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STRUCTURAL EQUATION MODELING: USES AND ISSUES
LISABETH F. DILALLA
Sc ho ol o f M edicine, S ou ther n Illinois University. Ccirboncicile. Il in ois
Structural equation modeling (SEM) has become a popular research tool
in the social sciences, including psychology, m anag em ent, econom ics, sociol-
ogy, political science, marketing, and education, over the past two to three
decades. Its strengths include simultaneous assessment of various types of
relat ions am ong variables an d the abili ty to r igorously examine and com pare
similarities among and differences between two or more groups of study
partic ipants. H ow ever, one o f its m ajo r lim ita tio ns is the ease with which
researchers can misinterpret their results when anxious to prove" the
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440 LISABETH F. DILALLA
1. D EF IN ING STRU CT UR AL EQ UA TION MO DEL ING
Latent variable analysis (Bentler . 1980) involves the study of "hidden"variables that are not measured directly but that are estimated by
variables that can be measured. Latent variable analysis includes such
techniques as factor analysis, path analysis, and SEM. Factor analysis
(see also Cudeck, chapter 10. and Hoyle, chapter 16. this volume)
involves assessing a latent factor that is operationalized by measured
variables. The latent factor is identified by the variance that is shared
among the measured variables: i t is the "true" variable that affects the
m ea su red variables. Path analysis, first de scribed by Sewall Wrigh t (1934.
1960). determines the causal relations between a series of independent
and dependent variables. SEM encompasses both of these and is a
m eth od for testing carefully delinea ted m odels based on hyp otheses
about how observed and latent variables are interrelated (Hoyle. 1995b)
in order to meaningfully explain the observed relations among the
variables in the most parsimonious way (MacCallum. 1995).
The strength of SEM lies in its ability to rigorously test a hypothesized
model of relations among manifest and latent variables. The key to this is
that the model must be specif ied a priori and be theoretically based. SEM
can then provide a series of indices that indicate the extent to which the
specified model appears to f i t the observed data. The results cannot beused to assert that a given model with a good fit must therefore precisely
identify the mechanisms through which the variables are intercorrelated
because , a lthough th a t one m odel m ay fit. th e re will also be countless o th ers
that might fit equally well or even better. Therefore, as long as the model
is theoretically strong, a goodfitting model can be said to be supported by
the data and to be sensible, but. as with any other statistical technique,
SEM cannot be used to "prove the model .
Clearly, careful forethought is essential in developing a model. The
first step is to develop a theoretical model delineated by the latent variables(drawn in ovals), which are the constructs that are of interest theoretically
(see Figure 15.1). This part of the model is the structural model and is
composed only of the latent, unmeasured variables. Then the variables that
are actually measured (drawn in rectangles) and are used as indicators of
the latent variables can be add ed to the m odel (see Figure 15.1). This pa rt
of the model that specif ies the relations between the latent and manifest
variables is called the measurement model. The final model, including both
the structura l and the measu rem ent pa rts of the model, has the adv antage
of allowing the researcher to explore the relations between the latent
variables th at are of interest theoretically by including the o perationa lized
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15. STRUCTURAL EQUATION MODELING 441
F IG U R E 15.1 Structural mode! and i ts e laborat ion into a me asurem ent model.
and Discipline in Figure 15.1) and several measures of aggression torepresent an aggression factor , and then the causal relat ion between the
two latent variables can be assessed using these measures.
II. C OM M ON U SES OF STRU C TU RA L EQ U A T I ON M OD EL IN G
A path diagram can be created to test a set of relat ions among variables
simu ltaneou sly. Thus, a variable can be regressed on se veral o ther variables
and can in turn simultaneously predict another outcome variable. This set
of relat ions cannot be tested using standard regression analysis because
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442 USABETH F. DILALLA
com par isons be tw een groups , to run growth curve models , and to com pare
nested models, to name some of the more common uses.The most important rule to bear in mind is that a s trong theoretical
m odel m ust be posited prior to model testing. W ithin this theoretical fram e-
w ork. a series of nested m odels can be com pared to come up with a parsim o-
nious ex planation for the relations amo ng the variables. Of course, exp lor-
atory analyses can be run, but then the structure of the relations among
the variables is being assessed for that particular data set. and replication
using an ind epend ently assessed sam ple is essential in orde r to draw co nclu-
sions with confidence. As with any statistical analysis, analyses that capital-
ize on the chance relations among variables in a given data set are not
useful for advancing our understanding of any of the sciences. This wouldresult if analyses are conducted without a theoretical framework to guide
the model fitting.
A. Exploratory Factor Analyses
One of the typical uses of SEM is to conduct exploratory factor analyses.
In exploratory factor analysis , no preexisting model is hypothesized.
Rather, the relations among the variables are explored by testing a series
of factors that account for shared variance among the variables. This isfeasible with SEM because there is a very specific type of model included
in the exploration. Only models with no causal l inks between factors and
with paths only between the factors and the manifest variables are tested
(Loehlin, 1992). Therefore, there are only a specif ic number of models
tested. This method is useful in allowing the researcher to determine the
simplest factor model ( the one with the fewest number of factors and the
fewest non zero paths in the pa ttern m atrix) tha t will adeq uately explain the
observed intercorrelations among the various manifest variables (Loehlin.
1992). In an SEM factor analysis, the latent variables are the factors, the
manifest variables are the variables that make up the factors, the loadings
be tw een th e m anif est and la te n t varia b le s form th e facto r p a tte rn m atr ix ,
and the residuals on the m anifest variables are the specif ic or uniqu e factors.
Thus, SEM can be used to explore the latent factor s tructure of a data set
by consid ering su ccess ively in creasin g or decreasin g num bers o f facto rs and
com par ing each new model to the p receding one to de termine the op t im al
and most parsimonious model that best accounts for the manifest variable
interrelations.
B. Confirmatory Factor Analyses
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15. STRUCTURAL EQUATION MODELING 443
variables are assumed to be related sufficiently to submit to factor analysis.
A dd ition ally, confirmatory factor analysis may sti ll involve a small am ou nt
of tweaking of the model to imp rove i ts fit. W ith confirm atory factor analysis
(see also Hoyle, chapter 16. this volume), a path model is designed that
describes the interrelations among the manifest variables by hypothesizing
a specific set of latent factors that account for those relations. These factors
are based in theory, as are the items that load on them. This last fact is
essential because i t both justif ies the model and requires that changes not
be m ade to the m odel w it hout reconside ring th e th eo ry behin d th e analy sis .
A com m on example of misuse of SEM is to modify the fac tor s t ruc ture o f
the model based on modification indices without regard to the underlying
theory on which the model was based. Assuming the theory is sound, i tp robab ly is w is er to m ain ta in th e orig in a l ite m s as lo ng as th ere are eno ugh
items with high loadings to define the latent factor.
Co nf i rmatory factor ana lyses can be cond ucted us ing SEM . The m odel
is based on theory: that is. a hyp othesized struc tural m odel is fit to the d ata
to determ ine how well the interrelation s am ong the variables are accoun ted
for by the a priori model . Thus , the im po r tant di f fe rence be tween conf i rma-
tory an d exp loratory factor analysis with SEM is that the factors that accou nt
for the variable intercorrelations are specif ied up front with confirmatory
analyses ra the r than becoming revea led thro ugh explora t ion o f the var iance
accounted for with a different number of factors, as with exploratory
analyses.
The measurement model of an SEM is a conf i rmatory fac tor ana lys is
because it reflects the th eo retically designed config urati on o f m anife st v a r i-
ables as they load on the latent factors. For this reason, it is important that
changes not be made to the measurement model l ight ly when a t tempting
to achieve a parsimonious and significant fit. It is prudent to maintain the
original measurement model even when some of the manifest variables
load m ore highly on a different la tent factor than the o ne originally specif ied
or when a manifest variable does not add significantly to the definition ofa la tent factor because the m odel supp osed ly was based on sound theoretical
underpinnings. Using SEM in an exploratory fashion is acceptable and can
be enorm ously usefu l as one explo res the ways in which certa in variab les
relate to each o ther, but it is essen tial that a t the end of this ex plo rato ry
foray the researcher is aware of the exploratory nature of the analyses and
describes them as such. U nd er these circum stances, the confirmatory aspect
of the modeling is lost and further confirmation with another data set is
necessary to determine that the final mode! is not simply a reflection of an
unusual data set .
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444 USABETH F. DILALLA
variables. For exam ple, i f we hy pothesize that socioeconomic status (SES)
p red ic ts th e hom e en v iro n m en t, which in tu rn p redic ts ch ild ren 's p la y b e -
haviors with peers, then a model can be designed with paths from SES tohome envi ronment and f rom home envi ronment to p lay behaviors , and
these p aths can be tested simultaneously. This analysis cannot be c ond ucted
with a standard regression analysis. Addit ionally, i f we want to measure
each of these variables with several i tems, then with SEM we can form
three factors SES. home e nvi ronment , and play behaviors and a num ber
of items that load on each of the factors to define them. So for instance,
p a ren ta l educa tion and occu pa ti on and fa mily in com e can load on SES. a
set of items assessing parental discipline and sociability can load on home
environment, and i tems based on videotaped analyses of children's play
behav io rs can lo ad on the pla y behav iors facto r. T hen each o f these th ree
factors can be regress ed a cco rding to the theory tha t was originally specified,
and these interrelat ions can be assessed direct ly and simultaneously.
D. M oderator-Mediator Analyses
Th e use of m od era tor and m ediato r var iables also can be tested using
SEM. For instance, with the above example, we can ask whether home
environm ent m ediates the re la t ion between SES and children 's p lay behav -
iors or whether SES has a direct inf luence on play behaviors even af ter the effects of the home environment are included in the model . Thus, we
can test the model specif ied above and then add the direct path from SES
to play behaviors to d eterm ine w heth er this path significantly improves the
overal l model f i t . The use of moderator var iables is more complex, but
these variables also can be tested using SEM . M ethods for assessing l inear ,
quad rat ic, and stepwise effects of the m oderat ing variable on the dep en de nt
variable are descr ibed in Baron and Kenny (1986) and can be applied
to SEM.
III. P LANN ING A STR UC TU RA L EQUAT ION
M O D E LIN G A N A L Y S IS
Clearly, SEM has a nu m be r of unique m ethodological advantages, such as
using mult iple measures as both independent and dependent var iables.
However , one dist inct disadvantage, as with many of the procedures de-
scribed in this volume, is that it has become so easy to use many of the
SEM programs th at a user can run analyses without being aw are of some
of the basic assumptions that are necessary for conducting the analysescorrect ly. Incorrect results can be reported if the user does not read and
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15. STRUCTURAL EQUATION MODELING 445
necessary steps for understanding the basics of perform ing and in terp reting
an SEM analysis.
Prior to setting up an SEM analysis, the variables must be examinedto determine their appropriateness for the analysis. Certain assumptions
m ust be met. Sample size is im porta nt b ecause SEM requires larger sam ples
than mo st other statistical procedures. T he form at of da ta input also must
be considered. E stim atio n pro ced u res m ust be chosen based on th e types
of variables in the model. Each of these issues is considered in the follow-
ing sections.
IV . DAT A REQU IREMEN TS
Basic assumptions common to all regressiontype analyses include data
multivariate normality and a sufficiently large sample size. SEM also as-
sumes continuous data , a l though there are ways to model ordinal data by
using tetrachoric o r polvchoric correlation s, and ways to m odel catego rical
d ata as well (Mislevy, 19S6; M uthe n, 1984; Pa rry & M cA rdle, 1991). A
num ber of boots trapping and M onte Carlo techniques have been used
in an attempt to determine the ramifications of violations of these basic
assumptions. There is not a clear consensus on the seriousness of relaxing
these constraints, but parameter estimates appear to be fairly robust evenwhen fit statistics may be severely compromised by this relaxation (Loeh-
lin, 1992).
A. Multivariate Normality
Multivariate normality of the variables in the model is necessary for a well
behav ed analys is . M ultivariate n o rm ality is sim ply a generalization of th e
b ivaria te norm al sit uation. W ith a m ultivariate norm al d istribu tio n , each
variable is normally distributed when holding all other variables constant,each pair of variables is bivariate normal holding all other variables con-
stant, et cetera, and the relation between any pair of variables is l inear. In
or de r to tes t for m ultivaria te n orm ality , the th ird and fourth o rder m om ents
of the variables must be examined fo r m ultivariate skewness and m ultivari-
ate kurtosis. Mardia (1970) has devised measures to assess these that are
available in the EQS (Bentler, 1992) and PRELIS (Joreskog & Sorbom.
1993b) computer software packages for data preparation.
It is im portant to assess the m ultivariate normality of the data, be cause
in its absence model fit and standard errors may be biased or irregular
(Jaccard & Wan, 1996: West, Finch, & Curran, 1995). However, there are
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446 USABETH F. DILALLA
can be located and el iminated from the data. However, i t is imperat ive
that this be do ne carefully and with a great d eal o f thou ght by the researcher.
If the outliers are truly errors, for instance, if someone were tested under
improper conditions, then the outlier can be eliminated and the resulting
sample is stil l the one that was originally expected. However, there are
times w hen an o utlier occurs und er reaso nab le circumstanc es, such as when
a person behaves very differently from expectation and from how others
in the sam ple beh ave , but there are no noticea ble d ifferences in test adm inis-
trat ion, and the person did not suffer from a known condit ion that would
affect test perform an ce. In these cases, it is not a ccep table to elim inate this
person fro m the to ta l sam ple because he o r she is a random ly ascertained
member of the populat ion. In other words, i t is not acceptable to drop
respondents simply because the researcher is not happy with the results
that were obtained. If such atypical participants are dropped from the
sample, then the resea rche r must be clear that resu l ts of the SEM may not
general ize to extreme members of the populat ion.
A second acceptable procedure for deal ing with nonnormali ty is to
transform the variable that is causing the problem. Several transformation
options exist, and it depends on the type of variable and the degree of
skewness or kurtosis present as to which transformation is preferred. For
instance, a linear transformation will not affect the distribution of the
variable, whe reas a nonlinear t ransform ation m ay a l ter the variable 's d istr i-
bu tio n and it s in te rac tio ns and curvil inear effec ts (W est et al., 1995). C en-
sored v ariab les (variables that have ceiling ' or floo r' effects such that
a large m ajority of the re spond ents receive the top o r bo ttom possible score,
and only a small percent of the respondents score along the continuum of
possib le v alu es) als o may bias p a ram ete r estim atio n (v an den O ord &
Rowe, 1997).
It is important after transforming a variable to reassess the skewness
or kurtosis to determine whether this has improved. It also is important to
be aw are th a t the in te rp re ta tio n of the varia b le follow in g a transform ation
cannot be the same as prior to the t ransformation. Transformation causes
a loss of metric so the variable cannot be easily compared across studies
or to other, comparable variables (West et al. . 1995). Examples of typical
and appropriate t ransformations are provided in Cohen and Cohen (1975)
and West et al. (1995).
A third approach to the violation of multivariate normality is to use a
routine other than maximum likelihood (e.g. . weighted least squares) to
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IS. STRUCTURAL EQUATION MODELING 447
across the samples . This al lows an est imation of whether the normali ty
violat ion is a prob lem for the part icular model. T he new v ers ion of LISRE L8 makes this method fairly simple.
B. Sample Size
The difficult part of choosing an appropriate sample size is that there is no
clearcut rule to follow. D ifferent researchers , experim enting w ith different
types of da ta and m odels , have found varying results in term s of the neces-
sary sample size for obtaining accurate solutions to modelfitting analyses
(Bentler & Chou, 1987; Guadagnoli & Velicer. 1988; Hu & Bentler. 1995;
Loeh lin, 1992). O ne con sideration is the num ber of m anifest variables usedto measure the latent factors. The larger this is, and the larger the loadings
are for the indicators , then the smaller the necessary sample can be. For
instance, Guadagnoli and Velicer (1988) found that as long as the factors
were sufficiently saturated with loadings of at least .80, the total number
of variables was not important in determining the necessary sample s ize,
and the sa m ple size could possibly go as low as 50. (W ith fac tor loadings
less than .80, how ever, the sam ple size requ ireme nts we re m ore s tringent.)
Another important consideration is the multivariate normali ty of the
m easures . Sm aller sam ples may be adeq uate if all me asures are m ultivariate
norm al. B en tler and C hou (1987) suggested that if all variables are normally
dis tr ibuted , a 5 :1 ra t io of respondents to num ber of f ree param eters may
be su ff ic ie nt. W ith ou t m u lt ivariate norm ality , a sa m ple size as la rg e as 5000
may be n ecessary to ob tain accurate results (Hu & B en tler . 1995). Fo r most
s tudies , An de rson and G erbing (1988) suggest that sam ple s izes of at least
150 should be adequate.
All of these sample s ize suggestions assume continuous data sampled
randomly from a population. I t is possible that sample s izes need to be
different when these assumptions are violated, but there are no definit ive
answers as yet. Of course, the basic concern with a small sample is howwell the sample represents the population. Quirks specif ic to the sample
may greatly affect the analysis, and that is more likely if a smaller sample
hap pen s to m isreprese nt the popu lation. I f the sample is truly random ly
ascertained and is an acc urate rep resentation of the large r pop ulation of
interest , then some of the concerns about small sample s ize may become
less problematic (Loehlin. 1992). An excellent discussion of the important
issues of sample s ize and the accompanying concern about power can be
found in Jaccard and Wan (1996).
V PREPAR ING DA TA FOR ANALYS IS
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448 USABETH F. DILALLA
A. Input Data Matrix
Most scholars recommend the use of a covariance matr ix for analysis be-
cause the m ethods tha t are m ost comm only used to solve s tructural equat ion
models (i .e. . maximum likelihood and generalized least squares) are based
on theories that were der ived using covariance m atr ices ra th er than c orre la-
tion m atrices (Lo ehlin, 1992). Also, when a corre lation m atrix is com puted,
the var iables are s tandard ized based on that sample . W hen those s tandard -
ized scores are used for the SEM, the sample s tandard deviat ion must be
used to calculate the standardized variable, resulting in the loss of one
degree of freedom (Loehlin, 1992). The effects of this are most serious
with small samples. I t is imperative to use covariance matrices as input
when a multiplegroup analysis is conducted because otherwise variance
differences across groups cannot be considered. Instead, the groups are
treated as though they did not differ in variance, which may be quite
misleading. Thus, covariance matrices are necessary for comparison across
groups and across t ime, but corre la t ion matr ices may be used when the
analysis focuses exclusively on withingroup variations.
B. Missing Data
A second issue conc erns the handling of missing data. Th ere are three main
options for dealing with this problem. First , variables can be deleted in a
listwise fashion, thereby excluding all participants who are missing any of
the variables that a re p ar t of the analyses. If variables are m issing random ly
througho ut the sam ple , however , th is procedu re of ten redu ces the analysis
sample to a size that is too small for reliable estimates to be produced.
Furtherm ore, the sm aller sample may be less representat ive of the general
p opu latio n th a t w as o rigin ally sam ple d, and th ere fo re is sues o f generaliz
ability become a concern.Second, variables can be deleted in a pairwise fashion, thereby only
omitting participants for those elements in the covariance matrix for which
they are missing variables. When pairwise deletion is used, the elements
in the resulting covariance matrix are based on different participants and
different sample sizes. This can result in impossible relations among vari-
ables and th erefo re in a m atrix that is not positive definite (Jac card & Wan.
1996: Kap lan. 1995). Th e m atrix may then be u ninv ertable, or th e pa ram eter
estimates m ay be theo retically impossible. Add itionally, th ere is a difficulty
in determining what sample size to claim for the analyses. Because ofthese shortcomings I recommend against using pairwise deletion for SEM
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15. STRUCTURAL EQUATION MODELING 449
be accurate . E ven if the new valu es are reasonab le, there m ay be p ro b lem s
with nonnormali ty and error var iance heteroscedast ici ty (Jaccard & Wan,1996). Problem s also m ay arise if the imp utation is bas ed on the va riables
used in the model , because then the same variables are used for est imating
an oth er var iable 's values an d for es timat ing the re la tions am ong the var i-
ables. This type of redundancy increases the probabil i ty of a Type I error .
R ovine (1994) describes several m etho ds of data est imation tha t may avoid
som e of these problem s. T hese m ethod s are too complex to descr ibe here
and may prove challenging for beginning users, but they may prove useful
if missing data are a problem.
D espite its drawbacks, the safest me thod for deal ing with m issing data
in SEM is to use listwise deletion. However, as long as the data are missing
completely at random, the sample size remains large, and no more than
10% o f the da ta are missing, there is not a large pract ical difference b etw een
the m ethods of deal ing with m issing data. I f an impu tat ion m ethod is chosen ,
i t is important to have a strong rat ionale for the type of imputat ion used
an d no t to confound the im pu tat ion with the modeling itself . M ore resea rch
is necessary on the various ways to handle missing data before vve can
recommend one method over another unequivocal ly .
C. Construction of Input Matrix
O nce the appropr ia te type of inpu t m atrix has been chosen and the problem
of missing data handled, the sam ple m atrix must be const ructed . I f L ISR E L
(Jores kog & Sorbom, 1993a) is the program of choice, a package cal led
PRELIS (Joreskog & Sorbom, 1993b) can compute the input matr ix in a
form that is ready to be read by LISREL. This is deceptively simple; al l
th at the user needs to do is pro vide a raw data f ile and specify the m issing
values and type of matr ix to compute. As with al l canned stat ist ical pack-
ages, the user must be diligent in specifying the desired options. It is very
simple to request the w rong type o f matr ix (e.g. , a correlat ion m atr ix rath er
than a covariance m atrix) or to misspecify the types of var iables th at have
b een m easured (e .g ., identi fy in g a con ti n uo u s variable as o rd inal). P R E L IS
will compute the requested matrix, even if it is not necessarily sensible,
LISREL wil l analyze that matr ix, and the naive user may think that the
m odel has been tested ad equ ately. PR E L IS is an excellent tool for pre parin g
da ta for input to LISR EL as long as the us er correct ly identifies the variables
and the desired type of input matr ix.
D. Estimation Procedures
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450 L1SABETH F. DILALLA
choice of est imation pro ced ure depend s on the samp le and the m odel
bein g estim ate d.
The MLE procedure i s the defaul t opt ion in LISREL and EQS. Thism ethod is based on the a ssum ption of m ult ivar iate n orm ali ty, and i t requires
a relat ively large samp le size to perform adeq uately. W hen these two as-
sumptions are not met. it may be wise to consider a different estimation
m ethod. A second m etho d is gene ral ized least squares, which also assumes
mult ivar iate normali ty and zero kurtosis. A number of other methods have
been develo ped, m any o f w hic h are available th roug h th e sta n d ard SEM
pro gram s. In p ra ctic e , th ere m ay be li tt le d if ference in ou tcom e betw een
the various m ethod s (C hou & B entler . 1995). but when the re is do ub t about
the appropriateness of a method, I recommend that the invest igator use
several methods and compare the results.
VI . MULT IPLE GROUPS
One huge advantage of SEM analyses over other types of analyses is the
abil i ty to compare two or more independent groups simultaneously on the
same model to determine whether the groups differ signif icantly on one or
more parameters. There are a number of instances when i t is useful to
model more than one group s imul taneously . A researcher may want tocompare exper imental groups to determine whether a t rea tment had an
effect , or gender groups to determine whether the relat ions among a set
of var iables are comparable for boys and gir ls, or small versus large busi-
nesses to determine whether shareholder earnings affect worker productiv-
ity comparably. The basic methodology is to hold al l or a subset of the
param eters co n stan t acro ss g roups and assess m odel fits fo r all groups
simultaneously. Then these equal parameters can be freed and the models
comp ared to determine w heth er holding the param eters equal across groups
pro vid es a sig nif icantly w orse fit. If it does, th en the p a ram ete rs dif fe r
significantly across groups; if not, then the parameters do not differ signifi-
cantly across groups and can be set equal across groups without loss of
model fit.
Certain regulat ions must be observed when conducting mult iplegroup
SEM. First , it is im po rtant to inp ut covariance m atr ices ra ther than correla-
tion matr ices, as descr ibed ear l ier , because variances may not be compara-
ble acro ss gro ups. Second, th e laten t variable s m ust be on a com m on scale
across groups, which means that the same manifest var iable(s) must be
used to set the scale for the latent variable(s) in all groups. These two
pra ctices allow com p ariso ns across groups as well as a te s t fo r w h e th er th ei l i f i i ( J k & S b
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IS. STRUCTURAL EQUATION MODELING 451
V II. AS SESS ING M O DEL F iT
Once the m odel has been specified properly and the data have been e ntered
correctly, the fit of the data to the hypothesized model must be evaluated.
A number of tes ts are used to evaluate how well the model descr ibes
the observed re la t ions among the measured var iables: d ifferent modeling
p ro g ram s prov ide d iffe ren t o u tp u t. T h ere is no consensus as to w hic h one
is "b e st becau se each test statistic has advantages and disadvantages. A lso,
there is no consensus regarding the effect of factors such as sample size
and normality violations on different fit indices (e.g.. Hu & Bentler, 1995;
M arsh. Ballad & M cD onald, 1988).
It is im perative to exa m ine several fit indices when e valuating a m odeland never rely solely on a single index (Hoyle, 1995b). Table 15.1 summa-
rizes some of the tes ts and the s i tuations under which m ost researchers
agree that they are most and least useful. The following descriptions and
recom m enda tions are by no m eans def ini tive or exhaust ive , but they incor-
p o ra te the m ost recent suggestions in th e li terature. H oyle (ch ap te r 16, th is
volume) and Tracey (chapter 22, this volume) also provide discussions of
fit indices.
A. Absolute Fit Indices
Th ese indices compare observ ed versus expected variances and covariances ,
thereby measuring absolute model fit (Jaccard & Wan, 1996). The earliest
m eas ure and o ne th at is stil l freq ue ntly rep orted is the chisquare fit index.
This index was designed to test whether the model fit is perfect in the
p o p u la tio n (Jaccard & W an. 1996). It com pares th e observed covariance
m atr ix with the expected covariance m atr ix given the re la tions am ong the
variables specified by the model. The chisquare will be zero when there
is no difference between the two matrices (i.e., there is perfect fit), and
the chisquare index will increase as the difference between the matrices
increases. A significant chisquare value signifies that the model predicts
relations that are significantly different from the relations observed in the
sample , and that the model should be re jected.
A problem with the chisquare statistic is that SEM requires the use
of large sam ples, and u nd er tho se co nditions the chisquare test is pow erful
and rejects virtually all models. Also, the chisquare statistic may not be
distributed as a chisquare when sample size is small or when the data
are nonnormal, and under these conditions the significance test is not
ap pro pria te (Jaccard & W an, 1996). However, the chisquare test is usefulh i d d l Th f I d h hi i i b
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15. STRUCTURAL EQUAT ION MODELING 453
bu t which ad ju sts fo r th e degre es o f freedom in th e m odel. A th ir d index
is the centrali ty index (Cl: McDonald, 1989). Scores on all three indices
can range from 0 to 1.0. with values closer to 1.0 being preferable. Manyresearchers have suggested that values greater than .90 on these indices
can be inte rp rete d as signifying acceptable mo del fit , bu t there is no em piri-
cal support for this . Although Gerbing and Anderson (1993) found the Cl
to be particularly robust with Monte Carlo simulations, i t is not provided
by L IS R E L and is used less frequently in general, m akin g it less use fu l fo r
comparing results across studies. The final two absolute fi t indices are
the standard ized roo t m ean square residual (R M R), which is the average
discrepancy b etw een the o bserved and the ex pected co rrelat ions across all
p a ram ete r estim ate s, and th e root m ean square e rro r o f approxim ation
(RM SE A; S teiger & L ind, 1980). which adjusts for parsim on y in the m odel.A p erfect fit will yield an RM R o r RM SE A of zero; sco res less than .08
are co nsid ered to be ad equ ate, and scores o f less tha n .05 are considered
to be goo d (Jacc ard & W an, 1996). The R M R is a function of the metric
used in m easuring the variables and is most interp retable with standardized
variables. The R M SE A is useful because it adds a pen al ty for including
too many parameters in the model .
B. Com para tive Fit indices
Th ese indices com par e the absolute fit of the m odel to an alternative model.
The co m parat ive fit index (CFI; Be nt ler , 1990). the D E L TA 2 or incremental
fit index (IFI; Bo llen. 1989a), the no rm ed fit index (NF I; Bentler & Bon ett ,
1980). and the n on no rm ed fit index (NN FI: Be ntler & B on ett , 1980), which
is a general ized version of the Tucker and Lewis index (TLI; Tucker &
Lewis, 1973). are the most widely used comparative fit indices (see Table
15.1). Each compares the fi t of a target model to a baseline model.
The CFI compares the tested model to a nul l model having no paths
that l ink the v ariables, therefore making the variables indepe nden t of each
other. The CFI appears to be quite stable, especially with small samples.It can range from 0 to 1.0: scores less than .90 are co nsid ered to be unaccep t-
able. The IFI also tends to be quite consistent even with small samples.
The IFI typically ranges from 0 to 1.0. although values can exceed 1.0,
which makes this more difficult to interpret than the CFI. Again, higher
values indicate better model fi t . There is some debate about the sensitivity
of the TLI and the NNFI to sample size; Marsh et al . (1988) suggest that
they are relatively indep end ent of sample size, whe reas Hu and B ent ler
(1995) suggest that their values may not stay within the easily interpreted
0 to 1 ran ge if the sa m ple size is small. Th e NF I clearly is m ore sensitive
to sample size, does not perform as consistently with smaller samples, andt d t b d ti t d (i l b t bl ll f
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454 USABETH F. DILALLA
C. Sum mary
Many indices have been proposed for evaluating the fit of a model (seeByrne. 1995; Hu & Bentler. 1995; Jaccard & Wan, 1996). and many cutoff
values have been suggested for interpreting these indices (e.g.. Schu
macker & Lomax, 1996). However, there is much discussion among SEM
users as to wh ether these cutoffs are app ropriate. No unam biguous inter pre -
tation exists whereby model fit can be described as "definitely good" or
definitely bad." Instead, fit indices are interpreted fairly subjectively, al-
though the cutoff values suggested in Table 15.1 will be of some help. The
best w ay to de te rm in e w h eth er a m odel is acceptable is to use several of
the indices listed in Table 15.1 and to look for agreement across indices.
Confidence can be placed in the m odel if m ost or all indices are a cceptable,
b u t the m odel should be considered a p o o r fit to the da ta if severa l o f th e
indices are unacceptable. A good general practice is to report the chi
square and A G FI statistics, but to rely more on the com parative fit indices
for interpreting the model fit .
V III. C H E C K I NG T H E O UT P U T F O R P R O B LE M S
I hav e insufficient space in this ch ap ter to do mo re than highlight aspectsof the SEM output that may cause readers some confusion or lead to
erroneous interpretations if not examined carefully. Therefore, in this sec-
tion I will focus on doublechecking output, using standardized scores,
interpreting parameter estimates, using model modification indices, and
comparing nested models . An example of LISREL output is provided in
Tables 15.2 and 15.4 through 15.6 to help clarify the points made below.
This model hypothesizes that day care experience and child temperament
are c auses of childhoo d ag gression (D iLalla, 1998; see Figure 15.2). SES and
sex are the two exogen ous variables, and all oth er variables are regressed onthem. Aggression is measured by parent report and by lab ratings during
a pe er play enco unter. Aggression of the pee r also is rated in the lab. Thus,
child lab aggression is correlated with parentrated aggression and peer
aggression, and is regressed on day care experience, child temperament.
SES, and sex.
A. Ensuring Accurate fnput
It is obvious that errors in input will lead to inaccurate output. Th erefo re,p rio r to in terp reting th e analy sis results , it is im perative that the user
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15. STRUCTURAL EQUATION MODELING 455
FS G U R E I 5 .2 Model depicting the relations betwe en day care experience, child tem pe ra-
ment. and child sociabil i ty (from DiLalla. 1998). and a nested version of the full model, with
the day care experience parameter f ixed to zero.
p rio r to exam inin g the ou tp u t for m odeling results . A n exam ple o f L IS R E L
"P ara m eter Specification output is provided in Table 15.2. All num bered
p a ram ete rs are supposed to be free and all zero p aram ete rs sh o u ld be
fixed. N ote , for instance, that B eta (4,1) is free (p ara m ete r 3) beca use lab
aggress ion is regresse d on day care ex perien ce. Psi (4,1) is fixed (set at zero)
b ecause lab aggre ssio n and day care experience are no t free to co rre la te .
Also, doub le check the input matrix before exam ining the m odel results.This requires that the user be familiar with the data set and understand
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456 IISABETH F. DILALLA
T A B L E 15.2 LI S R E L Outpu t: Par am eter Specifications for Model in
Figure 15.2
Beta matrix
Day care
experience
Tempera-
ment
Parentrated
aggression
Child lab Pe er lab
aggression aggression
Day care experience 0 0 0 0 0
Temperament 1 0 0 0 0
Paren trated aggression 2 0 0 0 0
Ch ild lab aggression 3 4 0 0 0
Pe er lab aggression 0 0 0 0 0
Gamma matrix
Socio-
economic Sex of
status child
Day care experience 5 6
Temperament 7 8
Pa rentra ted aggression 9 10
Child lab aggression 11 12
Peer lab aggression 0 13
Psi matrix
Day care Tempera- Parentrated Child lab Peer lab
experience ment aggression aggression aggression
Day care experience 14
Temperament 0 15
Par entra ted aggression 0 16 17
Child lab aggression 0 0 18 19
Peer lab aggression 0 0 0 20 21
B. Check for Warnings and Errors
A fter ensuring the accuracy of the mo del and data , check the o utp ut for
any warnings or errors. Fatal errors are easy to detect because they cause
the program to crash and make i t c lear that something is wrong, but there
are a number of other types of error messages that , i f ignored, can lead to
false confidence in the o utpu t. Table 15.3 lists some of the erro rs that
begin nin g users tend to ig nore o r m is understa nd th e m ost frequen tly .
One of the most common mistakes made when specifying a model is
inverting the direction of causality. It is necessary to com pletely un de rstan d
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S STRUCTURAL EQUATION MODELING 457
fac tor to the mani fes t var iables used to approximate i t . Thus , in LISREL
nota t ion, the lambda mat r ices a re not symmetr ic and must be spec i f ied
careful ly so that the rows and columns are not reversed. Similarly, in EQS,equat ions are specified using regression semantics: the predicted variable
(the arrow head ) is regressed on (eq ua ls) the pre dictor ( the tai l of the arrow ).
A re la ted e rror tha t may he more indica t ive of a misunders tanding of
the en t i re process of SEM is the m isinterp retat io n o f causal ity. SE M an aly-
ses are based ori the correlat ion or covariance between the variables, and
i t is axiomatic that causali ty can no t b e inferred from correlat ions. A test
of the m odels with the direct ion o f cau sal i ty reverse d would yield the same
fi t index because the f i t of the model depends only on how well the model
recaptures the original covariance matrix. If two variables are correlated,
i t does not m at te r wh ether the f irst var iable is regressed on the secon d, or
vice versa: the resulting fit will be the same. As noted earlier, this under-
scores the importance of firmly grou nd ing the m odel on theory and pr ior re-
search.
Final ly, poorly defined latent variables wil l cause the program to be
underident i f ied, to fai l to converge, or to yield a very poor f i t value. The
latent factor cannot be measured adequately unless the variables used to
m easure i t are fair lv highly inte rco rrela ted . W hen the la tent variables are
i A B LE 15.3 Selected LI SR EL Warning s or Errors
Warning /e r ro r
message" Interpretat ion What to do
Solution written on
Dump f i l e
Solut ion was unable to converge: parameter
estimates are saved in a f i le called
"D um p ; li t indices and pa ram eter est i-
mates are provided because they may
help locate the problem. These es t imatesshould n o t b e i n t e r p r e t e d t h ey a re not
meaningful.
Dump va lues can
be used as sta rt
values for the
next run.
Inadmissibili ty error Solution is nonadrnissible. This happens if a
Lam bda matr ix has a row of zeroes ,
meaning a latent variable is not defined
by any m anif est v ari ab le s (e .g ., oft en
used in behavior genetic analyses).
Turn of f the ad miss i-
bi li ty te st if a ro w
of zeroes is in-
tended.
Sigma not positive
definite
The covar iance matr ix based on the model
(not the data) is not invertible: this may
resul t if one or m ore pa ram eters have
been sta rte d a t ze ro .
Change some of the
start values and
rerun the
p ro gra m .
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458 LISABETH F. DILALLA
poorly defin ed, th e lo adin gs of each of th e m anif est varia ble s on the la te n tfactor will be small, and the latent factor will not be well defined. Poorly
defined factors are of little use in predicting another factor and the result
will be a poo r m od el fit. very small and n onsignificant par am ete r estim ates,
and a model that accounts for virtually no variance in the data. Thus, one
model check is to ensure that the variables used to measure a factor are
sufficiently intercorrelated that the factor will be defined.
! X . INTERPRET ING RESULTS
Only after the user is confident that there are no input errors should the
outp ut be exam ined to add ress the substantive issues guiding the research.
Joreskog and Sorbom (1993a) describe a logical sequence to follow in
examining the analysis output.
A. Examine Parameter Estimates
First , exam ine the p aram eter estim ates to ensure that they are in the right
direction and of reasonable size. In many programs (e.g. . LISREL). the
estimates can be presented in standardized (see Table 15.4) or unstandard-
ized form. M ost users will f ind the standardized o utp ut to be m ore inte rpr et-
able. For example, the partial regression of day care experience ( .17) and
lab aggression ( .16) on SES can be seen in the G am m a m atrix o f Table
15.4. As hypothesized, SES positively predicts day care quality and nega-
tively predicts child aggression.
Also, the user should examine the standard errors of the parameters
to ensure that they are not so large that the estimate is not reasonably
determ ined. A dditionally, the squared m ultiple c orrelations and the coeffi-
cients of determ ination indicate how well the laten t variables are m easured
by the observed varia ble s. These values should range from zero to one.
with larger values indicating better fitting models.
B. Examine Fit Indices
Second, exam ine the m easures of overall fit (see Tab le 15.5). R em em ber
to consider several indices, bearing in mind that interpretation of these
indices is subjective and that you are look ing for consistency across indices.
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15. STRUCTURAL EQUATION MODELING 459
T A B L E 15.4 LIS R EL O utput: Standardized Solution for Model in
Figure 15.2
Beta matrix
Dav care T em pera- P are ntra te d Child lab P eer lab
experience ment aggression aggression aggression
Day care experience __;
Temperam ent 0.03
Parentrated aggression 0.19
Child lab aggression 0.17 0.11
Peer lab aggression
Gamma matrix
Socio-
economic Sex of
status child
Day care experience 0.17 0.11
Tem peram ent 0 .33 0.12
Paren trated aggression 0.21 0.01
Child lab aggression 0 .16 0 .52
Peer lab aggression 0.17
Psi matrix
Day care Tem pera- Parentrated Child lab Peer lab
experience ment aggression aggression aggression
Day care experience 0.96
Temperam ent 0.87
Parentrated aggression 0.31 0.93
Child lab aggression 0.28 0.70
Peer lab aggression 0.14 0.97
TA BLE 15.5 LISR EL Output: Partial List of
Goodness-of-Fit Statistics for Model in Figure 15.2
Chisquare with 4 degrees of freedom = 7.40 (p = 0.12)
Root mean square error of approximat ion (RM SE A ) = 0 .11
Root mean square residual (RMR) = 0.057
Goodnessoffi t index (GFI) = 0.97
Adjusted goodnessoffi t index (AGFI) = 0.81
N orm ed fit in dex (N F I) = 0.91N onnorm ed fit in dex (N N F I) = 0 70
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460 LISABETH F. DILALLA
hand, the CFI (.94) and the 1FI (.96) are in the acceptable range. There is
not clear consensus among the f i t indices, and therefore the most prudent
interpretat ion is that the model requires fur ther ref inement.
C. Examine Individual Aspects of the Model
The next step is to examine the standardized residuals and modif icat ion
indices to determine what aspect or aspects of the model do not f i t the
data well . This step is important because the main function of SEM is to
test a theoret ical model and determine areas that require close scrut iny
in future theory development and empir ical research. Small standardized
residuals indicate that the observed and the expected correlat ions are very
simi lar and the m odel has done an adeq uate job o f account ing for the data
(H u & B en tler. 1995). Mod ification indices assess the value of freeing
p aram ete rs th a t are cu rre n tly fixed or constra in ed (fo r in sta nce, pa ram ete rs
that are forced to be equal to o ther param eters) . For example , the cor re la-
tion path between parentrated aggression and peer aggression is f ixed at
zero in Table 15.2. Modification indices (Table 15.6) show that the largest
change in chisquare would result f rom freeing this path (element 5,3 in
the Psi matrix). The path can be freed if this makes sense theoretically,
b u t it is essen tia l to re alize th at such m odif ications resu lt in p o st hoc analy-ses. They are useful in that they generate hypotheses for future research
(Hoyle, 1995b), but unti l crossvalidated on an independent sample there
is no way to be certain that they are not capitalizing on chance associations
in this par t icular data set (see Tinsley and Brown, chapter 1, this volume,
for a discussion of crossvalidat ion procedures) . In the example presented
ear l ier , i t makes no sense theoret ical ly to correlate a parent 's rat ing of
their own ch ild 's aggression with labo ratory rat ings of the aggression of an
unfamiliar peer in the lab. Therefore, this path should not be freed in the
model , even though i t would improve the model 's f i t .
D. Testing Nested Models
T here are two w ays to create n ested m odels for determ ining the best f it ting
model for the data. One is to hypothesize a priori a full model and certain,
more restr icted models that are based in theory. For instance, i t may be
reasonable to form a model that descr ibes the relat ions among day care
experience, chi ld temperament, and child aggression ( i .e . , the ful l model
depicted in Figure 15.2) , and then on theoret ical grounds to postulate anested m odel that includes only the path f rom tem peram ent to sociabil ity
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IS. STRUCTURAL EQUATION MODELING 4 61
T A B L E 15.6 L IS R E L Ou tput: Modification Indices for Model in
Figure 15.2
Modification indices for beta
Day care
experience
Tempera Parentrated
ment aggression
Child lab
aggression
Peer lab
aggression
Day care experience __ __ 0.04 0.04
Temperament 0.19 0.19Paren trated aggression 5.35 5.35Child lab aggression _
P eer lab aggression 0.15 0.00 4.24 4.90
Modification indices
for gamma
Socio-
economic Sex of
status child
Day care experience __
Temperament Parentrated aggression
Child lab aggression
Peer lab aggression 1.29
Modification indices for Psi
Day care T em pe ra P ar en t ra te d C hild lab Pee r lab
exp erience m en t ag gre ssio n aggression aggression
Day care experience
Temperament Paren trated aggression Child lab aggression Peer lab aggression 0.04 0.19 5.35
Maximum modification index is 5.35 for element (5.3) of Psi.
full model. This nested model is equivalent to a p o st ho c test that requires
crossvalidation.
Regardless of how they are created, nested models can be statistically
compared to determine whether dropping the paths f rom the ful l model
resulted in a statistically significant decrease in model fit. The chisquare
statistic is useful for com paring nested m odels. The difference b etw een chi
squa res (i .e. , chisquare (Full) minus chisquare (N ested )) is distributed as
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462 L1SABETH F. DILAILA
that the more restr ict ive model with the greater degrees of freedom can
be accepted as the b e tte r (m ore parsim onious) m odel.
X . CONCL US ION
SEM is a flexible analytic tool that can combine regression, correlation,
and factor analyses simu ltaneou sly to add ress im portant issues in the social
sciences, biological sciences, and humanities. Readers desiring a more de-
tailed introduction to the topic will find useful treatments in Byrne (1994).
H ayd uk (1987), H oyle (1995a). and Jacca rd and W an (1996). Re lated issues
are addressed in the journal devoted to SEM enti t led Structural Equation
M odeling: A M ult idiscip lin ary Journ al, and advice about specific issues is
available from SE M N ET , the lively discussion forum that is available on
an email list server ([email protected] ).
There is stil l much to be learned about SEM, including appropriate
uses of various fit indices and the interpretation of odd or impossible
pa ram ete r estim ates, b u t its valu e has been d em o n stra ted in n um erous
studies. Furthe rm ore, nove l m ethods for addressing unique m odeling issues
continue to emerge. Fo r instance, recent developm ents include the creat ion
of phantom variables that enab le one p aram eter est im ate to be equal to a
multiple of another or that allow a group of participants who are missing
a variable to be included as a separate group rather than to be omit ted
from analysis (e.g., Loehlin, 1992; Rovine, 1994). As these innovations
continue, the application of SEM to complex research questions will in-
crease.
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