Estimating Multilevel Linear Models as Structural Equation Models
Author(s): Daniel J. Bauer
Reviewed work(s):
Source: Journal of Educational and Behavioral Statistics, Vol. 28, No. 2 (Summer, 2003), pp.
135-167
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Journal of Educationaland Behavioral Statistics
Summer2003, Vol. 28, No. 2, pp. 135-167
Estimating Multilevel Linear Models
as Structural Equation Models
Daniel J. Bauer
North Carolina State University
Multilevel linear models (MLMs)provide a powerfulframeworkfor analyzing
data collected at nested or non-nested levels, such as students within classrooms. The current article draws on recent analytical and software advances
to demonstratethat a broad class ofMLMs may be estimatedas structuralequation models (SEMs).Moreover,withinthe SEMapproach it is possible to include
measurementmodelsfor predictorsor outcomes,and to estimatethe mediational
pathwaysamongpredictorsexplicitly,taskswhich are currentlydifficultwith the
conventional approach to multilevel modeling. The equivalency of the SEM
approach with conventional methodsfor estimating MLMs is illustrated using
empiricalexamples,includingan exampleinvolvingbothmultipleindicatorlatent
factorsfor the outcomesand a causal chainfor thepredictors. Thelimitationsof
this approachfor estimatingMLMsare discussedand alternativeapproachesare
considered.
Keywords: factor analysis, hierarchical linear models, mediation, multilevel models,
structuralequationmodels
Multilevel linear models and structuralequation models constitute two of the
dominantanalyticframeworksin contemporarysocial andbehavioralscience.Each
approachoffers distinctadvantages.Multilevel linearmodels (MLMs) generalize
simple regressionmodels to circumstancesin which the assumptionof independent
observationsis compromised,as occurswhen thereis a "nested"or "clustered"data
structure.A classic exampleis the observationof studentswithin schools. Students
withina schoolarelikelyto be moresimilarto one anotherthana simplerandomsample. Priorto the developmentof MLMs,this situationwas handledeitherby estimating an ordinaryleast squaresregressiondespitenonindependence,inflatingType I
errorrates,or by aggregatingthe dataat the highestlevel (i.e., schools),resultingin a
substantialloss of informationandpower(see Bryk& Raudenbush,1992;Goldstein,
1995). MLMs offer the opportunityof analyzingboth levels of the datasimultaneously (i.e., studentand school), therebyaccountingfor the nonindependenceof the
observations,andcan even be used when clustersareoverlapping(i.e., when exam-
This work was funded in part by grants DA06062 and DA13148. I would like to thank Patrick
Curran,Kenneth Bollen, Madeline Carrig, and the members of the Carolina StructuralEquations
Modeling Groupfor theirvaluableinput and assistancethroughoutthis project.
135
Bauer
ining individualswithinbothschoolsandneighborhoods).Further,MLMscapitalize
on this datastructure,allowingthe effects of predictorsto varyover clusters.
Despite these advantages,MLMs have severaltraditionallimitations.First,it is
difficultto incorporatea measurementmodelforthe outcomes;thatis, wheretheoutcomes aremultipleindicatorlatentfactorswith estimatedfactorloadings.Traditionally, a measurementmodel is addedas a thirdlevel of the model (theitem level), but
in this approachthe factor loadings of the items cannotbe estimatedand must be
specified a priori instead(Goldstein, 1995; Raudenbush,Rowan, & Kang, 1991).
A second traditionallimitationof MLMs, sharedwith otherregressionmodels, is
thatcomplexcausalprocesses,includingmediationalpathways,cannotbe modeled
directly.Currentapproachesforestimatingmediationalpathwaysinvolveeitherpostestimationmatrixmanipulationsof estimatesfroma noncausalmodel (Raudenbush
& Sampson, 1999) or the combinationof informationfrom multiplemodels (Krull
& MacKinnon,1999, 2001). Finally,inferentialtests of the overallfit of MLMs are
not routinelyavailable(Raudenbush,2001; but see du Toit & du Toit, in press).
The traditionallimitationsof the multilevellinearmodelaretheprimarystrengths
of structuralequationmodeling (SEM). SEMs are composed of two parts,an estimated measurementmodel relatingthe outcomes to the latentfactorsand a latent
variablemodelthatspecifiesthe causalrelationsamongthe latentfactors.One common structurefor these relationshipsis a mediationalpathway.In additionto obtaining coefficientestimatesfor the model, a key focus of SEM analysesis to assess the
overallfit of the model,for whichan asymptoticchi-squaretest (andnumerousother
fit indices) are available.However, use of SEMs has traditionallybeen limited to
simplerandomsamples.
Given the complementarystrengths and limitations of MLMs and SEMs, a
numberof investigators have worked to synthesize the two models (e.g., Goldstein & MacDonald,1988;Muth6n,1994;Muth6n& Sattora,1995;Rabe-Hesketh,
Skrondal& Pickles, 2002). Most of these approachesrequirean expansionof the
analytic model and highly specialized software.However, Rovine and Molenaar
(1998, 2000, 2001) recentlydemonstratedthatlinearmixed-effectsmodels (including MLMs) could be estimated as conventional SEMs. Bauer & Curran(2002)
extended this approach,showing that even MLMs for unbalanceddata could be
estimated as SEMs. This articleprovides an expandeddemonstrationof this approachto multilevel modeling. The first aim is to illustrate,both analyticallyand
with empirical examples, how MLMs for unbalanceddata can be estimated as
SEMs. Two approachesfor handlingunbalanceddataarediscussedandcontrasted.
The second aim of the article is to demonstratehow this approachmay be used
to integrate the strengthsof the two modeling frameworks. Specifically, a new
approachfor estimating multilevel structuralequationmodels will be illustrated
throughan empiricalexamplethatincludesboth a measurementmodel for the outcomes and a causal chain among the upper-levelpredictors.
The articlebegins by introducingthe structuralequationmodeling framework.
The multilevel linearmodel is then presentedas a linearmixed-effects model. Initially, severalimportantsimplifyingassumptionswill be made in orderto facilitate
136
EstimatingMultilevelLinearModels as StructuralEquationModels
translationof the multilevelmodel to an SEM. Specifically,no upper-levelcovariates will be included in the model, and the data will be assumed to be balanced.
However, these assumptionswill be progressivelyrelaxed, first to accommodate
unbalanceddata,then to accommodateupper-levelcovariates.I will then demonstratehow MLMs can be extendedwithinthe SEM frameworkto include multiple
indicator latent factors and complex causal structures.Finally, this approachto
multilevel SEM will be contrastedwith two alternativemethods.Empiricalexamples will be used throughoutto illustratethe majorpoints.
Structural Equation Modeling
Structuralequationmodels consist of a linear system of equationsof two basic
types. The firstset of equationsdescribesthe relationsbetweenthe latentvariables,
with a distinction made between endogenous latent variables which are pre1q,
dicted by other variables in the model, and exogenous latent variables g, which
are external predictors whose own causes are unmodeled. Using Jkreskog and
S~rbom's (1993) LISRELnotation,the latentvariablemodel is
Sl = ot + Blq + Ft + ?
(1)
where acis a vector of interceptterms for the equations,B is the matrixof co-efficients giving the impactof the endogenouslatentvariableson each other,F is the
coefficient matrixgiving the effects of the exogenous latentvariableson the endoof'q withcovariancematrix
genouslatentvariables,and? is thevectorof disturbances
'W.Themeanvectorandcovariancematrixof the exogenouslatentvariablesareconventionallydesignatedby Kand4, respectively.It is assumedthatthe expectedvalue
of ? is zero,thatg andCareuncorrelated,andthat(I - B) is nonsingular.
The second set of equationsdefines the measurementmodel for the latent factors. It is here thatthe observedvariablesof the model arerelatedto the latentvariables they are thoughtto represent.The set of observedvariablesrepresentingthe
exogenous latent variables is designated as x and is distinguished from the set
of observedvariablesrepresentingthe endogenouslatentvariables,which aredesignated as y. The measurementmodel then consists of two equationsof the same
basic form, one regressingx on g and anotherregressingy on lq:
x = vx + Ax + 8
(2)
y = v, + Ayq + E,
(3)
where Ax is the factor loading matrixrelatingx to g and Ay is the factor loading
matrixrelatingy to 1q.The interceptsof x andy are containedin vx and vy, respectively. Finally, the vectors6 and E representthe residualsof x and y, with covariance matricesOnand 0,, respectively.It is assumedthatthe expected values of 8
and e are zero and that8, E, ( and S are uncorrelatedwith one another.
137
Bauer
Severaladditionalfeaturesof the SEM are noteworthy.First,the generalmodel
reducesto manyotherfamiliarmodelingapproaches.Forexample,if only observed
variablesare presentin the model, then x and y can be substitutedfor g and ?qin
Equation1 andthe model reducesto a simultaneousequationmodel (or traditional
pathanalysis).Alternatively,if thereis a measurementmodelbutno specificcausal
structurefor the latentvariables(only correlations)then the model reducesto confirmatoryfactor analysis (CFA). Second, SEM is often referredto as covariance
structureanalysisbecause the model equationsimply a specific form for the population covariancematrixI and mean vector Fp.The CFA model will be important
for subsequentdevelopments,so it providesa useful example.If the CFA is modeled on the "y-side"thenthe model-impliedcovariancematrixandmeanvectorare
1(0) = A IPA-'+ 0
+ Ayo,
=L(O)= v
(4)
where 0 is the vector of model parametersfrom Ay,IW,0,, vy, and a.
A key objectiveof structuralequationmodelingis to assess the fit of the hypothesized moment structure.The null hypothesisis that
x(0)=
(5)
0 = ,
O()
that is, that the model perfectly reproducesthe populationvariances,covariances
and means of the observedvariables.This hypothesisis tested by of a likelihoodratioX2test, involving the estimationand comparisonof two nested models. The
firstis the structuralequationmodel of interest.The second is the saturatedmodel
whereno restrictionsareplacedon the estimatedcovariancematrixor meanvector.
The saturatedmodel providesa benchmarkof perfectfit for assessing the degreeof
misfit of the structuralequationmodel. Formally,using LLoto representthe loglikelihoodof the structuralmodel andLLIto representthe log-likelihoodof the saturated model, -2(LLo - LL1)is asymptotically distributedas a chi-square with
degrees of freedomequal to the differencein the numberof parametersfor the two
models. Small and nonsignificantchi-squaressuggest thatthe structuralequation
model fits well (i.e., cannotbe rejected).'
I now turnto themultilevellinearmodelandhow it maybe translatedintoan SEM.
A Simple Multilevel Linear Model
Considerthe special case of a two-level linearmodel thatincludes only Level 1
covariates.The Level 1 model may be writtenas
P
Yij=
138
+
Toj
E pj
p=l
p
+ rIj.
(6)
EstimatingMultilevelLinearModels as StructuralEquationModels
For ease of presentationassume that i indexes individualandj indexes group so
thatyj may be interpretedas the responseof individuali nestedwithingroupj. This
response is modeled as a function of a randominterceptterm (i0j), and P Level 1
covariateswith randomslopes (itj,
, pj). The residualsof yij,denoted rij,are
.....
assumedto be multivariatenormally
distributedas
r~ - MVN (0, 1•, ).
(7)
Often ,rjis assumedto be diagonal(reflectingconditionalindependence),andthe
residualvariancesare frequentlyconstrainedto be equal. These constraintson
Irj
are sometimes necessaryto estimatethe model while at other times they are not.
Given the absence of Level 2 covariates,the Level 2 model is simply
Ctj0
ntpj
=
=
P0 +
Oj
(8)
P,
+ Upj,
whereP0representsthe averageinterceptand p, representsthe averageeffect of x,,
while group-level deviations from these averages are representedby the disturbances uojand upj.These disturbancesare conventionally assumed to be multivariatenormallydistributedas
uj - MVN (0, T),
(9)
where T is typically an unrestrictedcovariancematrix.
Equation8 may then be substitutedinto Equation7 to obtainthe reduced-form
equationfor the model
P
Yij = (1o + Uo0) +
(PP +
p=1
pj) Xpij + ri.
(10)
By expandingand rearrangingthe terms,this equationmay be rewrittenas
Yij
=
3o
+
p=1
+
pXpJ+ Uoj
p=1
upxpYh+';j .
(11)
Rewritingthe model in this way segregatesthe fixed effects of the model (the first
bracketedterm)fromtherandomeffects (thesecondbracketedterm).Note thatwhile
the fixed effects areconstantover schools, the randomeffects aresubscriptedbyj to
indicatethatthey vary over schools. This illustratesthatthe model is a special case
of the generallinearmixed-effectsmodel given by Laird& Ware(1982) as
yj = X,J + Zjuj + r1,
(12)
139
Bauer
where yj is the response vector for groupj, Xj is the design matrix for the fixed
effects 0, Zjis the design matrixof the randomeffects uj,andrjis a vectorof residuals. Assuming the randomeffects and residuals are multivariatenormally distributed,the implied marginaldistributionfor yj is then
yj - MVN (Xj1, ZjTZ;" +
~r
).
(13)
As Verbeke& Molenberghs(2000) noted,it is often this marginalmodel for yjthat
is actuallyfit to the data.
A second reflection on Equation 11 is that the fixed and randomparts of the
model are strictly parallel.This parallelismfollows from the absence of Level 2
covariates.Importantly,because of this simplification,the design matricesfor the
fixed and randomeffects are equivalentand Xj may be substitutedfor Zj to yield
J + ,). r
y, - MVN (XjiP,X, TX
(14)
If the design is balanced(i.e., Xj = X for allj) thenthe marginalmodel may be further simplifiedto
y, - MVN (XP,,XTX' + Yr).
(15)
What is especially interestingabout Equation 15 is that it has the same basic
structureas the CFA model. Specifically, the LISRELmatricesin Equation4 can
be mappedonto the multilevel notationas follows
Ay=
X
v, = 0
y=
T
The key aspect of this translationis that the design matrixof the Level 1 covariates X is representedby the factorloading matrixAy. Thatis, the loadings for the
latent factors are fixed to the values of the covariates in X.2 In turn,the random
regression coefficients rrof the MLM are representedas latent factors 1qin the
SEM, with factor means equal to the fixed effects, and with factor variancesand
covariancesequal to the variancesand covariancesof the randomeffects.
An apparentlimitationof this translationis thatX can only be representedas
Ay
by assuminga balanceddesign.Use of this approachfor estimatingMLMsas SEMs
has thus been restricted to certain special cases, such as growth curve models
140
EstimatingMultilevelLinearModels as StructuralEquationModels
(Chou, Bentler, & Pentz 1998; Meredith& Tisak, 1984, 1990; MacCallumet al.,
1997; Willett & Sayer, 1994) and models for dyadic data(Newsom, 2002), where
balanced designs are common. As a startingpoint, this approachto estimating
MLMs with balanceddata as SEMs is illustratedbelow. Subsequently,I discuss
how unbalanceddesigns can be accommodatedwithin the SEM.
Fitting MLMs with Balanced Data as SEMs
Considera studyin which the languageproficiencyof 3 male and 3 female students was assessed in each of N schools. The primaryinterestis whetherthere is
a sex-difference in language ability, and whether this sex difference varies significantlyover schools. Sex has been dummycoded so thatmale = 0 and female =
1. The MLM then has two levels, a studentlevel and a school level. The Level 1
(student-level)model is writtenas
lang11=
+ Etjsexij+ rij,
t0oj
(16)
with constantresidualvariancea. The Level 2 (school-level) model is:
7t0j =
tElj
P + u0j
(17)
= p, + ulj,
where
T =T 00
.
(18)
The same model can be parameterizedas an SEM. Assume thatthe datavector
for each school is orderedso that the three male students' scores always precede
the threefemale students'scores. Then the measurementmodel may be writtenas
1
lang,,,
rmlY
1 0
langm2
langm3j
0
=
10
rm2j
o
j+
langflj
rm3,
(19)
1|7i
langf2
1 1
langf3J
11
1
)xl
rflij
rf2j
rf3j
where for convenience the subscriptson the variables(andtheirdisturbances)designatethe sex of the student(m for male andffor female) as well as the orderof the
observationswithin sex (1, 2, or 3). The design matrixfor the fixed and random
141
Bauer
effects X is representedby Ay, which is composed of a column vector of ones for
the interceptparameterand a column vector containingthe dummy coded values
of the sex covariatefor the slope parameter.
The latentvariablemodel for this MLM can then be written
=oj
1cj
oj +
(20)
uo
Ulj
where 0O
and pi are capturedas latentvariablemeans (a), and the randomcoefficients uojand uv1arerepresentedas disturbancesfromthose means (c). The covariance matrixof the latentfactorsis equivalentto the covariancematrixof the random
effects from the multilevelmodel, or
S= T =
110
Tll1
(21)
.
A path diagram showing the SEM representationof this model is presented in
Figure 1.
It is importantto recognize thatthe orderof the observationswithin sex is arbitrary.Thatis, althoughthe scoresof studentsm1, m2, andm3 arerepresentedas dif-
rml
Langm
rm2
rm3
Langm2
Langm3
1
0
101
1
0
(Po)
100
fl
rf2
rf3
angf
Langf2
angf3
71
(01)
T11
FIGURE 1. A path diagram of a multilevel linear structural equation model for
balanced data with a categorical Level 2 covariate.
142
EstimatingMultilevelLinearModels as StructuralEquationModels
ferent observed variables in the SEM, there is nothing in the model that distinguishes one male studentfrom anotherwithin a given school; they are viewed as
interchangeable.The arbitraryorderingof the observationsmust be accountedfor
by employingequalityconstraintson the elementsof the residualcovariancematrix
(this is trueof the multilevellinearmodel as well). In this case, constantvarianceis
assumedso that
0, =
r, = al.
(22)
Such constraintsensurethatthe model-impliedvariances,covariances,and means
are identicalover studentsof a given sex, so that the orderingof the observations
has no impacton the parameterestimates, standarderrorsor fit of the model. This
can be seen in the model-impliedcovariancematrixandmeanvectorfor this example, given in Table 1.
The arbitraryorderingof the observationsalso requiressome modificationof the
usual X2 test of overallmodel fit typicallyused with SEMs. Specifically,the appropriatesaturatedmodelforjudgingthe fit of thehypothesizedmodelmustalso be estimatedwith equalityconstraints,again following the notionthat studentsof a given
sex areessentiallyinterchangeable.
saturatedmodel
Forthisexample,theappropriate
is given in the lower half of Table 1, where elementsdesignatedby the same letter
are constrainedto be equal. The lettera is the variancein languagescores among
males, and b is the covarianceamongmale studentswithin schools (accountingfor
the nonindependenceof the observations).Similarinterpretations
may be given to d
ande for femalestudents.The letterc thencapturesthe covariancebetweenmale and
female studentswithinschools andfand g arethe meansof the male andfemale students, respectively.I will refer to this model as the constrainedsaturatedmodel.
Comparingthe model-impliedmoment matricesbetween the hypothesizedmodel
andthe constrainedsaturatedmodel shows thatthe two models areformallynested.
To demonstratethe equivalencyof the two approachesto fittingthe model,I reanalyze datapreviouslypresentedby SnijdersandBosker(1999, p. 46) on the language
proficiencyof 8th-grade(about 11 year-old) students.Between 4 and 35 students
were sampledfromeach of 131 elementaryschools in the Netherlands.To meet the
constraintsof a balanceddesign, only schools with languagescores on at least three
male andthreefemalestudentswereincludedin the analysis(N= 116 schools);when
more male or female studentswere available,three scores were randomlyselected
(bringingthe totalnumberof studentsto 696). Languagescoresrangedfrom 8 to 58,
andshowedlittle skew or kurtosis.This model, andthosethatwill be presentedsubsequently,was fitusingPROCMIXEDin SAS, a standardsoftwarepackagefor conductingmultilevelanalyses,and Mplus, a commonlyused SEM softwarepackage,
in both cases using the full-informationmaximumlikelihoodfittingfunction.3
of Gelman,Pasarica,& Dodhia(2002),
Figure2, drawnaftertherecommendations
the
estimates
and
standard
errorsobtainedfromthe two modelcompares parameter
ing approachesfor each of three estimated models. The two left-most entries in
each panelcorrespondto the estimatesfor the currentmodel with balanceddata(the
143
)I-r
TABLE 1
The CovarianceMatrixand Mean VectorImpliedby the MultilevelLinearModel and the Constrained
for the Balanced Design Example
CovarianceMatrixand Mean Vector Impliedby the M
Variance/covariance
lang,,oo
langm2
langm3
langfl
langf2
M
langf3
langmi
langm2
langm3
langfl
+ (Y
Too
00T
+ +o
ooio+
0oo Tlo
too+ 410
PO0
0a+
0oo
Toooo00
+
~%o
+
"oo+
ooo
0oo-o 1
o0o
+
oo+ o
oo +
00+
T10
oo+ 210 + til + a
oo + 20 + 11
+ 2t10+
0ooPo+
o
'11
Pi1
CovarianceMatrixand Mean Vector for the Constra
Variance/covariance
,,
langm
langm2
.b
langm3
b
M
_
a
C
langfl
lang,
lang,3
i
b
C
ad
C
C
f
f
C
f
Note. Elementsdesignatedby the same symbols or letters are constrainedto be equal. Symbols defined in text.
Estimating Multilevel Linear Models as Structural Equation Models
42
3C
OPROCMIXEDEstimates
0 MplusEstimates
41
S
E
25
39
4)15.
S38.
37
D
10
1
Balanced- Sex
Unbalanced- Sex
5
Unbalanced- IQ
Balanced- Sex
Unbalanced- Sex
Unbalanced- IQ
Balanced- Sex
Unbalanced- Sex
Unbalanced- IQ
5-20
4 -15
UJ
10
C4C-4
+1 3
+1
E
-.
L
Uw
1
0C..
Balanced- Sex
.
..
Unbalanced- Sex
..
.
Unbalanced- IQ
-5
-10
S750
0-1010
a
- 15
-SexUnbalanced
-SexUnbalanced
-10
-SexUnbalanced
-SexUnbalanced
-IQ Balanced
Balanced
Balanced- Sex
Unbalanced- Sex
Unbalanced- IQ
Balanced- Sex
Unbalanced- Sex
Unbalanced- Q10
FIGURE 2. A comparison of the parameter estimates (? 2 standard errors) obtained
byfitting threedifferentmultilevellinear models to the language data of Snijders& Bosker
(1999) using the SEM software Mplus and the multilevel software SAS PROC MIXED.
Thefirst model includes the categorical covariate sex and was restrictedto balanced data
by selecting three male and threefemale studentsper school, the second model includes
the categorical covariate sex and was fit to thefull sample of students using the missing
data approach to unbalanced data, and the third model includes the continuous covariate verbal IQ and was fit to thefull sample of students using school-varyingfactor loadings to accommodate the unbalanceddata.
remainingestimateswill be discussed shortly).As can be seen, the parameterestimates and standarderrorsarevirtuallyidenticalover the two modelingapproaches.4
In bothcases, thereis a smallbut significantsex differencein languagescoresfavoring female studentsby an averageof about3 points,andno significantschool-level
variabilityin this effect. The log-likelihoodsobtainedby the two approachesarealso
equivalentwithinroundingerror.To test the overallfit of the model to the data,this
valuewas comparedto thelog-likelihoodof the constrainedsaturatedmodel,yielding
145
Bauer
X2(1) = 1.18;p = .72. Thus we may conclude that the hypothesizedmodel fits the
observeddataquitewell. Note thatsuch a test of model fit is routinein SEM analyses, but atypicalusing traditionalapproachesto multilevelmodeling.
To summarizeto this point, when the design is balancedan MLM can be represented as an SEM quite easily. The example providedabove simply generalizesa
relationshipbetweenthe two modelsthatis well-understoodin certaincontexts,such
as growthcurvemodelingandthe modelingof dyadicdata,wherebalanceddesigns
arecommon. The primarycomplexity involved in estimatingthe MLM as an SEM
when the design is balancedis thatthe chi-squaretest of model fit mustbe modified
if the orderingof the observationsis arbitrary.Whenthe designis unbalanced,additionalcomplexitiesarise,butgiven recentanalyticalandsoftwaredevelopmentsthey
areno longeran impedimentto parameterizingMLMs as SEMs, as I now illustrate.
Fitting MLMs with Unbalanced Data as SEMs
Recall thatthe primaryreasonthat a balanceddesign was requiredto make the
translationfrom the MLM to the SEM was so thatXj could be simplified to X in
Equation 14. Given this, X could easily be incorporatedin the SEM in the factor
loading matrixAy. In the case that the data are unbalanced,however, this simplification cannot be made. How then, can the model be fit as an SEM? Mehta &
West (2000) consideredthis issue for growthmodels with unbalanceddata.They
reviewed two approachesfor handling unbalanceddata in the SEM framework,
both of which are generalizableto a wide variety of MLMs. The first approachis
to act as thoughthe design is balanced,but that certainobservationsare missing.
A full-informationcase-basedlikelihood is then used to estimatethe model using
only the availableobservations.This approachis availablein most SEM software
but can be cumbersometo implement.A second and more naturalapproachis to
accommodateXj directly by allowing the factor loading matrix to vary over
Ay
schools. Again, a case-based likelihood function is evaluated,where the modelimplied means and covariancesare updatedfor each case given their specific Ay.5
Both approacheshave distinct advantages and disadvantages,so I demonstrate
each in turnusing empiricalexamples.
Treating UnbalancedData as Missing
In the precedinganalysis,3 male and 3 female studentswere sampledfrom each
school so thatthe design would be balanced.However,in actuality,between 2 and
18 male studentsand 0 and 19 female studentswere availablefor analysisover the
131 schools. This situationis easily handledin the multilevelmodelingframework
wherethe marginalmodel for the outcomesnaturallyvariesover schools per Equation 14. The multilevel representation of the model is thus unchanged from
Equations16 and 17. By contrast,the representationof the model as an SEM must
be changedto accommodatethe unbalanceddata.
To fit the modelto the unbalanceddata,I firstact as thoughthe designis balanced
andthat 18 male and 19 female studentswere sampledfrom each school (the maximum numberfromeach sex). The totalnumberof possible observationsfromeach
146
EstimatingMultilevelLinearModels as StructuralEquationModels
school is then 37. Second, if a school has fewer than 18 males or 19 females, I code
the remainingobservationsas missing. Again placing the male language scores
before the female languagescores, the measurementmodel can be modifiedto
SlangmI,
1 O)
0l
rmj
langm2,
1 0
rm2z
langml8,
1
0
1 1
langflj
(
•noj++
(23)
rflj
lj
1 1
langf19
rml8j
rf2j
andthe latentvariablemodel remainsas before.I continueto assumeconstantvariance for the residualsand a full and freely estimatedcovariancematrixfor the random effects (latentfactors).A path diagramof the model is providedin Figure 3.
Note thatby settingup the model this way, a single common to all schools is
Ay
sufficient.The model is then fit by constructinga single 1(0), with 37 rows and
columns,anda single Pi(0),with 37 rows.As the likelihoodfunctionis evaluatedfor
each case j, rows and columns are droppedfrom these matricesto leave only the
rml
rm2
Langm, Langm2...
toO
01
1
rfl9
Langf2 ... Langfl9
Langm18s Langf,
1)1 1
(PO)
f2
rfl
rml8
1
1
71
/
FIGURE 3. A path diagram of a multilevel linear structuralequation model with a categorical Level 2 covariate where unbalanced data are treated as missing.
147
Bauer
elements of the covariancematrixand mean vectorthatcorrespondto the available
observationsforthe case (Arbuckle,1996;Wothke,2000). Thus,althoughA, is constant,the implied marginaldistributionfor the observations[summarizedby i(0)
and Pi(0)]is neverthelessupdatedcase-wiseto accommodatetheunbalanceddesign.
The SEM approachand multilevelapproacharehence analyticallyequivalent.
To demonstratethis point, the model was once again fit to the data of Snijders
and Bosker (1999), but this time includingthe language scores of all the students
(a total of 2,287 students).The results of fitting the model as an MLM and as an
SEM are presentedin the center of each panel of Figure 2. Note that the parameter estimates,standarderrors,andlog-likelihoodsareagainessentiallyequivalent
(within roundingerror).In comparisonto the priormodel, the standarderrorsfor
this model are much smaller, reflecting the largersample of studentsincluded in
the analysis. To test the fit of the model, the constrainedsaturatedmodel was also
fit to the data,andthe resultinglikelihood-ratiotest again indicatedgood fit for the
hypothesized model [x2(1) = 2.41; p = .12].
Therearetwo primaryadvantagesto this strategyfor handlingunbalanceddata.
First, a test of overall model fit is possible because a single (0O)and i(0O)can be
used to summarizethe data. Second, it is easy to allow the residual variances of
the observed variablesto differ over values of the Level 1 covariate(s)(e.g., constrainingthe residual varianceto different values for male and female students).
The primarydrawbackof this strategyis thatthere are many models for which a
single (0O)and ji(0) cannotbe constructed(at least not without substantialdifficulty). For instance,suppose thatthe covariateof interestwas verbalintelligence.
Unlike sex, verbalintelligence is a continuouscovariatethatcan, in principle,take
on an infinitenumberof differentvalues. In this case, it would be difficultto treat
the design as balanced,and the numberof "missing"values would far exceed the
numberof observed values, increasingthe computationaldifficulty of estimating
the model. What is needed in such cases is the ability to allow the factor loading
matrix to vary over the Level 2 units. This constitutes the second strategy for
accommodatingunbalanceddata in SEM, which I now demonstrate.
TreatingUnbalancedData with Case-VaryingFactor Loadings
Allowing the factorloadingmatrixto takeon differentfixed values for each case
j directlyparallelsthe multilevel approachto unbalanceddata (Neale et al., 1999).
That is, Xj is representedas Ayj.To demonstratethis approach,I extend the language proficiencyexample, this time using verbalintelligence (IQ) as the Level 1
covariate.The MLM consists of the Level 1 (student-level)model
langi =
JtOj
+ 7tljlQi + r,,
(24)
where ~, =
The Level 2 (school-level) model is identical in form to Equations 17 ando•.
18, only this time the coefficients reflect the regressionof language
ability on IQ (not sex).
This model can be representedequivalentlyas an SEM with the following measurementmodel
148
EstimatingMultilevelLinearModels as StructuralEquationModels
lang
1
IQj
r(j
+IQ2j
lang;,
lang2.
l
1
2j
j(25)
IQII/\r
where I representsthe total numberof students in school j, and the values IQij,
IQ2j, ... IQj in Ay are updated as the likelihood function is evaluated for each
schoolj. Note thatbecauseAyis permittedto vary overj, thereis no particularneed
to sortthe observations,unlikethe missingdataapproach.Consistentwiththe multilevel model, the covariancematrixof the residualsis constrainedto 0, = C1= oI.
The latentvariablemodel is of the same form as Equations20 and 21. A path diagramfor this model is presentedin Figure4.
This modelwas againfit to the dataof Snijders& Bosker(1999), including2,287
studentsfrom 131 schools.To makethe resultsof theanalysismoreinterpretable,
the
verbalIQ scores(obtainedfromthe ISI test) were standardizedpriorto the analysis.
r2
rl
Lang
Lang2
"2
r3
ri
Lang3
Langi
Q3j
Ij
Too
.11
FIGURE4. A path diagram ofa multilevel linear structuralequation model with a continuous Level 2 covariate accommodated by allowing the factor loading matrix to vary
over Level 2 units.
149
Bauer
The estimatesandstandarderrorsobtainedby fittingthe modelas an MLMandas an
SEM aredisplayedin Figure2 as the right-mostentriesof each panel.The parameter
estimatesareagainessentiallyidentical(as arethe log-likelihoodsof themodels),and
the standarderrorsdiffer only slightly.Both models indicatethat,on average,each
standarddeviationincreasein a student'sIQ is associatedwith a 2.5 unitincreasein
languagescores.Thoughnot readilyapparentfromFigure2, significantschool-level
differenceswere also detectedin the size of thiseffect.Finally,a basic comparisonof
theresidualvarianceestimatesfromeachof thefittedmodelssuggeststhatIQexplains
much moreof the individualvariabilityin languagescoresthansex. Note, however,
thatit is not possible to formallytest the fit of the IQ model using the conventional
likelihood-ratiostatisticbecausethismodelcannotbe summarizedby a singlecovariancematrixandmeanvector(Raudenbush,2001). This constitutestheprimarydrawback of using case-varyingfactorloadingsto accommodateunbalanceddatarather
thanthe missingdataapproachwhen the latteris feasible.
The issue of how to treat unbalanceddata when fitting multilevel models in
SEM is more thananythinga datamanagementproblem.As I have shown, recent
developmentshave made it possible to fit MLMs with categoricalandcontinuous
Level 1 covariatesas SEMs, even when the data is unbalanced.What I have not
yet addressedis the ability to incorporateupper-level predictorsof the random
coefficients in the model. It is to this topic thatI now turn.
Adding Upper-Level Predictors to the Model
Upper-level predictorswere initially excluded from the model to simplify the
problem of translatingthe MLM to an SEM. Specifically, the absence of upperlevel predictorsimplied that the design matrix for the fixed and randomeffects
would be identical (Xj = Zj) in Equation 12. This design matrixwas then easily
translatedinto the factorloadingmatrixof an SEM, allowing the fixed andrandom
effects to be capturedas factormeans and disturbances.The problemwith adding
Level 2 covariatesto the model is that this simplificationcan no longer be made.
The fixed effects will out-numberthe random effects, and the design matrix Xj
will thus requiremore columns than the design matrixZj. There are two analytically equivalent solutions to this problem.The first was formulatedby Rovine &
Molenaar(1998, 2000, 2001), andfollows the linearmixed-effectsmodel most literally. The second extends the approachused with conditionallatent curve models where upper-levelpredictorsare incorporateddirectly as observed variables.
These alternativesare contrastedin the next section.
TheLatent VariableSolution
Rovine & Molenaar(1998, 2000, 2001) parameterizedthe linearmixed-effects
model in Equation12 as an SEM by definingtwo types of latentvariables,one to
representthe fixed effects and anotherto representthe randomeffects. Using the
"y-side"of the SEM for consistency, I designatethe "fixed-effectsfactors"as 1l,
and the "random-effectsfactors"as 'r2,where the complete vector of endogenous
latent variables is then 'l' = l' I lq2. The means for mIare freely estimatedbut
150
EstimatingMultilevelLinearModels as StructuralEquationModels
their variancesand covariancesare constrainedto zero. In contrast,the means of
areconstrainedto zero but theirvariancesandcovariancesarefreely estimated.
'12
The
design matricesfor the fixed andrandomeffects can thenbe representedsimul= I1Z.6The means of 1qIare then
taneously in the factor loading matrix as
Ay,, Xj
equivalentto 0Iandthe factorvariancesandcovariancesof areequalto T. Note
q12
thatcross-level interactionscan be representedin 1qwith corresponding
columns
in Xj equal to the productof the Level 1 and Level 2 covariates.
To demonstratethis technique,considerthe simpleexamplegiven earlierof three
maleandfemalestudentssampledperschool,butsupposenow thatclass size hasbeen
addedas a Level 2 predictorof the sex effect (t1lj).The reduced-formmodel is then
Yij = [IP + Plosexij + 0ll(sexij * sizej)] + [u0j + uljsexij] + rj,
(26)
wherethe fixed andrandomeffects aresegregatedin the bracketedterms.The measurementmodel of the equivalentSEM can then be written
0
0
0
1
1 0
0
1 0
langm3j
1 0
0
1 0
langf j
1 1 sizej
1 1
sizej
1 s1
izej
langmj
langm2j
langfz2
langf3j
1
rl
rml•
rm2
T12
3
-
1 1
1 1 Ti4
1 1 T\5
+
rm(27)
,
(27)
rf
rf2
rf3
where the factorloading matrixhas been partitionedinto XjIZjandthe latentvariable vector has been partitionedinto 'I
representingthe fixed and random
I•y1'2,
of the factorloadingmatrixis speceffects, respectively.Note thatthe thirdcolumn
ified to capturethe cross-level interactionof class size with sex in the prediction
of languageability. Due to the inclusion of this effect, the model would have to be
estimatedusing a case-varyingfactorloading matrix.
The mean vector and covariancematrixof the latentvariablesare then
oa'= (
Pio
P0 P11
(28)
0)
0
00
W =0
0
O 0
0
0
O0
0 "10
lo T
,
(29)
00oo
illustratingthatthe latentfactorsin a1,capturethe fixed effects (withnonzeromeans
but zerovariances)andthe factorsin rq2capturetherandomeffects (withzeromeans
but nonzerovariancesand covariances).
151
The ObservedVariableSolution
The observed variable solution to the problemgeneralizes the approachused to
incorporatepredictorsin latentgrowthcurve models. Specifically,Level 2 predictors are incorporatedas exogenous observedvariables(in contrastto drawingthe
values of the predictorsinto the factor loading matrix).These observed variables
areassumedto be fixed andmeasuredwithouterror,so they may be substitutedfor
g in the latent variablemodel (Equation 1). The directeffects of the Level 2 predictors on vi(representingthe randominterceptsand slopes) are then capturedas
regressioncoefficients in the r matrix.As Curran,Bauer & Willoughby (now in
of themodel,the cross-levelinterpress)havepointedout,withthisparameterization
action of the standardmultilevellinearmodel is computedas the indirecteffect of
the predictoron the outcomevia the latentfactors,or the productof F andAj. Note
that with this approachthe randomcoefficients can still be representedas latent
factorswith both meansandvariances(i.e., no partitioningof iq is required).
To illustratethis approach,considerthe same model as above, where class size
is added as a Level 2 predictor of the sex effect on language ability. Using the
observedvariablesapproach,the measurementmodel can be left unchangedfrom
Equation 19. To estimate the model, the latent variablemodel in Equation20 is
simply modifiedto
=j Pi
O+Q
0
(sizej)+
uj.
(30)
Note thatclass size is representeddirectlyas an observedpredictorvariablerather
than via the factorloadings. This allows the cross-level interactionto be captured
as a coefficient in the F matrixratherthanas a latentvariablemean. The Wlmatrix
remainsunchangedfrom Equation21.
Comparingthe TwoApproaches
While trueto the formof the linearmixed-effectsmodel, therearethreeprimary
drawbacksto the latent variable approachof Rovine & Molenaar (1998, 2000,
2001). First,representingfixed andrandomeffects as separatesets of latentfactors
is nonintuitiveand can greatlyexpandthe size of the model. Second, the values of
the upper-levelpredictors(like the lower-levelpredictors)mustbe incorporatedinto
the factorloadingmatrix,increasingthe difficultyof parameterizingthe model and
potentiallyslowing the evaluationof the likelihoodfunction.This may also makeit
difficultto conducttests of overallmodel fit (i.e., use the missing dataapproachfor
unbalanceddata). Third,by representingthe upper-levelcovariates in the factor
loading matrix,there is little possibility to elaboratethe model for the predictors.
The observedvariablesolutionresolves these problems.It providesa parsimonious
andintuitiverepresentationof the model.Moreover,becausethe upper-levelcovariates are treatedas observedvariablesit is easy to extend the model by includinga
measurementmodel for the predictorsor mediationalpathwaysbetweenpredictors.
152
EstimatingMultilevelLinearModels as StructuralEquationModels
Both approachesarereadilygeneralizableto three-levelmodels andbeyond,though
the parameterization
of the model becomes increasinglycomplex.
In the following section I presentan empiricalexample of a three-level model
that includes a mediationalmodel for the predictorsat the highest level using the
observed variablesapproach.This example provides a vehicle for demonstrating
some of the advantagesof estimatingMLMs as SEMs.
Extending the Model
To this pointI have consideredhow MLMsmay be parameterizedandestimated
as SEMs even when the dataareunbalanced.The only advantageI have shown for
doing so is thatundercertaincircumstancesan inferentialtest of overall model fit
can be obtained.However, the primarymotivationfor estimatingMLMs as SEMs
is to take full advantageof the flexibility the SEM framework,includingthe ability to estimatemeasurementmodels for the outcomes or predictorsand the ability
to estimatethe pathsin a causal chain directly.I now consideran empiricalexample thatinvolves the estimationof a threelevel model wherethe firstlevel is a factor model for psychometricscale data.I follow this by addinga mediationalmodel
for the upper-levelpredictorsof the latentfactors.
MultilevelFactor Analysis
The datais drawnfromthe High School andBeyond AdministratorandTeacher
Survey: TeacherQuestionnaireconductedin 1984 and includes 10,365 teachers
from 456 schools. I centermy analysis on the TeacherControlscale, as described
by Pallas (1988), which includes 9 items, reproduced in Table 2, which were
designed to assess teacherperceptionsof controlof school policy (4 items) andthe
classroom (5 items). All items were ratedon six-point Likert scales, and for the
purposes of this analysis were grand-meancentered at zero. The data is highly
unbalancedat the teacherlevel: The numberof teacherssampledper school ranged
from 1 to 30. Therewas also a small amountof missing dataat the item-level (1%
of responses).All availabledatawere includedin the analyses (92,334 responses).
It is interestingto note thatthe originalmeasurementpropertiesof the Teacher
Questionnairewere assessed by conducting exploratoryfactor analysis without
regardto thenestedstructureof the data(Stem & Williams,1986,p. 241-245). More
recently,takinga confirmatoryapproach,Raudenbush,Rowan, & Kang(1991) reanalyzedthe TeacherSurveydatato demonstratehow measurementmodels could
be incorporatedin conventionalmultilevelanalysesby using a three-levelmodel. It
is useful to contrastthis approachwith the estimationof the model as an SEM.
Considera two-factormodelfor the ControlScale. Inthe multilevelapproach,the
firstlevel of the model constitutesthe measurementmodel andwould be written
Yijk =-tljkX
+ C2jkX2
+
ijk,
(31)
where i, j and k index items, teachers, and schools, respectively, and x1 and x2
are specified to hold the values of the factor loadings for the two latent factors,
153
Bauer
TABLE 2
Items Selectedfor Analysisfrom the High School and BeyondAdministrator
and TeacherSurvey:TeacherQuestionnaire(1984)
QuestionsAssessing TeacherPerceptionsof Controlof School Policy
How much influencedo teachershave over school policy in each of the areasbelow?
1. Determiningstudentbehaviorcodes
2. Determiningthe contentof in-service programs
3. Settingpolicy on groupingstudentsin classes by ability
4. Establishingthe school curriculum
Questions Assessing TeacherPerceptionsof Controlover Classroom
How much controldo you feel you have in your classroomover each of the following
areas of your planningand teaching?
5. Selecting textbooksand otherinstructionalmaterials
6. Selecting content,topics and skills to be taught
7. Selecting teachingtechniques
8. Disciplining students
9. Determiningthe amountof homeworkto be assigned
Note. All items answeredon a 6-point Likertscale.
represented as rTljkand nt2jk.Note that this approach reverses the procedure illustrated here for parameterizing MLMs as SEMs. Here the latent factors are represented as random regression coefficients and the factor loadings as the values of
the Level 1 covariates. The key difference is that in the multilevel approach the
values of the covariates are assumed to be known, so the factor loadings contained
in x, and x2 must be fixed a priori. Conventionally, values of is and Os are assigned
to designate whether the item loads on the factor or not. This practice parallels the
use of "dummy" covariates to toggle between outcomes in multivariate MLMs
(Goldstein, 1995).
Using this approach, the 2-factor model for the 9 items may be written
Yljk
1
Y2jk
1 0
r2jk
Y3jk
1
0
r3jk
Y4jk
1
0
r4jk
0
1
Y6jk
0
1
r6jk
Y7jk
0
1
r7jk
Y8jk
0
1
r8jk
(Y9jk
(O10
Y5jk
154
=
0)
rjk
k
+
r5
sjkk,
r9jk
(32)
EstimatingMultilevelLinearModels as StructuralEquationModels
or
Yjk = Xjk'njk+ rjk,
where 7nlrepresentsthe "controlover school policy" factor and nt2representsthe
"controlover classroom"factor. I assume thatthe residualsare independentwith
differentvariances,or that r
a9).
"=DIAG(ol1,....,
The teacher-levelmodel is then
Plk + Uljk
=
Tlljk
(33)
tE2jk = P2k + U2jk,
where Plkand P2kcapturethe averagesof the latentfactorswithin schools and uljk
and u2jkrepresentdeviations of the factor scores of individualteachersfrom their
school averages,having covariancematrix
T =
l
xr22
Srt21
.
(34)
Similarly,the school-level model is
Plk =
P2k
=
lk
F2k(
where no interceptsare needed because the data are centered,and ElkandE2krepresent variabilityin the mean factor scores of the schools, with covariancematrix
Tp
(36)
r22
T
"
jr.ll
•T021
Of particularinterestin this model is the proportionof factorvariancethatis due
to between-schooldifferencesversus within-schooldifferences.This is expressed
by the intra-classcorrelationcoefficient
ICC =
To + tl
,
(37)
or the between-school varianceover the total varianceof the factor. A high ICC
suggests that there are systematicdifferences between schools in teacherperceptions of controlandthatthese differencesmightbe productivelymodeledas a function of school characteristics.
155
Bauer
The same model can be estimatedas an SEM, with the importantexceptionthat
the factor loadings can be estimated and need not be fixed a priori. That is, the
design matrixXjkfrom Equation32 can be redefinedas
Xjk =
•,
X2
0)
X3
0
X4
0
0
0
h 5.
0
0
0
(38)
X6
X7
X8
To identify the model and to set the metricfor the latentvariables,the factorloadings X1and k5will be fixed to 1 (see Bollen, 1989).
To parameterizethe model as an SEM, I utilize the missing data approachfor
unbalanceddata.The datais sorted so thateach row correspondsto a school. The
firstnine entrieson each row correspondto the responsesof the firstteacherin that
school on the nine items of the Control scale, the second nine entries are for the
second teacher,and so on for up to J = 30 possible teachersor 9 x 30 = 270 variable entries. The model is then parameterizedin the same way as a second-order
latent factor model, similar to a second-order latent curve model (see Sayer &
Cumsille,2002). The matrixnotationfor the model is unwieldy,so I do not present
it here, but a pathdiagramis given in Figure 5.
There are two primarydifferences between the path diagramin Figure 5 and
those presentedpreviously.First,there is a measurementmodel for the outcomes,
or the two controlfactors.This extends the SEM to a three-levelmodel where the
lowest level is the measurementmodel for the items, with the differencethat here
the factorloadings for the items are estimated.A second key differenceis thatthe
model in Figure5 is multivariate;thatis, therearetwo outcomefactors.The key to
parameterizingthese aspectsof the model is thatthe measurementmodelfor the two
sets of indicatorsis repeatedover the J teachersof the model with equality constraintson the factorloadings(X,, . . . .
), residualvariancesof the items (al,...,
09), and within-schoolvariancesand covariancesof the factorscores for the teachers (•IlI, t.21 andtZ22).The commonvarianceamongteacherswithina school on the
two factorsis then drawninto the second-orderfactorsPlk and P2kwith variances
andcovariancesequalto rpll,
andt'22. No meansor interceptswere includedin
T•21 centeredat zero
this model because the items were
priorto the analysis.
First, I contrastthe multilevel approachto measurementmodels with the SEM
approach.Using the ControlScale data, a three-levelmultilevel linearmodel was
156
Estimating Multilevel Linear Models as Structural Equation Models
11 l
S4
)ll
Yl21?r421
T11
1
Y83
49
3
31
Y41 4
1-r491
kK2 2
4
T21
Y3J
r3J938
Y4J
r4Jr
Y51
T51
61
4
(5
(Y6
r61
281
9X81(8
n
Tit2
•02
5J
T222
T
r5J•05
•
lt2J
r 7J
X7
I8 4-r
Tx2
y4J
J
-tr9J
T7
O
8
n9
FIGURE 5. A path diagram of a 2-Factor multilevel linear structural equation model
with estimatedfactor loadings and unbalanced data treated as missing.
157
Bauer
estimatedwhere the factor loadings for the items were set to is and Osas is conventionalfor this modeling approach.An SEM was then estimated,parameterized
accordingto the pathdiagramin Figure5; thatis, with freely estimatedloadingsfor
the nonscaling indicators. The estimated loadings from the SEM analysis, presentedin Figure6, indicatedthatfactorloadings of is areadequatefor some items
(items 2 and 6) but not others(items 3, 4, 7, 8 and9). The apparentconsequenceof
using unitloadingsin the MLMwas to distortthe estimationof the factorvariances,
as can be seen in Figure 7 which compares the estimated within- and betweenschool factorvariancesandcovariances.Forinstance,the totalvarianceof the controlover policy factorwas estimatedat .89 using unitloadingscomparedto .73 with
estimatedloadings.Similarly,the totalvariancein the controlover classroomfactor
was estimatedat .34 usingunitloadingscomparedto .80 withestimatedloadings.Of
particularinterestwere the ICCs,which for the SEM analysiswere estimatedas .28
for the policy factor and .19 for the classroom factor. These are sizeable values,
reflectingthe disattenuationof the ICC due to the removal of measurementerror
(Raudenbush,Rowan & Kang, 1991).
To validatethe resultsof the SEM analysisempirically,a second MLMwas estimated, only this time the factor loadings were set to their estimatedvalues from
1.4
.&-I
cm 0.8
L
0.2
Factor Loading
x
X2
X3
4
X•
X6
X7
X8
k9
FactorLoading
FIGURE 6. Factor loading estimates ? 2 standard errors obtained byfitting the model
displayed in Figure 5 in Mplus. Starredfactor loadings werefixed to 1 to set the scale of
the latentfactors. The dotted line shows the reference unit loading value of I assumed in
many multilevel linear modeling approaches to multilevelfactor analysis.
158
Estimating Multilevel Linear Models as Structural Equation Models
0.8
* PROCMIXED- UnitLoadings
Mplus- EstimatedLoadings
* PROCMIXED- MplusLoadings
-1
0 0.7
C\V
(D 0.4
cu
0.3 -
0.1
0
in 11
T
11
ZT22
'Tic
21
T 22
Za
Parameter
21
FIGURE7. Parameter estimates ? 2 standarderrors of the within-school and betweenschool variances and covariances of the latentfactors of the model displayed in Figure 5
as obtained by fitting the model in PROC MIXED using traditional unit loadings, in
Mplus,freely estimating loadings X2throughX4and X6throughX9,and in PROC MIXED
fixing the loadings to the values estimated in Mplus.
the SEM analysis. Note thatby makingthis change the remainingparameterestimates for the MLM are almost identicalto the estimatesfrom the SEM, as shown
in Figure 7. It is worthnoting, however, thatthe standarderrorsfor the parameter
estimatesarenot equivalent.The standarderrorsfor the MLM fail to take account
of the uncertaintyin the estimation of the factor loadings and are thus slightly
smallerthanthey shouldbe.
Finally, the fit of the model was tested. For this data set the covariancematrix
for the constrainedsaturatedmodel consists of
IT(9x9)
1(270x270)
=
YT(9x9)
B(9x9)
SB(9x9)
(9,
...
"
B(9x9)
(39)
T(9x9)
where yT is the within-teacherscovariance matrix for the 9 items and is constrainedto be equal over the 30 blocks on the diagonalof I (correspondingto the
30 possible teacherswithineach school). Note thatlT representsthe totalvariance
159
Bauer
and covariancein the set of nine items over teachersand hence takes on the same
values as one would obtainby calculatinga covariancematrixon the total sample
of teachersaggregatedover schools. The covariancebetween items among different teachersof the same school is capturedin IB, which accountsfor the dependence introducedby systematicbetween-school differencesin teacherresponses.
No mean vector is necessary since means were not modeled. Unfortunately,the
large size of this model, involving 90 free parameters,a 270 x 270 covariance
matrix, and over 92,000 responses, exceeded the memoryresourcesof two SEM
softwarepackages (Mplus version 2.1 and Mx version 1.52a).7
Thus, to test the fit of the hypothesizedmodel, both the multilevel factormodel
and constrainedsaturatedmodel were fit to a reduceddata set, consisting only of
the scores fromthe firstseven teachersof each school. Althoughnot ideal, this provided a good indicationof the fit of the hypothesizedmodel. The likelihood-ratio
test calculated by fitting the models to the reduced sample was X2(68) = 1352;
p < .001, indicatingquite poor fit. Note thatthis informationon the poor fit of the
model typically would not be available if a multilevelframeworkwere used to fit
the model. Within the SEM tradition,however, asignificant X2test of model fit
often initiates the search for an alternative,hopefully better-fitting,theoretical
model. For instance, an alternative 2-factor model might group items into
curriculum-orientedquestions (e.g., items 4, 5, 6, 7) and discipline-orientedquestions (e.g., items 1 and 8). For didactic purposes, however, I retain the current
model (as depictedin Figure 5) in the subsequentanalysis despite its poor fit.
Prediction and Mediation
I now extend the multilevel factor analysis presentedpreviously to a full SEM
including school-level predictors,following the observed variables approachto
incorporatingupper-level predictors.A mediationalmodel is posited where the
effect of one school-level predictor,whetherthe school is public or private,is partially explainedby anotherschool-level variable,school size. The school-level data
was obtainedfrom the High School and Beyond First Follow-Up: School Questionnaireadministeredin 1982, two years priorto the Administratorand Teacher
Survey. Only data for the 456 schools in the preceding analysis were included
(using all availableteacherdata).The two predictorsof interestwere whetherthe
school was public (82%) or private (18%), and school size, which ranged from
21 to 5,342 with a mean of 1,254. To ease computations,school size was rescaled
by dividing by 100, so that each unit represents100 students.There was missing
datafor 13 schools for the public/privatevariableand missing dataon school size
for 42 schools. Because the SEM approachutilizes a joint likelihood function for
the predictorsand outcomes, missing datacan be accommodatedfor both types of
variables.8All 456 schools were thus retainedin the analysis.
One interestingquestionis whetherschool size completely explains the difference between public and privateschools in teacherperceptionsof control,or only
partiallyexplains the difference. To test between these possibilities, two models
160
EstimatingMultilevelLinearModels as StructuralEquationModels
were estimated.The pathdiagramin Figure8 presentsthe predictiverelationships
of the full model, allowing both direct and indirect effects of school status on
teacher perceptions of control (note that the teacher-level and item-level of the
model are not reproducedin Figure 8 since they are unchangedfrom Figure5). In
the reducedmodel, the dashedarrowsrepresentingthe directeffects of school status independentof school size were removed. The full and reduced models are
nested,allowingfor a formaltest of the hypothesisthatthe differencebetweenpublic and private schools in teacherperceptionsof control is entirely due to differences in school size. The likelihood-ratiotest, X2(2)= 70; p < .001, rejected this
hypothesis in favor of the view that school size only partiallymediatesthe effect
of employmentat a public versus privateschool.
The parameterestimatesfor the mediationalpathsin the full model arealso presentedin Figure8 (the parameterestimatesfor the remainderof the model changed
little from Figures 6 and 7 and are not reported).An examinationof these values
reveals the same generalpatternfor the two controlfactors,but also subtle differences. First,teachersin privateschools perceivedthemselves to have morecontrol
over school policy and their classrooms than teachers in public schools. Specifically, on a six-pointscale, teacherperceptionsof controlover school policy were on
average.6 unitshigherin privateschools thanpublic schools [wherethe totaleffect
is calculated as the direct effect plus the indirecteffect, or .51 + (-7.50)(-.01)].
This difference accountedfor 28% of the between-school variancein the Policy
pi
.15
.51
-.01
Private
SchoolI
-7.50
School
Size
.07
1.54
-.02
.29
P
.11
FIGURE 8. A path diagram of the latent variable modelfor a 2-Factor multilevelstructural equation model where the effect of the School-Level predictor Private School is
mediatedby School Size (The teacher and item levels of the model are identical to Figure 5
so are not displayed).Paths are labeled with theirestimatedparametervalues (all displayed
parameter estimates significantly differfrom zero at p < .05).
161
Bauer
factor. Similarly,teachersin privateschools perceivedthemselves to have greater
controlover theirclassroomsby an averageof .4 units [calculatedas .29 + (-7.50)
(-.02)], a difference that explained 33% of the between-school variance in the
Classroomfactor.
Second, the results supportedthe hypothesis thatthe differencesbetween public and privateschools are partiallydue to differencesin school size. On average,
750 fewer studentswere enrolled at privateschools than public schools. Further,
for each 100 students enrolled in a school, teacher perceptions of control over
school policy droppedby a tenth of a point, while teacherperceptionsof control
over the classroom droppedby two-tenths of a point (again on six-point scales).
Given these values, only 14%of the total difference between public and private
school teachers'perceptionsof controlover school policy can be attributedto differences in school size. In contrast,a muchlargerpercentage,34%,of the total differencebetweenpublic andprivateschool teachers'perceptionsof controlover the
classroomcan be explainedby the fact thatprivateschools tend to have fewer students. Thus the results suggest that while school size partiallymediatesthe effect
of employmentin a privateversus public school for both factors, its role is much
more prominentfor teacherperceptionsof control over the classroom.9Note that
these same results generally could not be obtaineddirectly using a conventional
multilevel approach,thoughindirectproceduresareavailablefor calculatingsome
of them (Krull& MacKinnon,1999, 2001; Raudenbush& Sampson, 1999).
Limitations and Alternative Approaches
The approachillustratedhere for estimatingMLMs as SEMs integratesmany of
the strengthsof the two modeling frameworks.Threekey extensions of the basic
multilevel linearmodel made possible by the SEM frameworkare formaltests of
model fit, measurementmodels for the outcomes, and mediationalmodels among
predictors.Many other extensions are also possible. These include measurement
models for the predictors,models for multiplesamples(J6reskog,1971), and finite
mixturemodels (Arminger& Stein, 1997;Jedidi,Jagpal,& DeSarbo,1997;Muthdn
& Shedden,1999), amongotherpossibilities.Thereis nothingin principlethatprevents the estimationof similarmodels within the multilevel framework,but currentlymany of these featuresare only availablewithinthe SEM.
Likewise, many features of the multilevel modeling frameworkare currently
unavailablewithinthe SEM. For instance,many multilevelnonlinearmodels cannot be parameterizedas SEMs, since structuralequationmodeling is predicatedon
a linear system of equations.10Similarly,generalizedlinearmixed-effects models
with link functions for binary,ordinal,or count response scales cannot currently
be estimatedas SEMs. Traditionally,SEM has takena much differentapproachto
handlingthese differentresponsescales. Specifically,it is assumedthatbinaryand
ordinal data representcrude categorizationsof underlyingcontinuous variables.
The relationshipsof these underlyingcontinuousvariablesare capturedin a polychoriccorrelationor covariancematrix,andthe model is fit to this matrixwith special modificationsto the fit function (Muthdn,1984).
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EstimatingMultilevelLinearModels as StructuralEquationModels
Otheraspects of the two modeling frameworksalso differ. For instance,not all
estimatorsareavailablein each modelingframework(e.g., the restrictedmaximum
likelihoodestimatorcommonlyusedwithmultilevelmodelscurrentlyhasno counterpartin SEM). Also, while Wald tests of the variancecomponentsaretypical of the
SEM approachand some MLMs, otherMLMsutilize a chi-squaretest of the residuals to account for the skewed sampling distributionsof the parameters(Bryk &
Raudenbush,1992, p. 55). These differences largely reflect modeling traditions
ratherthan the constraintsof the analytic models themselves. It is hoped that by
illustratingthe commonalitiesbetweenthe two approaches,advancesmadein each
modelingframeworkwill informthe other,as has been the case in the growthmodeling literaturewherethe identitybetween the two approacheshas been known for
many years. As one example, it is common for MLMs to be presentedwith formal
assumptionsof normalityfor the residuals and randomeffects, as was done here
(e.g., Raudenbush,2001; Verbeke & Molenberghs, 2000). However, it is widely
known in the SEM community that the maximum likelihood estimator retains
many of its desirablepropertiesunderthe less stringentassumptionof no excessive
kurtosisfor the endogenousobserved variables(Browne, 1984).
Multilevel latentvariablemodels such as the two factormodel presentedabove
have been the topic of manyotherrecentanalyticdevelopments.Therearetwo primaryalternativeapproachesto the one illustratedhere. The first,originatingwith
Goldstein& MacDonald(1988), involves partitioningthe total populationcovariance matrixyT (aggregatedover all individuals) into two additive components:
1w representingwithin-groupscovariancein the set of items (or the covarianceof
individualdeviationsfromtheirgroupmeans)andlB representingbetween-groups
covariance (or the covariance of the group means of the items). Muthen (1994)
demonstratedhow sample estimates of the within- and between-groupscovariance matricescan be obtainedandproposeda quasi-maximumlikelihoodestimator
(MUML)for estimatingthe modelon the two componentcovariancematricessimultaneously.The primaryproblemwith MUML is thatit assumes a balanceddesign
to make the estimationof the model more tractable.More recently,however, fullinformationmaximumlikelihood (FIML)routineshave become availablethatcan
accommodateunbalancedesigns (Bentler & Liang, 2003; du Toit & du Toit, in
press;Muthen& Muth6n,2002). In comparison,the approachillustratedhere also
accommodatesunbalanceddesigns but models yT and lB directly, with no prior
decompositionof IT, as can be seen in Equation39. It thus provides a more naturalanalogto conventionalMLMs. Furtherresearchis neededto morethoroughly
compare these alternatives both analytically and in terms of their relative performance in finite samples. It is importantto note, however, that the within- and
between-groupsapproachto multilevel SEM, unlike the one illustratedhere, cannot currentlybe combinedwith some of the otherfeaturesof the SEM framework
(e.g., multiplegroups analysis, mixturemodeling).
A second alternativeapproachto multilevel SEM was recently proposed by
Rabe-Hesketh and colleagues in their Generalized Linear Latent and Mixed
Models (GLLAMM) STATA macro (Rabe-Hesketh, Pickles, & Taylor, 2000;
163
Bauer
Rabe-Hesketh,Pickles, & Skrondal,2001; Rabe-Hesketh,Skrondal,& Pickles, in
press). The key advantageof GLLAMMis thatit permitslink functionsto be used
for outcomes on differentresponse scales, such as binaryor count data.However,
because the derivativesin the likelihood functionareapproximated,it is very slow
and is not recommendedfor continuousdata(Rabe-Hesketh,Pickles, & Skrondal,
2001). GLLAMMmay best be viewed as complementaryto other approachesto
multilevel SEM, to be used when outcomes are not continuous.The rapidpace of
these developmentsatteststo the continuingconvergence of multilevel modeling
and SEM, and the new modeling possibilities that an integrativeframeworkmay
offer to appliedresearchers.
Notes
Any pairof nestedmodels can be comparedby the likelihood-ratiotest (subject
to certainregularityconditions), makingpossible formalcontrastsbetween alternative models.
2 This differs from the traditionalCFA model in which factorloadings are estimatedand relect the regressionof the observedvariableson the latentvariable.
3 SAS code and Mplus input files for all of the example analyses can be downloaded from the authorat http://www4.ncsu.edu/-djbauer.Note that these same
analyses could be conductedin othermultilevel or SEM softwareprogramsoffering the same capabilities.
4 Tabledvalues for the parameter
estimates,standarderrors,andlog-likelihoods
of this and subsequentmodels may also be downloadedfrom the authorat http://
www4.ncsu.edu/-djbauer.
5 Currently, only two SEM software programs, Mx and Mplus have this
capability.
6 AlthoughRovine & Molenaar(1998, 2000, 2001) do not do so, to achievemaximum generality,I have subscriptedthe factorloadingmatrixby j to indicatethatit
may vary case-wise. This subscriptcould be removedfor balanceddataor if using
the missing dataapproachto unbalanceddata.
7 This reflectsthe fact thatSEM softwarewas not initiallydesignedto fit models
of this type and so does not always efficiently analyze partitionedmatrices as
in Equation 39. Future developments are likely to overcome this difficulty
(L. Muthdn,personalcommunication).
8 This contrastswith the conditional likelihood function typically used in the
multilevel frameworkwhich requirescomplete dataon the predictors.
9 Of course,these conclusionsmustbe temperedby the poor fit of the underlying
factormodel.
10To be more precise, if the response is a nonlinear function of the random
effects (e.g., a negatively acceleratedexponential function with randomparameters)the model cannotcurrentlybe parameterizedas an SEM. Alternatively,if the
nonlinearityis only in the fixed effects partof the model, then the model can be
parameterizedas an SEM by using nonlinearconstraints(see du Toit & Cudeck,
2001). Further,manynonlinearfunctionscan be reasonablyapproximatedby poly1
164
EstimatingMultilevelLinearModels as StructuralEquationModels
nomial functions(which arelinearin theirparametersand hence can be estimated
as SEMs).
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Author
DANIEL J. BAUER is AssistantProfessor,Psychology Department,NorthCarolinaState
University,Raleigh, NC 27695-7801; [email protected] areasof specialization
are quantitativemethodologyand developmentalpsychology.
ManuscriptReceived August, 2002
Revision Received October,2002
Accepted December,2002
167