A General Multilevel SEM Framework for Assessing Multilevel Mediation


 Gwendoline McDowell
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1 Psychological Methods 1, Vol. 15, No. 3, America Psychological Associatio X/1/$1. DOI: 1.137/a141 A Geeral Multilevel SEM Framework for Assessig Multilevel Mediatio Kristopher J. Preacher Uiversity of Kasas Michael J. Zyphur Uiversity of Melboure Zhe Zhag Arizoa State Uiversity Several methods for testig mediatio hypotheses with level ested data have bee proposed by researchers usig a multilevel modelig (MLM) paradigm. However, these MLM approaches do ot accommodate mediatio pathways with Level outcomes ad may produce coflated estimates of betwee ad withilevel compoets of idirect effects. Moreover, these methods have each appeared i isolatio, so a uified framework that itegrates the existig methods, as well as ew multilevel mediatio models, is lackig. Here we show that a multilevel structural equatio modelig (MSEM) paradigm ca overcome these limitatios of mediatio aalysis with MLM. We preset a itegrative level MSEM mathematical framework that subsumes ew ad existig multilevel mediatio approaches as special cases. We use several applied examples ad accompayig software code to illustrate the flexibility of this framework ad to show that differet substative coclusios ca be draw usig MSEM versus MLM. Keywords: multilevel modelig, structural equatio modelig, multilevel SEM, mediatio Supplemetal materials: Researchers i behavioral, educatioal, ad orgaizatioal research settigs ofte are iterested i testig mediatio hypotheses with hierarchically clustered data. For example, Bacharach, Bamberger, ad Doveh (8) ivestigated the mediatig role of distress i the relatioship betwee the itesity of ivolvemet i workrelated icidets ad problematic drikig behavior amog firefightig persoel. They used data i which firefighters were ested withi ladder compaies ad all three variables were assessed at the subject level. Usig data from customer service egieers workig i teams, Mayard, Mathieu, Marsh, ad Ruddy (7) foud that teamlevel iterpersoal processes mediate the relatioship betwee teamlevel resistace to empowermet ad idividual job satisfactio. Both of these examples ad may others ivolve data that vary both withi ad betwee higher level uits. Traditioal methods for assessig mediatio (e.g., Baro & Key, 1986; MacKio, Lockwood, Hoffma, West, & Sheets, ; MacKio, Warsi, & Dwyer, 1995) are iappropriate i these multilevel settigs, primarily because the assumptio of idepedece of observatios is violated whe clustered data are used, leadig to dowwardly biased stadard errors if Kristopher J. Preacher, Departmet of Psychology, Uiversity of Kasas; Michael J. Zyphur, Departmet of Maagemet ad Marketig, Uiversity of Melboure, Melboure, Victoria, Australia; Zhe Zhag, W. P. Carey School of Busiess, Arizoa State Uiversity. We thak Tihomir Asparouhov, Begt Muthé, Vice Staggs, ad Soya Sterba for helpful commets. We also gratefully ackowledge Zhe Wag for permittig us to use his data i Examples ad 3. Correspodece cocerig this article should be addressed to Kristopher J. Preacher, Departmet of Psychology, Uiversity of Kasas, 1415 Jayhawk Boulevard, Room 46, Lawrece, KS ordiary regressio is used. For this reaso, several methods have bee proposed for addressig mediatio hypotheses whe the data are hierarchically orgaized. These recommeded procedures for testig multilevel mediatio have bee developed ad framed almost exclusively withi the stadard multilevel modelig (MLM) paradigm (for thorough treatmets of MLM, see Raudebush & Bryk,, ad Sijders & Bosker, 1999) ad implemeted with commercially available MLM software, such as SAS PROC MIXED, HLM, or MLwiN. For example, some authors have discussed models i which the idepedet variable X, mediator M, ad depedet variable Y all are measured at Level 1 of a twolevel hierarchy (a desig, adoptig otatio proposed by Krull & MacKio, 1), 1 ad slopes either are fixed (Pituch, Whittaker, & Stapleto, 5) or are permitted to vary across Level uits (Bauer, Preacher, & Gil, 6; Key, Korchmaros, & Bolger, 3). Other mediatio models have also bee examied, icludig mediatio i 1 desigs, i which both X ad M are assessed at the group level (Krull & MacKio, 1; Pituch, Stapleto, & Kag, 6), ad mediatio i 11 desigs, i which oly X is assessed at the group level (Krull & MacKio, 1999, 1; MacKio, 8; Pituch & Stapleto, 8; Raudebush & Sampso, 1999). Each of these approaches from the MLM literature was suggested i respose to the eed to estimate a particular model to test a mediatio hypothesis i a specific desig. However, the MLM paradigm is uable to accommodate simultaeous estimatio of a 1 We slightly alter the purpose of Krull ad MacKio s (1) otatio to refer to data collectio desigs rather tha to models. The models we propose for various multilevel data desigs are ot cosistet with models that have appeared i prior research. 9
2 1 PREACHER, ZYPHUR, AND ZHANG umber of other threevariable multilevel mediatio relatioships that ca occur i practice (see Table 1; models for some of these desigs were also treated by Mathieu & Taylor, 7). Desigs 1 ad i Table 1 caot be accommodated easily or completely i the MLM framework, because multivariate models require sigificat additioal data maagemet (Bauer et al., 6), ad MLM does ot fully separate betweegroup ad withigroup effects without itroducig bias. Desigs 3 through 7 i Table 1 caot be accommodated i the MLM framework at all because of MLM s iability to model effects ivolvig upper level depedet variables. I this article, we first explai these two reasos i detail ad show how they are circumveted by adoptig a multilevel structural equatio modelig (MSEM) framework for testig multilevel mediatio. Because a geeral framework for multilevel mediatio i structural equatio modelig (SEM) has yet to be preseted, we the itroduce MSEM ad show how Muthé ad Asparouhov s (8) geeral MSEM mathematical framework ca be applied i ivestigatig multilevel mediatio. We show how all previously proposed multilevel mediatio models as well as ewly proposed multilevel mediatio models ca be subsumed withi the MSEM framework. Third, we preset several empirical examples (with thoroughly aotated code) i order to itroduce ad aid i the iterpretatio of ew ad former multilevel mediatio models that the MSEM framework, but ot the MLM framework, ca accommodate. The primary cotributios of this article are to show (a) how MSEM ca separate some variables ad effects ito withiad betweegroup compoets to yield a more thorough ad less misleadig uderstadig of idirect effects i hierarchical data ad (b) how MSEM ca be used to accommodate mediatio models cotaiig mediators ad outcome variables assessed at Level (e.g., bottomup effects). We advocate the use of MSEM as a comprehesive system for examiig mediatio effects i multilevel data. First Limitatio of MLM Framework for Multilevel Mediatio: Coflatio of Betwee ad Withi Effects (Desigs 1 to 3 i Table 1) I twolevel desigs, it is possible to partitio the variace of a variable i clustered data ito two orthogoal latet compoets, the Betwee (cluster) compoet ad the Withi (cluster) compoet (Asparouhov & Muthé, 6). Variables assessed at Level have oly Betwee compoets of variace. Variables assessed at Level 1 typically have both Betwee ad Withi compoets, although i some cases a Level1 variable may have oly a Withi compoet if it has o betweegroup variatio.,3 If a variable has both Betwee ad Withi variace compoets, the Betwee compoet is ecessarily ucorrelated with the Withi compoet of that variable ad the Withi compoets of all other variables i the model. Similarly, the Withi compoet of a variable is ecessarily ucorrelated with the Betwee compoet of that variable ad the Betwee compoets of all other variables i the model. We refer to effects of Betwee compoets (or variables) o other Betwee compoets (or variables) as Betwee effects ad to effects of Withi compoets (or variables) o other Withi compoets (or variables) as Withi effects. Because Betwee ad Withi compoets are ucorrelated, it is ot possible for a Betwee compoet to affect a Withi compoet or vice versa. Ordiary applicatios of multilevel modelig do ot distiguish Betwee effects from Withi effects ad istead report a sigle mea slope estimate that combies the two (see, relatedly, Cha, 1998; Cheug & Au, 5; Klei, Co, Smith, & Sorra, 1). This is a importat issue for mediatio researchers to cosider; the use of slopes that combie Betwee ad Withi effects ca easily lead to idirect effects that are biased relative to their true values, because the compoet paths may coflate effects that are relevat to mediatio with effects that are ot. If the Withi effect of a Level1 predictor is differet from the Betwee effect (i.e., a compositioal or cotextual effect exists; Raudebush & Bryk, ), the estimatio of a sigle coflated slope esures that regardless of the level for which the researcher wishes to make ifereces the resultig slope will mischaracterize the data. The problem of coflated Withi ad Betwee effects i MLM is already well recogized for ordiary slopes i multilevel models (e.g., Asparouhov & Muthé, 6; Hedeker & Gibbos, 6; Kreft, de Leeuw, & Aike, 1995; Lüdtke et al., 8; Macl, Leroux, & DeRoue, ; Neuhaus & Kalbfleisch, 1998; Neuhaus & McCulloch, 6), but has bee uderemphasized i literature o multilevel mediatio. 4 To illustrate why this issue is problematic for mediatio research, we will cosider the case i which X is assessed at Level ad M ad Y are assessed at Level 1 (i.e., 11 desig). The effect of X o Y must be a strictly Betwee effect; because X is costat withi a give group, variatio i X caot ifluece idividual differeces withi a group (Hofma, ). 5 Therefore, ay mediatio of the effect of a Level X must also occur at a betweegroup level, regardless of the level at which M ad Y are assessed, because the oly kid of effect that X ca exert (whether direct or idirect) must be at the betweegroup level. I fact, ay mediatio effect i a model i which at least oe of X, M, ory is assessed at Level must occur strictly at the betweegroup level. Now cosider the 1 portio of the 11 desig. The first compoet of the idirect effect (a, the effect of X o M) is estimated properly i stadard MLM. That is, the effect of X o M ad the effect of X o Y are ubiased i the MLM framework. (However, it is ot geerally recogized that at a iterpreta The Betwee/Withi variace compoets distictio pertais to latet sources of variace. This cocept is related to, but distict from, the group mea ad grad mea ceterig choices commoly discussed for observed variables i the traditioal MLM literature. I the MSEM approach, observed Level1 variables are typically either grad mea cetered or ot cetered prior to aalysis. I either case, i MSEM all Level1 variables are subjected to implicit, modelbased group mea ceterig by default uless costraits are applied to the model. 3 I rare cases it is possible for a Level1 variable to have a egative estimated Level variace (Key, Maetti, Pierro, Livi, & Kashy, ), but we do ot discuss such cases here. 4 Exceptios are Krull ad MacKio (1), MacKio (8), ad Zhag, Zyphur, ad Preacher (9), who explicitly separated the Withi ad Betwee effects i the 11 portio of a mediatio model for 11 data. 5 It is assumed here that, if X is a treatmet variable, idividuals withi a cluster receive the same dosage or amout of treatmet. If iformatio o dosage or compliace is available, it should be icluded i the model.
3 MULTILEVEL SEM FOR TESTING MEDIATION 11 Table 1 Possible TwoLevel Multilevel Mediatio Desigs No. Desig Methods proposed Ivestigated by Limitatios of MLM MLM Key et al. (1998, 3); Krull & MacKio Coflatio or bias of the idirect effect (1999, 1); MacKio (8); Pituch & Stapleto (8); Pituch et al. (6); Raudebush & Sampso (1999); Zhag et al. (9) MLM Bauer et al. (6); Key et al. (3); Krull & Coflatio or bias of the idirect effect MacKio (1); MacKio (8); Pituch et al. (5) MSEM Raykov & Mels (7) Noe Noe MLM caot be used; Level depedet variables ot permitted OLS MLM Krull & MacKio (1); Pituch et al. (6) Twostep, osimultaeous estimatio; Level depedet variables ot permitted SEM Bauer (3) Noe Noe MLM caot be used; Level depedet variables ot permitted Noe Noe MLM caot be used; Level depedet variables ot permitted 71 Noe Noe MLM caot be used; Level depedet variables ot permitted Note. MLM traditioal multilevel modelig; MSEM multilevel structural equatio modelig; OLS ordiary least squares; SEM structural equatio modelig. tioal level these are actually Betwee effects because X may affect oly the Betwee compoets of M ad Y.) Now cosider the 11 compoet of the 11 desig. The secod portio of the idirect effect (b, the effect of M o Y), arguably, is ot estimated properly i stadard MLM. This effect is separable ito two parts, oe occurrig strictly at the betweegroup level ad the other strictly at the withigroup level. The use of slopes that combie betwee ad withigroup effects is kow to create substatial bias i multilevel models (Lüdtke et al., 8). Whe the true Withi effect is greater tha the true Betwee effect, there is a upward bias i the coflated effect. Whe the true Withi effect is less tha the true Betwee effect, dowward bias is preset. Because the b compoet is part of the Betwee idirect effect that is of primary iterest i 11 desigs, the idirect effect is similarly biased. This is true of ay desig ivolvig effects likig Level1 variables (e.g., 111, 11, ad loger causal chais cotaiig at least a sigle 11 likage). For example, cosider a 11 desig i which the Level variable X is the treatmet variable traiig o the job, with oe group receivig traiig ad aother ot receivig traiig; the Level1 mediator M is jobrelated skills; ad the Level1 outcome variable Y is job performace (i.e., traiig iflueces jobrelevat skills, which i tur iflueces job performace). I this case, X varies oly betwee groups (people as a group either do or do ot receive traiig), whereas both M ad Y vary both withi ad betwee the groups (people differ from each other withi the groups i their skills ad performace, ad there are differeces betwee the groups i skills ad performace). Whe oe estimates the ifluece of traiig o jobrelated skills, traiig iflueces idividual skills but does so for the group as a whole, makig the traiig effect a betweegroup effect. Because traiig was provided to the etire group without differetial applicatio across people withi the group, it caot accout for withigroup differeces of ay kid. Agai, that is ot to say that traiig has o impact o Level1 skills; it does, but oly because idividuals belog to groups that either did or did ot receive traiig. 6 For the same reaso, traiig ca impact job performace oly at the level of the group. Traiig caot accout for idividual differeces withi a group i skills ad performace (i.e., deviatios of idividuals from the group mea), because traiig was applied equally to all members withi each group. This logic is similar to that of the ANOVA framework, i which a treatmet effect is purely a betweegroup effect. Therefore, the idirect effect of traiig (X) o job performace (Y) through jobrelevat skills (M) may fuctio oly through the betweegroup variace i M ad Y. It follows that, i a mediatio model for 11 data, whe the b effect estimate coflates the Withi ad Betwee effects, the idirect effect that ecessarily operates betwee groups is cofouded with the withigroup portio of the coflated b effect. This is a importat issue, because multilevel mediatio studies have ofte used MLM without separatig the Betwee ad Withi compoets of the 11 likage (for models for 11 data, see Krull & MacKio, 1999, 1, ad Pituch & Stapleto, 8; for models for data, see Key et al., 3, ad Bauer et al., 6). I these studies, the Betwee ad Withi effects of M o Y are coflated (i.e., oly oe mea b slope is estimated). However, because each of these studies ivolved data geerated i a way that 6 It is of course possible that idividuals may respod differetly to traiig. However, this differetial respose does ot reflect the effect of traiig (which has o withicluster variace ad thus caot evoke variability i idividual resposes withi clusters). Rather, it reflects the ifluece of umeasured or omitted persolevel variables that moderate the ifluece of clusterlevel traiig.
4 1 PREACHER, ZYPHUR, AND ZHANG guarateed that the Betwee ad Withi compoets were idetical (i the datageeratig models, oly a sigle coflated effect of M o Y was specified), this led to low bias for the coditios examied. This situatio of equal b parameters across levels, we cojecture, is rare i practice. If the Betwee ad Withi b effects differ, the sigle estimated b slope i the coflated model will be a biased estimate of both. The stadard solutio for the problem of coflatio i MLM has bee to disetagle the Withi ad Betwee effects of a Level1 variable by replacig the Level1 predictor X ij with the two predictors X ij X.j ad X.j, where X ij X.j represets the withigroup portio of X ij ad X.j is the group mea of X ij (Hedeker & Gibbos, 6; Kreft & de Leeuw, 1998; Sijders & Bosker, 1999). We term this approach the ucoflated multilevel model (UMM) approach, emphasizig that the Betwee ad Withi compoets of the effect of X o Y are o loger coflated ito a sigle estimate. Two prior multilevel mediatio studies have used the UMM approach to reduce bias due to coflatio (group mea ceterig M ad icludig both the cetered M ad the group mea of M as predictors i the equatio for Y; MacKio, 8; Zhag, Zyphur, & Preacher, 9). However, eve though the UMM procedure separates Betwee ad Withi effects, a problem remais. Usig the group mea as a proxy for a group s latet stadig o a give predictor typically results i bias of the betweegroup effect for the predictor. This effect ca be uderstood by drawig a aalogy to latet variable models; Level1 uits ca be uderstood as idicators of Level latet costructs, so i the same way that few idicators ad low commualities ca bias relatioships amog latet variables i factor aalysis, few Level1 uits ad a low itraclass correlatio (ICC) ca bias Betwee effects (Asparouhov & Muthé, 6; Lüdtke et al., 8; Macl et al., ; Neuhaus & McCulloch, 6). Lüdtke et al. (8) showed that the expected bias of the UMM method i estimatig the Betwee effect i a 11 desig is Eˆ Y 1 B W B 1 1 ICC X ICC X 1 ICC X, (1) where ˆ Y 1 is the Betwee effect estimated usig observed group meas; W ad B are the populatio Withi ad Betwee effects, respectively; is the costat cluster size; ad ICC X is the itraclass correlatio of the predictor X. The expected bias of the Betwee idirect effect i a model for 11 data usig UMM (separatig M ito Betwee ad Withi compoets by group mea ceterig) is B M Eˆ M 1 ˆ Y 1 B B B XM M W X M XM X B M B, where ˆ M1 ad ˆ Y1 are, respectively, the Betwee effect of X o M ad of M o Y estimated with observed group meas; B is the populatio Betwee effect of X o M; X, M, ad XM are the Betwee variaces ad covariace of X ad M; ad M is the Withi variace of M. The Betwee effect ˆ Y1 is a weighted () average of the Betwee ad Withi effects of M o Y that depeds o the clusterlevel variaces ad covariace of X ad M, the withicluster variace of M, ad the cluster size (assumed to be costat over clusters; for full derivatio, see Appedix A). As icreases, the Withi variace of M carries less weight, yieldig a ubiased estimator. I most applicatios, UMM will yield a biased estimate of the true Betwee idirect effect. I ay mediatio model cotaiig a 11 compoet, the Betwee idirect effect will be similarly biased. If the Betwee ad Withi effects of M o Y i the 11 desig actually do differ, either the stadard MLM approach or the UMM approach will retur ubiased estimates of the Betwee idirect effect. To mitigate this bias the researcher must use a method that treats the clusterlevel compoet of Level1 variables as latet. The proposed MSEM framework does this. No prior multilevel mediatio studies have used the MSEM approach to retur ubiased estimates of the Betwee idirect effect. We recommed that MSEM be used to specify the model for 11 data ad, ideed, ay multilevel mediatio hypothesis i which ureliability i groups stadig exists. Usig group meas as proxies for latet clusterlevel meas implies that the researcher believes the meas are perfectly reliable. To the extet that this is ot true, relatioships ivolvig at least oe such group mea will, o average, be biased relative to the true effect (Asparouhov & Muthé, 6; Lüdtke et al., 8; Macl et al., ; Neuhaus & McCulloch, 6). The MSEM approach ot oly permits separate (ad theoretically ubiased) estimatio of the Betwee ad Withi compoets of 11 slopes but also icludes the coflated MLM approaches of Bauer et al. (6), Krull ad MacKio (1999, 1), ad Pituch ad Stapleto (8) as special cases, if this is desired, with the impositio of a equality costrait o the fixed compoets of the Betwee ad Withi 11 slopes. Additioally, as with the various MLM methods, the Withi slopes may be specified as radom effects. Secod Limitatio of MLM Framework for Multilevel Mediatio: Iability to Treat UpperLevel Variables as Outcomes (Desigs 3 to 7 i Table 1) MLM techiques are also limited i that they caot accommodate depedet variables measured at Level. Ideed, Krull ad MacKio (1) idicated that two basic requiremets for examiig multilevel mediatio with MLM are (a) that the outcome variable be assessed at Level 1 ad (b) that each effect i the causal chai ivolves a variable affectig aother variable at the same or lower level. I other words, upward effects are ruled out by the use of traditioal multilevel modelig techiques. Pituch et al. (6) ad Pituch, Murphy, ad Tate (1) also oted that MLM software caot be used to model  relatioships. Yet, theoretical predictios of this sort do occur. Effects characterized by a Level1 variable predictig a Level outcome are kow as bottomup effects (Bliese, ; Kozlowski & Klei, ), micromacro or emerget effects (Croo & va Veldhove, 7; Sijders & Bosker, 1999), or (crosslevel) upward ifluece (Griffi, 1997). Bottomup effects are commoly ecoutered i practice. For example, family data are itrisically hierarchical i ature: Paret variables are associated with the family (cluster)
5 MULTILEVEL SEM FOR TESTING MEDIATION 13 level ad childre are ested withi families, yet a great deal of research shows that childre ca impact familylevel variables (e.g., Bell, 1968; Bell & Belsky, 8; Kaugars, Kliert, Robiso, & Ho, 8; LugoGil & TamisLeModa, 8). To date, three approaches have bee used to circumvet this issue whe mediatio hypotheses are tested with multilevel data: twostep aalyses, aggregatio, ad disaggregatio. Each has limitatios, which we discuss i tur. TwoStep Aalyses Griffi (1997) proposed a twostage method for aalyzig bottomup effects i which the itercept residuals from a Level1 equatio (estimated with MLM) serve as predictors i a Level equatio (estimated with ordiary least squares [OLS] regressio). He illustrated the procedure by predictig grouplevel cohesio usig grouplevel empirical Bayes estimated residuals draw from the idividuallevel predictio of affect by cocurret ad previous group cohesio. Coversely, Croo ad va Veldhove (7) applied liear regressio aalysis predictig a Level outcome from group meas of a Level1 predictor adjusted for the bias that results from usig observed meas as proxies for latet meas. Applyig Griffi s (1997) or Croo ad va Veldhove s (7) methods i the cotext of mediatio aalysis would ivolve multiplyig the estimate thus obtaied for the 1 portio of the desig by aother coefficiet to compute a idirect effect. These methods could, i theory, be used i 11, 1, or more elaborate desigs. Lüdtke et al. (8) showed that Croo ad va Veldhove s procedure is characterized by less efficiet estimatio tha a oestage fulliformatio maximum likelihood procedure. Furthermore, use of these methods would ivolve a icoveiet degree of maual computatio, especially i situatios ivolvig more tha oe predictor ad more tha oe depedet variable (as i mediatio models). For predictors already assessed at Level, Krull ad MacKio (1) ad Pituch et al. (6) used OLS regressio to obtai the a compoet of the idirect effect ad MLM to obtai the b compoet for a model for 1 data (see Takeuchi, Che, & Lepak, 9, for a applicatio). Although this method is useful ad ituitive, it is more coveiet to estimate the parameters makig up the idirect effect simultaeously, as part of oe model, i which multiple compoets of variace could be cosidered simultaeously. The coveiece of MSEM relative to twostep aalyses becomes more apparet as the models become more complex (e.g., icorporatig multiple idirect paths ad latet variables with multiple idicators). Aggregatio Aggregatio ivolves usig group meas of Level1 variables i Level equatios, such that models for 1, 1, 1, or other multilevel desigs are all fit at the cluster level usig traditioal siglelevel methods i aalysis. Aggregatio of data to a higher level is highly problematic, i part because this assumes that the withigroup variability of the aggregated variable is zero (Barr, 8). As is well kow, aggregatio ofte (a) leads to severe bias ad loss of power for the regressio weights composig a idirect effect, (b) discouts iformatio regardig withiuit variatio, (c) drastically reduces the sample size, ad (d) icreases the risk of committig commo fallacies, such as the ecological fallacy ad the atomistic fallacy (Alker, 1969; Chou, Betler, & Petz, ; DiezRoux, 1998; Firebaugh, 1978; Hofma, 1997, ). Furthermore, aggregatio effectively gives small groups ad large groups equal weight i determiig parameter estimates. Fially, aggregated Level1 data may ot always fairly represet grouplevel costructs (Blalock, 1979; Chou et al., ; Klei & Kozlowski, ; Krull & MacKio, 1; va de Vijver & Poortiga, ). Disaggregatio Disaggregatio ivolves igorig the hierarchical structure altogether ad usig a siglelevel model to aalyze multilevel mediatio relatioships. However, disaggregatio fails to separate withigroup ad betweegroup variace, muddyig the iterpretatio of the idirect effect, ad also may spuriously iflate power for the test of the idirect effect (Chou et al., ; Julia, 1; Krull & MacKio, 1999). MSEM I cotrast to these three procedures (twostep, aggregatio, ad disaggregatio), MSEM does ot require outcomes to be measured at Level 1, or does it require twostage aalysis. As we oted earlier, for ay mediatio model ivolvig at least oe Level variable, the idirect effect ca exist oly at the Betwee level. So it is ot sesible to hypothesize that the effect of a Level1 X o a Level Y ca be mediated. Rather, it is the effect of the clusterlevel compoet of X that ca be mediated. Withicluster variatio i X ecessarily caot be related to betweecluster variatio i M or Y. Nevertheless, it may ofte be of iterest to determie whether or ot mediatio exists at the cluster level i desigs i which a Level1 variable precedes a Level variable i the causal sequece. Threevariable systems fittig this descriptio iclude 11, 1, 1, ad 11 desigs. Multilevel SEM MSEM has received little attetio as a framework for multilevel mediatio because aalytical developmets i MSEM have oly recetly made this approach a feasible alterative to MLM for this purpose. I the past, approaches for fittig MSEMs were hidered by a iability to accommodate radom slopes (e.g., Bauer, 3; Curra, 3; Härqvist, 1978; Goldstei, 1987, 1995; McArdle & Hamagami, 1996; Mehta & Neale, 5; Rovie & Moleaar, 1998,, 1; Schmidt, 1969) ad sometimes also a iability to fully accommodate partially missig data ad ubalaced cluster sizes (Goldstei & McDoald, 1988; McDoald, 1993, 1994; McDoald & Goldstei, 1989; Muthé, 1989, 199, 1991, 1994; Muthé & Satorra, 1995). These early methods typically ivolved decomposig observed scores ito Betwee ad pooled Withi covariace matrices ad fittig separate Betwee ad Withi models usig the multiplegroup fuctio of may SEM software programs.
6 14 PREACHER, ZYPHUR, AND ZHANG O the basis of these ad other developmets, several authors have specified causal chai models usig MSEM, but they did ot address idirect effects directly (e.g., Kapla & Elliott, 1997a, 1997b; Kapla, Kim, & Kim, 9; McDoald, 1994; Raykov & Mels, 7). Although a few authors have implemeted multilevel mediatio aalyses usig MSEM for a specific model ad have addressed the idirect effect specifically (Bauer, 3; Heck & Thomas, 9; Rowe, 3), they did ot allow geeralizatios for radom slopes, did ot provide a framework that accommodated all possible multilevel mediatio models, ad did ot provide comparisos to other (MLM) methods (as is doe here i the followig sectios). These shortcomigs of previous work may be attributed largely to techical ad practical limitatios associated with the twomatrix ad other approaches to MSEM. Recet MSEM advaces have overcome the limitatios of the twomatrix approach regardig radom slopes ad have also accommodated missig data ad ubalaced clusters (Asari, Jedidi, & Dube ; Asari, Jedidi, & Jagpal, ; Chou et al., ; Jedidi & Asari 1; Muthé & Asparouhov, 8; RabeHesketh, Skrodal, & Pickles, 4; Raudebush, Rowa, & Kag, 1991; Raudebush & Sampso, 1999; Skrodal & RabeHesketh 4). However, these methods vary i their beefits, drawbacks, ad geerality. For example, Raudebush et al. s (1991) method ivolves addig a restrictive measuremet model to a otherwise ordiary applicatio of MLM, so it easily allows multiple levels of estig but does ot allow fit idices, uequal factor loadigs, or idirect effects at multiple levels simultaeously. These disadvatages are overcome by Chou et al. s method. However, their method results i still other drawbacks: biased estimatio of grouplevel variability i radom coefficiets, o correctio for ubalaced group sizes, ad osimultaeous estimatio of the Betwee ad Withi compoets of a model. More geeral methods iclude those of Asari ad colleagues (Asari et al.,, ; Jedidi & Asari, 1), Rabe Hesketh et al. (4), ad Muthé ad Asparouhov (8). Of these, Muthé ad Asparouhov s (8) method is ofte the most computatioally tractable for complex models. 7 I compariso, Asari et al. s Bayesia methodology is challegig to implemet for the applied researcher; RabeHesketh et al. s methodology may ofte require sigificatly loger computatioal time (Bauer, 3; Kamata, Bauer, & Miyazaki, 8) ad does ot accommodate radom slopes for latet covariates limitig what ca be tested with latet variables ad Level1 predictors ad mediators. Therefore, we apply Muthé ad Asparouhov s (8) MSEM approach to the cotext of mediatio i the ext sectio. A Geeral MSEM Framework for Ivestigatig Multilevel Mediatio The siglelevel structural equatio model ca be expressed i terms of a measuremet model ad a structural model. Accordig to the otatio of Muthé ad Asparouhov (8), the measuremet model is Y i i KX i i, (3) where i idexes idividual cases; Y i is a pdimesioal vector of measured variables; is a pdimesioal vector of variable itercepts; i is a pdimesioal vector of error terms; is a p m loadig matrix, where m is the umber of radom effects (latet variables); i is a m 1 vector of radom effects; ad K is a p q matrix of slopes for the q exogeous covariates i X i. The structural model is i B i X i i, (4) where is a m 1 vector of itercept terms, B is a m m matrix of structural regressio parameters, is a m q matrix of slope parameters for exogeous covariates, ad i is a mdimesioal vector of latet variable regressio residuals. Residuals i i ad i are assumed to be multivariate ormally distributed with zero meas ad with covariace matrices ad, respectively. The model i Equatios 3 ad 4 expads to accommodate multilevel measuremet ad structural models by permittig elemets of some coefficiet matrices to vary at the cluster level. The measuremet portio of Muthé ad Asparouhov s (8) geeral model is expressed as Y ij j j ij K j X ij ij (5) where j idicates cluster. 8 Note that Equatio 5 is idetical to Equatio 3, but with cluster (j) subscripts o both the variables ad the parameter matrices, idicatig the potetial for some elemets of these matrices to vary over clusters. Like Y i,, ad i, the matrices Y ij, j, ad ij are pdimesioal. Similarly, j is p m (where m is the umber of latet variables across both Withi ad Betwee models ad icludes radom slopes if ay are specified), ij is m 1, ad K j is p q. X ij is a qdimesioal vector of exogeous covariates. The structural compoet of the geeral model ca be expressed as ij j B j ij j X ij ij, (6) where j is m 1, B j is m m, ad j is m q. Residuals i ij ad ij are assumed to be multivariate ormally distributed with zero meas ad with covariace matrices ad, respectively. Elemets i these latter two matrices are ot permitted to vary across clusters. Elemets of the parameter matrices j, j, j, j, B j, ad j ca vary at the Betwee level. The multilevel part of the geeral model is expressed i the Level structural model: j j X j j (7) 7 The model of Muthé ad Asparouhov (8) is implemeted i Mplus. Versios 5 ad later of Mplus implemet maximum likelihood estimatio via a accelerated EM algorithm described by Lee ad Poo (1998) ad icorporate further improvemets ad refiemets, some of which are described by Lee ad Tsag (1999), Betler ad Liag (3, 8), Liag ad Betler (4), ad Asparouhov ad Muthé (7). Ulike earlier methods, the ew method ca accommodate missig data, ubalaced cluster sizes, ad crucially radom slopes. The default robust maximum likelihood estimator used i Mplus does ot require the assumptio of ormality, yields robust estimates of asymptotic covariaces of parameter estimates ad, ad is more computatioally efficiet tha previous methods. 8 We use a otatio slightly differet from that of Muthé ad Asparouhov (8) to alig more closely with stadard multilevel modelig otatio ad to avoid reuse of similar symbols.
7 MULTILEVEL SEM FOR TESTING MEDIATION 15 It is importat to recogize that j i Equatio 7 is very differet from the ij i Equatios 5 ad 6. The vector j cotais all the radom effects; that is, it stacks the radom elemets of all the parameter matrices with j subscripts i Equatios 5 ad 6 say there are r such radom effects. Also, X j is differet from the earlier X ij. X j is a sdimesioal vector of all clusterlevel covariates. The vector (r 1) ad matrices (r r) ad (r s) cotai estimated fixed effects. I particular, cotais meas of the radom effect distributios ad itercepts of Betwee structural equatios, cotais regressio slopes of radom effects (i.e., latet variables ad radom itercepts ad slopes) regressed o each other, ad cotais regressio slopes of radom effects regressed o exogeous clusterlevel regressors. Clusterlevel residuals i j have a multivariate ormal distributio with zero meas ad covariace matrix. Together, Equatios 5, 6, ad 7 defie Muthé ad Asparouhov s (8) geeral model, which we adopt here for specifyig ad testig multilevel mediatio models. I the preset cotext we ca safely igore matrices j, j, j, ad ad we do ot require exogeous variables i X ij or X j or residuals ij, although all of these are available i the geeral model. Furthermore, i the models discussed here, j. Put succictly, the, Measuremet model: Y ij ij (8) Withi structural model: ij j B j ij ij (9) Betwee structural model: j j j (1) with ij MVN, ad j MVN,. Expadig the j vector clarifies that it cotais the radom effects, or elemets of the matrices j ad B j that potetially vary at the cluster level. Lettig the vec{.} operator deote a colum vector of all the ozero elemets (fixed or radom) of oe of these matrices, take i a rowwise maer, j vecb j vec j (11) The vector cotais the fixed effects ad the meas of the radom effects: B (1) Whe all the elemets of oe of the subvectors of j have zero meas ad do ot vary at the cluster level, the correspodig subvectors are elimiated from both j ad. Withi, oly B cotais parameters ivolved i the mediatio models cosidered here. Thus, oly B ad cotai estimated structural parameters, with B cotaiig the Withi effects ad cotaiig the Betwee effects. Throughout, we assume the Withi ad Betwee structural models i Equatios 9 ad 1 to be recursive (i.e., they cotai o feedback loops, ad both B j ad ca be expressed as lower triagular matrices). Whe the geeral MSEM model is applied to raw data, o explicit ceterig is required. Rather, the model implicitly partitios each observed Level1 variable ito latet Withi ad Betwee compoets. The MSEM models we discuss here do ot ivolve or require explicit ceterig of observed predictor variables, but researchers are free to grad mea or group mea ceter their observed variables as i traditioal MLM. Grad mea ceterig simply rescales a predictor such that the grad mea is subtracted from every score, regardless of cluster membership. Zero correspods to the grad mea i the cetered variable, ad other values represet deviatios from the grad mea. Group mea ceterig ivolves subtractig the cluster mea from every score. Group mea ceterig thus removes all Betwee variace i a observed variable prior to modelig, rederig it a strictly Withi variable (i.e., all cluster meas are equal to zero i the cetered variable). O a practical ote, the researcher may wish to group mea ceter a Level1 predictor variable whe its Betwee variace is essetially zero. If a model is estimated usig a variable with a very small ICC, the estimatio algorithm may ot coverge to a proper solutio. More geerally, the researcher may simply wish to focus o the withicluster idividual differeces ad have o iterest i betweecluster differeces ad thus may choose to elimiate the Betwee portio of a predictor i the iterests of parsimoy. Group mea ceterig of Level1 predictors is ill advised i cases where betweecluster effects are of theoretical iterest, or are at least plausible i light of theory ad potetially of iterest; we suspect this accouts for the majority of cases i practice. More is said about ceterig strategies, ad the motivatios for ceterig, by Eders ad Tofighi (7), Hofma ad Gavi (1998), ad Kreft et al. (1995). The geeral MSEM model i Equatios 8 1 cotais SEM (ad thus factor aalysis ad path aalysis) ad twolevel MLM as special cases. Although we restrict attetio to the siglepopulatio, cotiuous variable MSEM to simplify presetatio, Muthé ad Asparouhov (8) preseted extesios to this MSEM accommodatig idividual ad clusterlevel latet categorical variables, as well as cout, cesored, omial, semicotiuous, ad survival data. Future ivestigatios of these extesios i the cotext of multilevel mediatio aalysis are certaily warrated. Additioal discussio of the geeral model ca be foud i Kapla (9) ad Kapla et al. (9). Specifyig Multilevel Mediatio Models i MSEM We ext describe how models discussed previously i the multilevel mediatio literature, as well as ew mediatio models for hierarchical data, ca be specified as special cases of this geeral MSEM model. Example Mplus (Muthé & Muthé, ) sytax is provided o the first author s website for each of the models discussed here (for Versio 5 or later). 9 We begi with the simplest mediatio model ad follow with expositio of the multilevel mediatio model for 11 data. We address several additioal models (for 1, 111, 11, 1, 1, ad 11 data) i Appedix B. Ordiary SigleLevel Mediatio It is possible to employ the framework described here whe there is o clusterlevel variability ad oly measured variables are used. 9 See
8 16 PREACHER, ZYPHUR, AND ZHANG Whe cofroted with such data, most researchers use OLS regressio, or more geerally path aalysis, to assess mediatio (Wood, Goodma, Beckma, & Cook, 8), followig the istructios of umerous methodologists who have ivestigated mediatio aalysis (e.g., Baro & Key, 1986; Edwards & Lambert, 7; MacKio, 8; MacKio et al., ; Preacher & Hayes, 4, 8a, 8b; Shrout & Bolger, ). Siglelevel path aalysis is a costraied special case of MSEM with o Betwee variatio, o latet costructs, ad o error variace or estimated itercepts for the measured variables. It is ot ecessary to ivoke a multilevel model to assess mediatio effects whe there is o cluster variatio i the variables, but it is iformative to see how the geeral model may be costraied to yield this familiar case by ivokig Equatios 8, 9, ad 1 ad lettig, j, j,, ad.weca further grad mea ceter all variables ad let to reder the path aalysis model. I the case of siglelevel path aalysis, the geeral MSEM model reduces to Y ij I B 1 ij (13) The elemets of B cotai the path coefficiets makig up the idirect effect: B B MXj (14) B YXj B YMj Thus, Y ij X M Y ij 1 B MXj 1 B YXj B YMj 1 B MXj B YMj B YXj B YMj 1 1 Xij Mij Yij 1 B MXj 1 Xij Mij Yij (15) The total idirect effect of X o Y through M is give by (Bolle, 1987): Thus, F I B 1 I B. (16) F B MXj B YMj, (17) i which B MXj B YMj represets the idirect effect of the first colum variable (X) o the last row variable (Y) via M. I the more elaborate multilevel models described below ad i Appedix B, the same procedure may be used to obtai the idirect effect expressios. Mediatio i 11 Data Mediatio i 11 data (also called upperlevel mediatio) has become icreasigly commo i the multilevel literature, especially i prevetio, educatio, ad orgaizatioal research (e.g., Key, Kashy, & Bolger, 1998; Key et al., 3; Krull & MacKio, 1; MacKio, 8; Pituch & Stapleto, 8; Pituch et al., 6; Tate & Pituch, 7). For example, Hom et al. (9) foud that job embeddedess (Level 1) mediates the ifluece of mutualivestmet employee orgaizatio relatioship (Level ) o lagged measures of employees quit itetio ad affective commitmet (Level 1). Similarly, Zhag et al. (9) hypothesized that job satisfactio (Level 1) mediates the effect of flexible work schedule (Level ) o employee performace (Level 1). I this example, because flexible work schedule is a clusterlevel predictor, it ca predict oly clusterlevel variability i employee performace. Therefore, the questio of iterest is ot simply whether satisfactio mediates the effect of flexible work schedule o performace but whether, ad to what degree, clusterlevel variability i job satisfactio serves as a mediator of the clusterlevel effect of flexible work schedule o the clusterlevel compoet of performace. The geeral MSEM model may be costraied to yield a model for 11 data by ivokig Equatios 8, 9, ad 1. I the case of mediatio i 11 data with radom b slopes, the geeral MSEM equatios reduce to Y ij X j M Y ij ij 1 1 Mij 1 1 Yij 1 Xj (18) Mj Yj Yij B YMj Yij ij Mij Xj Xj Xj Mj Yj Mj Yj Mj Mij YMj Xj j B Mj Xj Mj YjBYMj Xj MX Mj Yj YX YM BYMj Yij YjMij (19) Xj Mj YjBYMj Yj () The partitios serve to separate the Withi (above/before the partitio) ad Betwee (below/after the partitio) parts of the model. Here, Mij ad Yij are latet variables that vary oly withi clusters, ad Xj, Mj, ad Yj vary oly betwee clusters. Because there are oly two variables (M ad Y) with Withi variatio, there is o Withi idirect effect. The elemets of cotai the path coefficiets makig up the Betwee idirect effect. The total Betwee idirect effect of X j o Y ij via M ij are give by extractig the 3 3 Betwee submatrix of (call it B ) ad employig Bolle s (1987) formula, F I B 1 I B MX YM, (1) i which MX YM is the Betwee idirect effect of the first colum variable (X j ) o the third row variable (the Betwee portio of Y ij ) via the clusterlevel compoet of M ij. The simultaeous but coflated model of Pituch ad Stapleto (8) may be obtaied by costraiig BYMj YM. Specifyig the Withi b slope (B YMj )
9 MULTILEVEL SEM FOR TESTING MEDIATION 17 as fixed rather tha radom will ot alter the formula for the Betwee idirect effect, although this probably will alter its value ad precisio somewhat. Further MSEM Advatages Over MLM for Multilevel Mediatio Of course, the models discussed here ad i Appedix B (i.e., for 1, 11, 111, 11, 1, 1, ad 11 data) do ot exhaust the twolevel possibilities that ca be explored with MSEM. Whereas radom slopes are permitted to be depedet variables i MLM, the flexibility of MSEM permits examiatio of models i which radom slopes might also (or istead) serve as mediators or idepedet variables (Cheog, MacKio, & Khoo, 3; Dage, Brow, & Howe, 7). Additioally, whereas the additio of multiple mediators (MacKio, ; Preacher & Hayes, 8a) becomes a burdesome data maagemet ad model specificatio task i the MLM framework, it is more straightforward i MSEM. I additio, whe MSEM is used, ay variable i the system may be specified as a latet variable with multiple measured idicators simply by freeig the relevat loadigs. Eve these loadigs may be permitted to vary across clusters by cosiderig them radom effects. I cotrast, it is ot easy to use latet variables with stadard MLM software. I order to icorporate latet variables ito MLM, the researcher is obliged to make the strict assumptio of equal factor loadigs ad uique variaces (Raudebush et al., 1991; Raudebush & Sampso, 1999). Researchers ofte compute scale meas istead. A atural cosequece of this practice is that the presece of ureliability i observed variables teds to bias effects dowward ad thus reduce power for tests of regressio parameters. Correctio for atteuatio due to measuremet error is especially importat whe testig mediatio i data at the withigroups level of aalysis, because a majority of the error variace i observed variables exists withi groups (because betweegroup values are aggregates, they will be more reliable; Muthé, 1994). Fially, withi the MLM framework it is ot easy to assess model fit, because for may models there is o atural saturated model to which to compare the fit of the estimated model (Curra, 3), ad MLM software ad other resources do ot emphasize model fit. I the MSEM framework, o the other had, fit idices ofte are available to help researchers assess the absolute ad relative fit of models. Most of the models discussed here are fully saturated ad therefore have perfect fit. However, this will ot always be the case i practice, as researchers will frequetly wish to assess idirect effects i models with costraits (e.g., measuremet models, paths fixed to zero). It will be importat to demostrate that overidetified models have sufficietly good fit before proceedig to the iterpretatio of idirect effects i such models. A large array of fit idices is available for models without radom slopes (Muthé & Muthé, ), ad iformatio criteria (AIC ad BIC) are available for comparig models with radom slopes. Lee ad Sog (1) demostrated the use of Bayesia model selectio ad hypothesis testig i MSEM. 1 Work is uder way to develop strategies for assessig model fit i the MSEM cotext (Ryu, 8; Ryu & West, 9; Yua & Betler, 3, 7). Empirical Examples Example 1: Determiats of Math Performace (Mediatio i a 1(11)1 Desig) Our first example is draw from the Michiga Study of Adolescet Life Trasitios (MSALT; Eccles, 1988, 1996). MSALT is a logitudial study followig sixth graders i 1 Michiga school districts over the trasitio from elemetary to juior high school ( ), with two waves of data collectio before ad two after the trasitio to seveth grade ad several followups. The basic purpose i MSALT is to examie the impact of classroom ad family evirometal factors as determiats of studet developmet ad achievemet. This extesive data set combies studet questioaires, academic records, teacher questioaires ad ratigs of studets, paret questioaires, ad classroom eviromet measures. We cosidered studets (N,993) ested withi teachers (J 16) durig Wave of the study (sprig of sixth grade). We tested the hypothesis that the effect of studetrated global selfesteem (gse) o teacherrated performace i math (perform) was mediated by teacherrated talet (talet) ad effort (effort). Global selfesteem was computed as a composite of five items assessig studets resposes ( sort of true for me or really true for me ) relative to exemplar descriptios such as Some kids are pretty sure of themselves ad Other kids would like to stay pretty much the same. Performace i math was assessed by the sigle item Compared to other studets i this class, how well is this studet performig i math? (1 ear the bottom of this class, 5 oe of the best i the class). Talet ad effort were assessed with the items How much atural mathematical talet does this studet have? (1 very little,7 a lot) ad How hard does this studet try i math? (1 does ot try at all, 7 tries very hard), respectively. ICCs were gse (.3), talet (.11), effort (.13), ad perform (.4). Because all four variables were assessed at the studet level, this model could be described as a 1(1,1)1 model with two mediators with correlated residuals (see Figure 1). We specified radom itercepts ad fixed slopes, save that the direct effect of gse o perform was specified as a radom slope. Although a MSEM model with all radom itercepts ad radom slopes ca be estimated, such a model adds uecessary complicatios ad may reduce the probability of covergece. The MSEM approach permits us to ivestigate mediatio by multiple mediators simultaeously ad at both levels. The Withi idirect effect through talet (.14) was sigificat, 9% CI [.14,.178], 11 as was the idirect effect through effort (.98), 9% CI [.73,.15]. These 1 There is some ambiguity over whether to use the total sample size, the umber of clusters, or the average cluster size i formulas for fit idices i MSEM (Bovaird, 7; Selig, Card, & Little, 8). Mehta ad Neale (5) suggested that the umber of clusters is the most appropriate sample size to use, whereas Lee ad Sog (1) used the total sample size. Total sample size is used i Mplus. 11 All idirect effect cofidece itervals are 9% to correspod to oetailed,.5 hypothesis tests, which we feel are ofte justified i mediatio research; cotrast CIs are 95% itervals to correspod to twotailed hypothesis tests.
10 18 PREACHER, ZYPHUR, AND ZHANG Figure 1. Illustratio of a mediatio model for data from Example 1. For simplicity, the radom slope of perform regressed o gse is ot depicted. effort teacherrated effort; gse studetrated global selfesteem; perform teacherrated performace i math; talet teacherrated talet. two idirect effects were ot sigificatly differet from each other (cotrast.4), 95% CI [.11,.96]. The Betwee idirect effect of gse o perform via talet was sigificat (.96), 9% CI [.46,.633], as was the idirect effect via effort (.18), 9% CI [.4,.43], idicatig that teachers whose studets ted o average to have higher selfesteem ted to award their studets higher ratigs for both talet ad effort, ad these i tur mediated the effect of group selfesteem o collective math performace. These Betwee idirect effects were ot sigificatly differet (.114), 95% CI [.31,.589]. The aalysis was repeated with the UMM method, that is, by explicitly group mea ceterig gse, talet, ad effort ad usig the cetered variables ad their meas as predictors of perform.as expected, the Withi idirect effects through talet ad effort, ad the cotrast of the two idirect effects, were essetially the same as i MSEM, as were their cofidece itervals. The Betwee idirect effect of gse o perform via talet was sigificat (.19), 9% CI [.78,.381], as was the idirect effect via effort (.13), 9% CI [.8,.5]. This idicates, agai, that teachers whose studets have higher average selfesteem ted to award higher ratigs for both talet ad effort, ad these ratigs i tur mediate the effect of group selfesteem o collective math performace. These Betwee idirect effects were ot sigificatly differet (.9), 95% CI [.13,.31]. The oteworthy feature of the
11 MULTILEVEL SEM FOR TESTING MEDIATION 19 UMM aalysis is that, whereas the Withi idirect effects are comparable to those from the MSEM aalysis, the Betwee idirect effects are cosiderably smaller because of bias, although still statistically sigificat. The Betwee results are preseted i the upper pael of Table. Example : Atecedets of Team Potecy (Mediatio i a 1 Desig) I our secod example, we examie the relatioship betwee a team s shared visio ad team members perceived team potecy, as mediated by the team leader s idetificatio with the team. Multisource survey data were collected from 79 team leaders ad 49 team members i a customer service ceter of a telecommuicatios compay i Chia. The various scales measurig the variables were traslated ad backtraslated to esure coceptual cosistecy. Cofidetiality was assured, ad the leaders ad members retured the completed surveys directly to the researchers. The umber of sampled team members raged from 4 to 7 (excludig the leader), with a average team size of 5.6 members per team. Leaders reported teams shared visio (three items used i Pearce & Esley, 4) ad their idetificatio with the team (five items used i Smidts, Pruy, & va Riel, 1), ad members reported their perceived team potecy (five items used i Guzzo, Yost, Campbell, & Shea, 1993). All three measures were assessed with a 7poit Likert scale (1 strogly disagree, 7 strogly agree). Example items iclude Team members provide a clear visio of who ad what our team is (shared visio), I experieced a strog sese of belogig to the team (idetificatio), ad Our team feels it ca solve ay problem it ecouters (team potecy). We used scale items as idicators for all three latet costructs. The ICCs for team potecy idicators raged from.33 to.48. Usig both the MSEM ad covetioal twostep approach, we tested the hypothesis that shared visio would affect potecy idirectly via idetificatio. Because shared visio ad idetificatio are leaderlevel measures ad team potecy is a memberlevel measure, this is a 1 desig. We specified radom itercepts ad fixed slopes i both the MSEM ad the twostep approaches. Results based o the two approaches are show i the middle pael of Table. Although both approaches showed sigificat idirect effects, there are differeces i the estimates of the path coefficiets ad the idirect effect itself. I the twostep approach, the first step ivolves estimatig the  part of the 1 desig usig OLS regressio (J 79 team leaders). With leader idetificatio regressed upo leader visio, the ustadardized coefficiet (.387, SE.94, p.1) represets path a of the idirect effect. I the secod step, leader visio ad leader idetificatio both predict team members perceptios of team potecy i a hierarchical liear model with radom itercepts ad fixed slopes. The coefficiet of leader idetificatio (.377, SE.149, p.5) represets path b of the idirect effect. Whe this twostep approach is used, the idirect effect is a b.146 (p.5), 9% CI [.44,.69]. Note that, although the three variables i this example all have multiple idicators, i this twostep approach the mea of multiple items is typically used to measure the variable of iterest whe the iteral cosistecy coefficiet is satisfactory (e.g., greater tha.7). I the MSEM approach, i cotrast, multiple idicators are used to measure the latet variables of leader visio, leader idetificatio, ad team members perceptios of team potecy (as show i Figure ). MSEM permits us to ivestigate the () ad (1) likages simultaeously, rather tha i two steps as the covetioal MLM framework requires. The MSEM estimated idirect effect was higher tha that of the twostep approach (.18, p.1, vs..146, p.5) ad had a wider cofidece iterval, 9% CI [.55,.43]. I additio, the MSEM results showed a larger stadard error for the a path tha that i the OLS regressio of the twostep approach. The MSEM approach provides fit idices for the mediatio model (ot show i the table), ad these idices showed satisfactory Table MSEM Versus Covetioal Approach Estimates ad Stadard Errors for Path a, Path b, ad the Idirect Effect Approach a path b path Idirect effect [9% CI] Example 1: 1(1,1)1 desig UMM paths via talet.757 (.86).9 (.5).19 [.78,.381] UMM paths via effort.659 (.35).197 (.44).13 [.8,.5] MSEM paths via talet (.594).198 (.86).96 [.46,.633] MSEM paths via effort 1.19 (.656).153 (.65).18 [.4,.43] Example : 1 desig Twostep approach with maifest variables Step 1: OLS for a path Step : MLM for b path.387 (.94).377 (.149).146 [.44,.69] MSEM with latet variables.567 (.14).385 (.135).18 [.55,.43] Example 3: 11 desig Twostep approach with maifest variables Step 1: UMM for a path Step : OLS with aggregated variables for b path (.161).45 (.171).53 [.76,.387] MSEM with latet variables (.331).46 (.39).697 [., 1.459] Note. Stadard errors are show i paretheses. Cofidece itervals are based o the Mote Carlo method (available at MSEM multilevel structural equatio modelig; CI cofidece iterval; UMM ucoflated multilevel modelig; OLS ordiary least squares; MLM traditioal multilevel modelig. p.5. p.1. p.1 (oetailed tests).
12 PREACHER, ZYPHUR, AND ZHANG Figure. Illustratio of mediatio model for 1 data from Example. shared visio leaderreported team shared visio; idetificatio leaderreported member idetificatio with the team; potecy memberreported perceived team potecy. model fit (comparative fit idex.94, Tucker Lewis idex.9, RMSEA.47, SRMR B.8, SRMR W., where SRMR B ad SRMR W are the stadardized root mea square residuals for the Betwee ad Withi models, respectively). Although the estimates based o the two approaches differ i magitude, both idicate that teams with a strog shared visio are more likely to have leaders who idetify with their teams. This i tur leads to higher team potecy as perceived by team members. Example 3: Leadership ad Team Performace (Mediatio i a 11 Desig) Our third example examies a upward ifluece model i which team members perceived trasformatioal leadership (TFL) impacts team performace, as mediated by members satisfactio with the team. We utilized the data set from Example to examie this model for 11 data. Team members reported their perceptios of trasformatioal leadership (four subdimesio scores were based upo items i the Multifactor Leadership Questioaire; Bass & Avolio, 1995) ad satisfactio with the team (three items used i Va der Vegt, Emas, & Va de Vliert, ). Higher level maagers who are exteral to the teams operatios rated teams performace o five items (used i Va der Vegt & Buderso, 5). TFL was assessed usig a 5poit scale ( ever, 4 frequetly), ad performace ad satisfactio were measured o 7poit Likert scales (1 strogly agree,7 strogly disagree). Example items iclude My leader articulates a compellig visio of the future (TFL), I am pleased with the way my colleagues ad I work together (satisfactio), ad This team accomplishes tasks quickly ad efficietly (performace). All variables were agai treated as latet i the MSEM approach, with items or subdimesio scores as idicators. Variables were treated as maifest i the twostep approach. ICCs raged from.7 to.37. We specified radom itercepts ad fixed slopes for both the twostep approach ad MSEM. Results based o MSEM ad the twostep approach are show i the lower pael of Table. The two approaches showed highly differet estimates of the idirect effect, which would lead to very differet substative coclusios based o the same data. I the twostep approach, the first step estimated the 11 part of the 11 desig usig MLM with radom itercepts ad fixed slopes. Job satisfactio was regressed upo both the group mea cetered TFL ad the aggregated TFL at the team level. The ustadardized coefficiet of the aggregated TFL (1.168, SE.161, p.1) represets the a path of the idirect effect. I the secod step, we aggregated TFL ad job satisfactio to the team level ad used them to predict leader rated team performace usig OLS regressio (J 79 teams). The coefficiet of aggregated job satisfactio (.45, SE.171, p.8) represets path b i the idirect effect. Whe this twostep approach was used, the idirect effect was ot sigificat, a b.53 (p.85), 9% CI [.76,.387]. Although the three variables i this example all have multiple idicators, i this twostep approach the meas of the multiple items are used to measure the variable of iterest whe the iteral cosistecy coefficiet is satisfactory (e.g., greater tha.7).
13 MULTILEVEL SEM FOR TESTING MEDIATION 1 I the MSEM approach, i cotrast, multiple idicators were used to measure the latet variables of TFL, job satisfactio, ad leaderrated team performace (as show i Figure 3). Paths a ad b were estimated simultaeously rather tha i two separate steps. The MSEM estimated idirect effect was sigificat ad much higher tha that of the twostep approach (.697, p.1, vs..53, p.85). I additio, the MSEM results shows larger estimates for both the a path ad b path. The MSEM approach provides fit idices for the mediatio model (ot show i the table), which showed satisfactory model fit (comparative fit idex.97, Tucker Lewis idex.96, RMSEA.5, SRMR B.7, SRMR W.3). I sum, the MSEM results show a sigificat idirect effect of trasformatioal leadership perceptio o maagerrated team performace via members satisfactio, a b.697, 9% CI [., 1.459]. The MSEM approach allowed us to examie this upward mediatio model, whereas the MLM framework that aggregates the memberlevel measures to the team level failed to provide support for a idirect effect. Summary Discussio Several recet studies have proposed methods for assessig mediatio effects usig clustered data. These studies have provided methods that target specific kids of clustered data withi the cotext of MLM. However, these MLM methods ow i commo use have three limitatios critically relevat to mediatio aalysis. First, these methods frequetly assume that Withi ad Betwee effects composig the idirect effect are equivalet, a assumptio ofte difficult to justify o substative grouds (e.g., Bauer et al., 6; Krull & MacKio, 1; Pituch & Stapleto, 8), or they separate the Withi ad Betwee effects composig the idirect effect i a maer that risks icurrig otrivial bias for the idirect effect (MacKio, 8; Zhag et al., 9). Secod, other MLM methods ivolve the icoveiet ad ad hoc combiatio of OLS regressio ad ML estimatio iteded to model idirect effects with Level M ad/or Y variables (e.g., Pituch et al., 6). A third limitatio that we have ot emphasized, but that evertheless exists, is that multilevel mediatio models have, to date, each bee preseted i isolatio ad each bee geared toward a sigle desig. Therefore, applied researchers have had difficulty establishig a itegrated uderstadig of how these existig models relate to each other ad how to adapt them to test ew ad differet hypotheses. To address these issues, we proposed a geeral MSEM framework for multilevel mediatio that icludes all available methods for addressig multilevel mediatio as special cases ad that ca be readily exteded to accommodate multilevel mediatio models that have ot yet bee proposed i the literature. Ulike most MSEM approaches described i the literature, the method we use ca accommodate radom slopes, thus permittig structural coefficiets (or fuctio of structural coefficiets) to vary radomly across clusters. To demostrate the flexibility of the MSEM approach, we demostrated its use with three applied examples. We argue that the geeral MSEM approach ca be used i the overwhelmig majority of situatios i Figure 3. Illustratio of mediatio model for 11 data from Example 3. TFL team members perceived trasformatioal leadership; satisfactio team members satisfactio with the team; performace maagerrated team performace.
14 PREACHER, ZYPHUR, AND ZHANG which it is importat to address mediatio hypotheses with ested data. Below, we address issues relevat to the substative iterpretatio of MSEM mediatio models, limitatios of the approach proposed here, specific cocers related to implemetig MSEMs, ad a series of recommeded steps for estimatig multilevel mediatio models with MSEM. Issues Relevat to Iterpretig Multilevel Mediatio Results Because the multilevel mediatio models described here ivolve variables ad costructs that spa two levels of aalysis, it is paramout that researchers do more tha simply cosider the statistical implicatios of multilevel mediatio aalyses usig MSEM. I order to substatively iterpret these models, oe should also cosider the theoretical implicatios of variables ad costructs used at two levels of aalysis via a examiatio of the likages amog levels of measuremet, levels of effect, ad the ature of the costructs i questio. Such likages have bee explored by may authors (e.g., Bliese, ; Cha, 1998; Hofma, ; Klei, Dasereau, & Hall, 1994; Klei & Kozlowski, ; Kozlowski & Klei, ; Rousseau, 1985; va Mierlo, Vermut, & Rutte, 9). These discussios emphasize the importace of establishig the relatioships amog measuremet tools ad costructs at multiple levels of aalysis, how higher level pheomea emerge over time from the dyamic iteractio amog lower level parts, as well as how similar costructs are coceptually related at multiple levels of aalysis. Although a idepth exploratio of the complexity of issues surroudig multilevel processes, costruct validatio, ad theorizig are beyod the scope of the curret paper, it is importat to make the followig ote. As described above, all multilevel mediatio models aside from those for data describe a mediatio effect fuctioig strictly at the betweegroup level of aalysis. Whether the desig suggests topdow effects (e.g., 11; 1), bottomup effects (e.g., 11; 1), or both (e.g., 11; 1), the mediatio effect is iheretly at the betweegroup level of aalysis. Because of this fact, researchers must pay close attetio to the meaig of lower level variables at the betweegroup level of aalysis ad how that meaig is implicated i a particular multilevel mediatio effect. For example, cosider idividual versus collective efficacy. Whereas idividual efficacy origiates at the level of the idividual, collective efficacy emerges upward across levels of aalysis as a fuctio of the dyamic iterplay amog members of a group (Badura, 1997). Therefore, simply aggregatig idividuallevel efficacy will ot effectively capture the collective efficacy of a group. A selfreferetial measure of efficacy, such as I am able to complete my tasks, will assess idividual efficacy both withi ad betwee groups. Although differeces betwee groups o this measure are at the level of the group, the costruct is still iheretly focused o the idividual. I cotrast to this, a groupreferetial measure, such as my group feels capable of completig its tasks, captures differeces withi a group i perceptios of the group s collective level of efficacy (which may be thought of as error variace; Cha, 1998), whereas at the betweegroup level it assesses collective efficacy (Badura, 1997). This simple illustratio is meat to highlight the potetial complexity of multilevel mediatio models ivolvig differet types of lower level variables. Cosider, for example, a mediatio model for 1 data, i which both higher level variables are cotextual i ature ad the lower level variable is idividual efficacy. I this case, latet group stadig o idividual efficacy mediates the relatioship betwee the higher level variables. Alteratively, cosider the same model, where the lower level variable is a measure of collective efficacy. I this case, a fudametally differet form of efficacy collective efficacy mediates the relatioship. Regardless of the specific form of the multilevel mediatio model, researchers should atted to the meaig of a source of variatio rather tha simply the techical aspects of multilevel mediatio poit estimates. Limitatios A limitatio of our approach is oe shared by most other socalled causal models. Hypotheses of mediatio iheretly cocer cojectures about cause ad effect, yet a model is oly a collectio of assertios that may be show to be either cosistet or icosistet with hypotheses of cause. Results from a statistical model aloe are ot grouds for makig causal assertios. However, if (a) the researcher is careful to ivoke theory to decide upo the orderig of variables i the model, (b) competig explaatios for effects are cotrolled or otherwise elimiated, (c) all relevat variables are cosidered i the model, (d) data are collected such that cojectured causes precede their cojectured effects i time, ad/or (e) X is maipulated rather tha observed, claims of causality ca be substatially stregtheed. MSEM more geerally also has some limitatios worth metioig. The geeral MSEM preseted here caot employ the restricted maximum likelihood estimatio algorithm commoly used i traditioal MLM software for obtaiig reasoable variace estimates i very small samples. It also caot easily accommodate desigs with more tha two hierarchical levels. However, may data sets i educatio research ad social sciece research are characterized by multiple levels of estig. Some work has bee devoted to mediatio i threelevel models (Pituch et al., 1, discussed 311, 31, ad desigs). The geeral MSEM could, i theory, be exteded to accommodate additioal levels (by, for example, permittig the ad matrices to vary at Level 3), but the relevat statistical work has ot yet bee udertake. This would be a fruitful aveue for future research. We did ot cosider models that iclude moderatio of the effects cotributig to the Betwee or Withi idirect effects, although such extesios are possible i MSEM. I models ivolvig crosslevel iteractios, it is possible to fid 1 effects moderated by Level1 variables. However, i this case 1 mai effects still may exist oly at the Betwee level. Fially, we are careful to limit our discussio to cases with low samplig ratios. That is, we assume twostage radom samplig, i which both Level uits ad Level1 uits are assumed to have bee radomly sampled from ifiite populatios of such uits. If the samplig ratio is high (i.e., whe the cluster size is fiite ad we select a large proportio of idividuals from each cluster), the maifest group mea may be a good proxy for group stadig (for cotiuous variables) or group compositio (for dichotomous variables such as geder). More discussio of why samplig ratio is importat to cosider i MSEM is provided by Lüdtke et al. (8).
15 MULTILEVEL SEM FOR TESTING MEDIATION 3 Implemetatio Issues To scaffold the implemetatio of the MSEM framework i practice, we ow tackle two issues that a researcher would cofrot whe coductig such a aalysis: sample size cocers ad obtaiig cofidece itervals for the idirect effect. Sample size. Questios remai about the appropriate sample size to use for MSEM. To date, o research has ivestigated the questio of appropriate sample size for Muthé ad Asparouhov s (8) model ad certaily ot i the mediatio cotext. Nevertheless, it is istructive to cosult the literature o sample size requiremets for precursor MSEM methods for guidace. Ivestigatios of appropriate sample size for MSEM address three issues: the recommeded miimum umber of uits at each level, the degree to which balaced cluster sizes matter, ad the dimiishig rate of retur for collectig additioal Level uits. Regardig the umber of uits to collect, Hox ad Maas (1) suggested that at least 1 clusters are required for good performace of Muthé s maximum likelihood (MUML) estimator used with Muthé s twomatrix method, echoig Muthé s (1989) recommedatio that at least 5 to 1 clusters should be sampled. Meulema ad Billiet (9) cocluded that as few as 4 clusters may be required to detect large structural paths at the Betwee level, whereas more tha 1 may be ecessary to detect small effects (for the models they examied). They also foud that havig fewer tha clusters or a complex Betwee model ca dramatically icrease parameter bias. Muthé (1991) ad Meulema ad Billiet suggested that, i geeral, it may be better to observe fewer Level1 uits i favor of observig more Level uits. Cheug ad Au (5) suggested that icreasig Level1 sample size also may ot help with regard to iadmissible parameter estimates i the Betwee model. So, o balace, it would seem that emphasizig the umber of Level uits is particularly importat i MSEM, although this questio has ot bee ivestigated specifically i the cotext of Muthé ad Asparouhov s model, which uses a full ML estimator rather tha MUML. Less is kow regardig balaced cluster sizes. Hox ad Maas (1) suggested that ubalaced cluster sizes may ifluece the performace of MSEM usig the MUML estimator. O the other had, Lüdtke et al. (8) examied ubalaced cluster sizes ad foud that they did ot oticeably affect bias, variability, or coverage whe ML was used. Regardig the utility of collectig additioal Level uits, Yua ad Hayashi (5) suggested that (whe the MUML estimator is used) collectig additioal groups may be couterproductive if they cotai oly a few Level1 uits, although this questio, too, has yet to be addressed for Muthé ad Asparouhov s method. Cofidece itervals. Our focus has bee o model specificatio rather tha o obtaiig cofidece itervals for the idirect effect. It is possible to use the Mplus MODEL CONSTRAINT commad to create fuctios of model parameters ad obtai delta method cofidece itervals, but such itervals do ot accurately reflect the asymmetric ature of the samplig distributio of a idirect effect. This limitatio is problematic, particularly i small samples, but the delta method may be a legitimate approach i large samples. To date, o other procedures have bee proposed for obtaiig CIs for the idirect effect usig MSEM. However, at least three approaches are available for costructig CIs for the idirect effect usig MLM the distributio of the product method, the parametric bootstrap, ad the oparametric bootstrap ad at least oe of them (parametric bootstrap) is geeralizable to the MSEM settig. MacKio, Fritz, Williams, ad Lockwood (7); MacKio, Lockwood, ad Williams (4); ad Williams ad MacKio (8) described the distributio of the product method for obtaiig cofidece limits for idirect effects. This method ivolves usig the aalytically derived distributio of the product of radom variables (i the preset case, the product of structural coefficiets). This method has the advatage of ot assumig symmetry for the samplig distributio of the idirect effect, ad therefore cofidece itervals based o this method ca be appropriately asymmetric. MacKio et al. (7) developed the software program PRODCLIN ow available as a stadaloe executable program ad i versios for SAS, SPSS, ad R to eable researchers to use this method. The parametric bootstrap (Efro & Tibshirai, 1986) uses parameter poit estimates ad their asymptotic variaces ad covariaces to geerate radom draws from the parameter distributios. I each draw, the idirect effect is computed. This procedure is repeated a very large umber of times. The resultig simulatio distributio of the idirect effect is used to obtai a percetile CI aroud the observed idirect effect by idetifyig the values of the distributio defiig the lower ad upper 1(/)% of simulated statistics. I the siglelevel mediatio cotext, this method is termed the Mote Carlo method (MacKio et al., 4). This method was proposed for use i the MLM cotext by Bauer et al. (6); a similar method kow as the empiricalm method was used i the MLM cotext by Pituch et al. (5, 6) ad Pituch ad Stapleto (8). A key feature of the parametric bootstrap is that oly the parameter estimates are assumed to be ormally distributed. No assumptios are made about the distributio of the idirect effect, which typically is ot ormally distributed. Pituch et al. (6) foud, i the MLM cotext, that empiricalm performed early as well as the biascorrected bootstrap, which is quite difficult to implemet i multilevel settigs. Advatages of this procedure are that it (a) yields asymmetric CIs that are faithful to the skewed samplig distributios of idirect effects, (b) does ot require raw data to use, ad (c) is very easy to implemet. Our empirical examples i this article are the first to implemet the parametric bootstrap for the idirect effect i MSEM. To do so, we used Selig ad Preacher s (8) webbased utility to geerate ad ru R code for simulatig the samplig distributio of a idirect effect. The utility requires that the user eter values for the poit estimates of the a ad b paths, their stadard errors (i.e., the square roots of their asymptotic variaces), a cofidece level (e.g., 9%), ad the umber of values to simulate. The oparametric bootstrap, o the other had, is a family of methods ivolvig resamplig the origial data (or Level1 ad Level residuals) with replacemet, fittig the model a large umber of times, ad formig a samplig distributio from the estimated idirect effects. A biascorrected residualbased bootstrap method was implemeted i SAS by Pituch et al. (6) for the coflated model for 11 data. However, as of this writig, there is o software that ca perform both MSEM ad the oparametric bootstrap. Although oly oe of the methods proposed for obtaiig CIs for the idirect effect i MLM has bee used with MSEM (the parametric bootstrap), some methods for obtaiig CI for the idirect effect i siglelevel models could potetially be geeralized to the MSEM settig i future research ad would be worth
16 4 PREACHER, ZYPHUR, AND ZHANG comparig. These iclude the multivariate delta method (Bolle, 1987; Sobel, 1986) ad oparametric bootstrap methods. Coclusio I this article we have advocated a comprehesive MSEM framework for multilevel mediatio for the purposes of (a) bias reductio for the idirect effect, (b) ucoflatig the Betwee ad Withi compoets of the idirect effect, ad (c) straightforward geeralizatio to ewly proposed multilevel mediatio models (Desigs 3, 5, 6, ad 7 i Table 1). Because of the complexities ivolved i MSEM, we coclude with the followig stepbystep guide (buildig o Cheug & Au, 5; Heck, 1; Muthé, 1989, 1994; Peugh, 6) for implemetig the MSEM framework for multilevel mediatio. 1. Idetify the mediatio hypothesis to be ivestigated. The first ad most critical step is to develop a mediatio model that is cosistet with theory ad icludes variables at the appropriate levels of aalysis. If there is at least oe Level variable i the causal chai, the idirect effect of iterest likely is a Betwee idirect effect. If all three variables are assessed at Level 1, it becomes possible to etertai mediatio hypotheses at both levels. As a cautio, whe researchers lack sufficiet theory to posit a appropriate Betwee model (Peugh, 6), they ofte ed up specifyig idetical Withi ad Betwee models without justificatio. This is problematic because the Betwee model almost always will be differet tha the Withi model (Crobach, 1976; Crobach & Webb, 1975; Grice, 1966; Härqvist, 1978; Key & La Voie, 1985; Sirotik, 198).. Esure that there is sufficiet betweecluster variability to support MSEM. If more tha a couple of the variables to be modeled have ICCs below.5, it is likely that covergece will be a problem or that estimatio of the idirect effect will be ustable with potetially large bias. Should this happe, the researcher should cosider group mea ceterig the offedig variables for the sake of stability, as multilevel techiques may ot be profitable or eve possible otherwise. 3. Fit the Withi model. A properly specified withicluster model is ecessary, but ot sufficiet, for a properly specified betweecluster model (Heck, 1; Hofma, ; Hox & Maas, 1; Kozlowski & Klei, ; Muthé & Satorra, 1995; Peugh, 6). Therefore, we recommed that the Withi model be ivestigated first. There are a few ways to do this. A somewhat crude method ivolves group mea ceterig all observed variables ad fittig the Withi model as a siglelevel model. Alteratively, the researcher might fit the full model, permittig the clusterlevel costructs to covary freely. If the latter approach is take, it is sesible to begi with few radom effects ad build i complexity util all theoretically motivated radom effects have bee icluded. 4. Fit the Withi ad Betwee models simultaeously. 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20 8 PREACHER, ZYPHUR, AND ZHANG Appedix A Bias Derivatio I this Appedix we formally derive the expected bias that results from computig a Betwee idirect effect i models for 11 desigs usig group mea ceterig for M ad eterig the group mea of M (M.j )asa additioal predictor of Y (what we term the UMM approach). The developmets i this Appedix parallel those i the Appedix of Lüdtke et al. (8). The followig populatio models will be assumed for X, M, ad Y: X j X U Xj M ij M U Mj R Mij Y ij Y U Yj R Yij, (A1) where the s are populatio meas, the U terms are deviatios from the meas for cluster j, ad the R terms are idividual deviatios from the cluster meas. Because X is a clusterlevel variable i the 11 desig, there is o withicluster deviatio R Xij. We assume the Us ad Rs to be idepedet. The Withi ad Betwee covariace matrices for X, M, ad Y are W B M MY Y X XM M XY MY Y We are iterested i estimatig the parameters of the followig model, (A). (A3) M ij M B U Xj Mj ε Mij Y ij Y W R Mij B U Mj B U Xj Yj ε Yij, (A4) where M ad Y are the coditioal meas of M ad Y, B is the Betwee effect of X o M, B is the Betwee effect of M o Y cotrollig for X, B is the Betwee effect of X o Y cotrollig for M, W is the Withi effect of M o Y, Mj ad Yj are clusterlevel residuals, ad ε Mij ad ε Yij are idividuallevel residuals. Followig the UMM strategy, the followig model is used to estimate the parameters of A4, M ij M M1 X j u Mj e Mij Y ij Y Y1 M ij M.j Y1 M.j Y X j u Yj e Yij, (A5) where M.j is the cluster mea of M, ˆ M1 is the estimator for B, ˆ Y1 estimates B, ˆ Y estimates B, ad ˆ Y1 estimates W. Assumig equal cluster sizes, the observed covariace matrix S of Y ij, M ij M.j, M.j, ad X j ca be represeted i terms of A ad A3 as (Appedices cotiue)
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