example, a study of a treatment effectmight include two different drugs being evaluated aswell as a placebo control condition. Or a study ofmood effects might involve comparisons among apositive, negative, and neutral mood induced in par-ticipants at three different times. And in these casesthere might also be a single concomitant variablemeasuring a stable characteristic or a varying con-comitant variable measured in each treatment condi-tion. With less detail, we show the generalizationfrom two to three levels of the within-subject treat-ment variable and describe the general principles thatalso apply to more than three repeated measures.

First consider the case of three repeated measureswithout concomitant variables:

Y2i =

Y3i = <*30 •

(19)

For illustrative purposes, we assume that Yu and Y2i

are measures of the dependent variable after admin-istration of one of two drugs, whereas Y3i comes fromthe placebo control condition.

Procedures for assessing the treatment effect fortwo repeated measures generalize to orthogonal con-trasts for three or more repeated measures. For ex-ample, the researcher might want to compare the twodrug conditions with the placebo control condition(l,l,-2)13 to determine whether there is an overalleffect of the drugs. Then he or she might want tocompare the two drug conditions with each other(1,-1,0). We use these two contrasts for illustrativepurposes; many other pairs of orthogonal contrastsare, of course, also possible. To obtain the treatment

13 We represent contrasts with their coefficients. Thecontrast (1,1 ,-2) represents Y}, + Y2i - 2 Y3i, and the contrast(1,-1,0) represents Yti - Y2i.


difference coded by each contrast, we take the corre-sponding differences among the Ys:

10 ' "20 - 2a30) + (e](- + e-u - 2e3/),

YD2i = Y\i- Y2i = ("10 - «2(20)

In other words, the two treatment differences of in-terest, except for error, are constant for all of theobservational units. The mean treatment effects ac-cording to these two contrasts are then

2a30,

= a10 ~ a

(21)20-

Next consider the case of a single stable concomi-tant variable. That is,

the relationship between X and Y varies across thelevels of treatment specified in the Y contrast. Equiva-lently, X moderates the magnitude of the treatmenteffect specified in the contrast, such that larger orsmaller contrast differences are associated with largeror smaller X scores. In addition, if X has been cen-tered, the intercept estimates the contrast effect at theaverage level of X.

Just as we did in the case of two treatment levels,we might also have an X variable whose values varyacross the treatment conditions. If we assume, as wedid in the case of two treatments, that this X variablemanifests stability and causal transience, then treat-ment differences in it are assumed to be causal effects,as they are in the outcome variable. In this case, wecan model each Y as a function of the X measured inthat condition:

(22)

Then the contrast differences are given by

YD\i = Y\i Y2i ~

= (Pio + £20 - 2P30) + (Pn + P21 - 2p31)X,+ (e,,. + e2,-2e3l.), (23)

YD2i =Y\i~ Y2i

= (Pio - P20) + (Pn - P2i)*,- + (e» - e2l-).

Now treatment differences according to each contrasthave two components (other than error). First, there isa constant effect for all units (represented by the in-tercept differences in the preceding equations). Sec-ond, the magnitude of each contrast difference de-pends on the unit's X value. Just as was the case whenthe independent variable had only two levels, if therelationships between Y and X are homogeneouswithin conditions (i.e., (3,, = (32i

= PsiX men tne

contrast effects do not depend on or are not moderatedby X. In this case, the estimates of the contrast effectsare the same whether or not X is controlled. However,if the relationships are not homogeneous, then thecontrast effects are moderated by X. The average con-trast effects are

(24)= (P,o + P20 - 2P30) + (Pn + P2i -

V-YD2 = M-n - M-re = (Pio - P2o) + (Pn -

Estimation is accomplished by regressing each con-trast difference on the X variable. A significant re-gression coefficient associated with X indicates that

Y2i =

Yi = 8

(25)

30 e3,.

Given these models, we can calculate the differencesdue to each of the contrasts of interest:

YDli~ Y2i ~

~ (810 + ^20 ~ 2830) + §! [X,,. + 822X2,. - 2833X3,.+ (eu + 82, - 2e3i) (26)

YD2i~ Yli~ Y2i

= (810 - S20) + 8UX,,. - 822X2,. + (e,(- - e2/).

As in the case of two treatment levels, these modelscan be equivalently reexpressed as two models, withthree predictors in each model: the sum of the Xs, thedifference in the Xs according to the first contrastweights (1,1, -2), and the difference in the Xs accord-ing to the second contrast weights (1,-1,0):

^30)

h28

8n + 822-:Xs,

33

Dli^11 -8;22

(27)

YD2i —

>io 820)

Xn,,+ %D2i+ (eli


These unwieldy expressions reduce considerably ifwe assume that all within-condition relationships be-tween X and Y are homogeneous, that is 8 t , = 822 =8,J33-

= (810 + 820-2830) +

+ (e,,- + s2(- — 2s3,-)

811+822•^Dli

(28)

= (810-820)8n +822

In other words, homogeneity implies that the contrastsamong the 7s are a function not of the sum of the Xsbut only of the contrast among them that correspondsto the contrast among the Fs that is being modeled. Asa result, whenever the sum of the Xs is found to be asignificant predictor of a Y contrast, moderation ofthat treatment contrast by the Xs is implied. If there isa mean contrast difference for both the Xs and the 7s,then the X contrast being predictive of the Y contrastimplies mediation. In this case, part of the overallcontrast difference in the Ys can be attributed to theoverall contrast difference in the Xs.

The preceding conclusions generalize to situationshaving more than three levels of a within-subjecttreatment variable. Estimation in such cases involvesdefining a series of orthogonal contrast scores amongboth the Fs and the Xs. Then the Y contrasts are re-gressed on the sum of the Xs and the full set of cor-responding contrast differences among the Xs. Mod-eration of a given contrast difference is indicated ifthe sum of the Xs is predictive of that given Y contrast.Mediation is indicated if the corresponding X contrastis significant, having already shown mean differencesin both the Y and X contrasts of interest.

Complications When Assumptions Are Relaxed

Following the lead of Rubin (1974, 1978) and Hol-land (1986, 1988), we have made the strong assump-tions of temporal stability and causal transience. Infact, within-subject experimental designs are fre-quently used in psychological research wherein theseassumptions are unlikely. There is considerable con-sensus that such designs should not be used in cases inwhich causal transience is violated (i.e., treatment ef-fects carry over and may affect observations at laterpoints in time). However, the assumption of temporalstability (that responses are stable over time in theabsence of treatment differences) is frequently not

tenable. Instead, to overcome the threats to internalvalidity that violations of this assumption imply (e.g.,history, maturation, regression toward the mean, in-strumentation, and testing threats; Campbell & Stan-ley, 1963; Judd & Kenny, 1981a), alternative ordersof treatment administration are typically used, andparticipants are randomly assigned to these alternativeorders.

In still other cases, within-subject analyses are fre-quently used in cases in which causal inference isclearly impossible; that is, "treatments" are in nosense under the control of the experimenter, and ex-perimental manipulation of their levels is not conceiv-able. Consider a researcher who is interested in gen-der differences in subjective happiness. He or sherecruits married couples for the research and asks thespouses about their subjective happiness. To assessgender differences in happiness in these couples, it isentirely appropriate to conduct a repeated measuresanalysis of variance or paired samples t test to askwhether the mean happiness of the men is differentfrom the mean happiness of the women. But clearlythe Rubin and Holland assumptions make no sense inthis context. Although we can ask whether the meandifference is different from zero, causal inference togender is impossible.

And yet, it is entirely possible in such a situationthat one might be interested in knowing whether someconcomitant variable (stable across the two individu-als in the couple, e.g., length of time they have beenmarried) moderates the within-couple gender differ-ence in subjective happiness. Perhaps couples whohave been married longer are less likely to show gen-der differences in subjective happiness. Certainly, onecan ask about this within-couple moderation using theprocedures developed earlier, even though we wouldnot want to refer to it as moderation of a "treatmenteffect."

Similarly, we could focus on some other concomi-tant variable that possibly varies between genderwithin a couple (such as overall mental health) andask whether differences in mental health "mediate"the gender difference in subjective happiness. In thiscontext, it seems that we should be particularly cau-tious in using the term mediation because there seemsto be an implicit assumption in this term that an "ef-fect" is "due to" the mediator. But certainly one canreframe the question, avoiding the term mediation,and conduct analyses to ask whether the gender dif-ference in subjective happiness is reduced once wecontrol for gender differences in mental health. In


light of this discussion, it seems appropriate to con-sider the ways in which the models that we haveoutlined need to be modified once we move outside ofthe experimental context.

In general, analyses either without a concomitantvariable or with a single stable concomitant variableare identical to the models presented earlier. That is,the overall mean difference in Y between the two"treatment" levels (e.g., male vs. female spouses) isestimated as the mean difference in Y. In addition, thatdifference can be regressed on some concomitantvariable, X, that is stable across the two "treatmentlevels" (e.g., length of marriage of the couple) to de-termine whether it moderates the magnitude of thedifference in Y. If it does, then that means that Xrelates to Y differently in the two "treatment levels."In terms of our example, it might be the case thatfemale spouses report, on average, significantly lesssubjective happiness than the male spouses. But thisdifference may be less pronounced among coupleswho have been married longer than among more re-cently married couples. This would imply that lengthof marriage relates differently to the men's self-reports of subjective happiness than to women's self-reports.

In the case of an X variable that varies across "treat-ment levels," however, the models necessarily be-come a bit more complicated, because it may be thecase that the X in one "treatment level" exerts effectson the Y in the other level. Again, consider our ex-ample. Suppose we were interested in each spouse'soverall mental health status as both a potential "me-diator" and moderator of the spousal difference insubjective happiness. It seems likely that eachspouse's mental health not only influences his or herown subjective happiness but also the subjective hap-piness of his or her marriage partner. In the previousmodels with a varying X, we could make the assump-tion that such crossed effects were zero. Now we nolonger can. Accordingly, each Y must be modeled asa function not only of X in the same "treatment level"but also of X in the other "level":

i ~ *2i Y\ i= (820 - 810) + (821 - 8, ,)XU + (822

Y2i = 820

(29)

In these models, 8,, and 822 represent the within-leveleffects, whereas 8:2 and 821 represent the crossed ef-fects.

Given these models, the model for the differencebetween the two Ys is more complicated:

(30)

So too is the model in which the difference in the Ysis modeled as a function of the sum of the Xs and thedifference in the Xs:

YD — Y2 '

= (820-S10), ,-812

X,•Si

8,, + 822-82, -812

(31)

Now the effect of the sum of the Xs on the Y differ-ence will be large to the extent that the two within-level effects differ and to the extent that the twocrossed effects differ (in the same direction). The ef-fect of the X difference on the Y difference will belarge to the extent that the crossed effects are smallerthan the two within effects (assuming all 8s are posi-tive).

The estimation of "mediation" and moderation canstill be accomplished by regressing the Y differenceon the X sum and X difference. If X serves as a mod-erator, then the X sum should predict the Y difference.Equivalently, moderation implies that within andcrossed effects of the X in one "treatment level" aredifferent from the within and crossed effects in theother. If the X difference predicts the Y difference,then X can be said to "mediate" the Y difference,assuming that there is a meaningful mean differencein X as well as in Y. Finally, if the X sum has beencentered as a predictor, then the intercept in this re-gression model estimates the mean residual "treat-ment effect" in Y when the "treatment effect" in Xequals zero, that is, over and above any difference inX. To clarify the nature of the within and crossedeffects in this context, we also recommend estimatingthe separate Y models, regressing each Y on the twoXs (one from the same "treatment level" and one fromthe other).

In terms of the example that we have been devel-oping, the female-male difference in subjective hap-piness within each couple would be regressed on thecentered sum of the two mental health scores and thefemale-male difference in the two mental healthscores. If greater within-couple differences in happi-ness are associated with lower average mental healthscores for the two individuals, then the sum predictorwould have a significant regression coefficient. This


would imply moderation. If greater within-couple dif-ferences in happiness are associated with greaterwithin-couple differences in mental health status, andif on average there was a female-male difference inmental health status in the same direction as the sub-jective happiness mean difference, then mental healthstatus might be said to "mediate" the subjective hap-piness difference. That is, part of the gender differ-ence in happiness might be attributable to the genderdifference in mental health status. This interpretation,of course, depends heavily on an implicit causalmodel suggesting that mental health status is some-how causally prior to subjective happiness. In addi-tion, we again issue the warning here that talkingabout "mediation" in this situation, where we clearlycannot talk about causal effects of a treatment vari-able, is highly suspect. At best, perhaps, we couldargue that gender differences are found in both mentalhealth and subjective happiness and that, when theformer difference is controlled, the magnitude of thelatter one is diminished. In the absence of clear con-ditions that establish the causal effect of a treatmentvariable, "mediational" analyses can ultimately donothing to help make the causal arguments that areimplicit in the terminology used.14

Summary and Conclusion

In this concluding section, we have three maingoals. First, we want to summarize the analytic rec-ommendations we have made for assessing mediationand moderation in within-subject designs. Second, inaddition to the complications discussed in the previ-ous section, we want to mention some caveats andconcerns that need to be raised whenever the analyseswe have outlined are conducted. Finally, we concludewith some general comments about ways in which theassessment of mediation and moderation in within-subject designs permits us to think more clearly aboutthese processes and how they are assessed in between-subjects designs.

Summary of Analyses

The assessment of overall treatment effects inwithin-subject designs involves estimating the meandifference between the two treatment conditions in Y,the outcome variable. To assess whether the magni-tude of this treatment effect is moderated by somestable concomitant variable, X, the Y difference isregressed on that concomitant variable. If the magni-tude of the treatment effect depends on X, then X will

be predictive of the Y difference. Equivalently, thismeans that X relates to Y more strongly in one of thetreatment conditions than the other. In the estimatedmodel, if X has been centered, then the intercept willequal the overall or average treatment effect, equiva-lent to the estimated treatment effect assessed inde-pendently of moderation.

When the concomitant variable is affected by thetreatment, then it may serve as a mediator of the treat-ment effect if it is causally prior to Y. In this case, Xis measured in each treatment condition, and there ison average a mean difference in X between the twoconditions. This concomitant variable, measured ineach treatment condition, may also serve as a mod-erator of the treatment effect, with treatment effects ofdifferent magnitudes depending on a participant's to-tal or average X score.

To assess both mediation and moderation due to aconcomitant variable that varies between treatmentconditions, one regresses the Y difference on both theX sum and the X difference. Assuming that there is anoverall treatment effect on X and that the X differencepredicts the Y difference, mediation of the treatmenteffect in Y by X is indicated (assuming that X and Yare scaled to have a positive relationship and the treat-ment effects in Y and X are in the same direction). Ifthe X sum is predictive of the Y difference, then X alsoserves as a moderator of the treatment effect and,equivalently, X relates to Y differently in the two treat-ment conditions. Finally, if the X sum has been cen-tered, then the intercept in this regression equationwill equal the magnitude of the residual treatmentdifference in Y, over and above mediation due to Xand at the mean value of X. In other words, it willequal the portion of the mean treatment effect that isnot mediated through X.

In cases in which the treatment variable has morethan two levels, this analytic approach can be accom-

14 Given the content of the example we have been pur-suing, it makes sense to link the "mediation" and modera-tion model we have been developing with the actor-partnereffect model that Kenny and colleagues have developed(Kashy & Kenny, 2000; Kenny, 1996). That model is in factgiven by Equation 29. In our model, wherein the differenceis regressed on the sum and the difference, the regressioncoefficient for the difference estimates the extent to whichthe partner effects are smaller than the actor effects. Thecoefficient for the sum estimates the extent to which theeffects on one target (both actor effect and partner effect)are greater than the effects on the other target.


plished by computing contrast differences in both Yand X. One then focuses on the analysis of individualY contrasts among the treatment conditions, regress-ing these on the X sum as well as the set of X contrastdifferences. Mediation of a particular contrast differ-ence in the Ys is indicated if there is a parallel Xdifference and if the Y contrast difference is predictedby that same X contrast difference.

Our approach, as just summarized, relies exclu-sively on estimates provided by ordinary least squaresregression models. Estimation can be conductedthrough any general purpose multiple regression pro-gram. As we noted in the introduction, much of whatwe have presented could also have been developedwithin the context of multilevel modeling, using pro-grams that rely on other estimation strategies (e.g.,HLM or Proc Mixed in SAS). We chose the ordinaryleast squares approach simply because most psycho-logical researchers are much more familiar with re-gression and analysis of variance procedures thanwith multilevel modeling.

Further Concerns

In the initial sections of this article, we made aseries of assumptions when we laid out the basicmodel. These included that any treatment differencesin Y and X were causal in nature and were estimatedwithout bias. An additional assumption is importantif, in fact, one wishes to use the proposed analyses toargue for mediation. It must be the case that X iscausally prior to Y; that is, changes in X induced bythe treatment cause changes in Y between the twotreatment conditions. Because X is a measured vari-able rather than one that is manipulated in the designswe have been considering, this assumption cannot becritically examined without additional research inwhich X is actually manipulated and its effects on Yassessed experimentally.

Implicitly, we have also been assuming that both Yand X are measured without error. Random errors ofmeasurement, particularly in the concomitant X vari-able, will result in biased assessments of both media-tion and moderation. In the case of moderation due toa single stable X, measurement error in X will attenu-ate the estimate of moderation. In the case of a con-comitant X that varies between the two treatment con-ditions, the situation is more complicated becauseboth mediation and moderation are assessed in thesame model. Random errors of measurement in thiscase can lead to either an underestimation or an over-estimation of these effects. All of the analyses that we

have outlined could be conducted within a structuralequation approach that allows for multiple indicatorsand controls for errors of measurement. Such an ap-proach would need to model the means of the vari-ables as well as their variances and covariances (seeKline, 1998, pp. 293-307).

Concluding Comments

Assessment of mediation and moderation in within-subject designs suggests some refinement in ourthinking about these processes in both these designsand designs in which the treatment variable variesbetween participants. We conclude with two suchthoughts.

First, the within-subject analysis with a varying Xthat can serve as both a moderator and a mediator isconducted by assessing whether the magnitude of thetreatment difference in Y depends on two differentlinear functions of the Xs: their sum and their differ-ence. Moderation is indicated if the treatment differ-ence in Y depends on the X sum. Mediation is indi-cated if the treatment difference in Y depends on theX difference, assuming overall treatment effects inboth X and Y. Thus, in a deeper sense, both modera-tion and mediation suggest that the magnitude of theY treatment difference varies and is predictable fromsome function of the Xs. Although we are convincedof the utility of thinking about moderation and me-diation as fundamentally different processes, theyboth imply treatment effects varying in magnitude inpredictable ways as a function of a concomitant vari-able.

Second, our analyses suggest that when X variesbetween treatment conditions and Y is a function ofthat X in each condition, the full model necessitatesincluding both the X sum and the X difference aspredictors. To assess the impact of one of these with-out the other also included in the model is to riskbiased assessments of either mediation or modera-tion.15 The same presumably should apply in be-tween-subjects assessments of mediation and modera-tion. Furthermore, in cases in which crossed effectsmay exist (i.e., the X in one treatment condition af-fects Y in the other), analyses in between-subject de-

15 If the Xs in the two conditions have equal variances, itcan be shown that their sum and difference are uncorrelated(Kenny, 1979, p. 21). Accordingly, in this equal variancecase, there will be no bias if estimation includes only thesum or only the difference as predictor.


signs may be misleading because such crossed effectscannot be estimated. In this sense, one can think ofwithin-subject designs as containing considerablymore information than between-subjects designs. Or,equivalently, one can think of between-subjects de-signs as within-subject designs with half of the datafrom each participant randomly missing.

As with any analytic approach to data, the modelswe have outlined should be applied only after carefulconsideration of their appropriateness and of the as-sumptions that underlie them. Mediation and modera-tion of treatment effects are clearly important analyticissues to be addressed in both between-subjects andwithin-subject designs. And we hope to have clarifiedmethods in the latter case. But these analyses makesense only if the many assumptions that underlie themcan realistically be made.

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Received February 3, 1999Revision received January 5, 2001

Accepted January 8, 2001 •

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Estimating and Testing Mediation and Moderation in Within ...subject treatment variables (e.g., Fiske, Kenny, & Taylor, 1982; Neuberg, 1989). In general, these ar-ticles have explored