dilalla - structural equation.pdf

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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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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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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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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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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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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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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


15/27

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16/27

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


17/27


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


18/27


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


19/27

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


20/27

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 .


21/27


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.


22/27


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


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


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


23/27


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


24/27

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


25/27

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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psych ological research : C oncep tu a l, s tr ate g ic , and sta ti st ic al consid erati on . Jou rn a l o f

Personality and Social Psychology, 51, 11731182.

B en tler. P. M. (1980). M ultivaria te analysis with latent variables: Causal mo deling. A n n u a l

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Bentler. P. M. (1990). Comparative fit indices in structural models. Psychological Bulletin ,

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Bollen. K. A. (1989b). Structural equations with latent variables. NY: Wiley.

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cepts. applications, an d prog ram ming . Thousand Oaks . CA Sage Publ icat ions .

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modeling: Concepts, issues, and applications (pp. 138157). Thousand Oaks. CA: Sage

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Ch ou. C.*P.. &. Bentler . P. M. (1995). Estimates and tests in structural equation modeling. In

R. H. Hoyle (Ed.) . Structural equation modeling: Concepts, issues, and applications (pp.

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D iLa lla. L. F. (1998). Influences on presc ho olers social beha viors in a pee r play setting. Child

Study Journal, 28, 223244.G erbin g. D. W ., & And erson. J. C. (1993). Mo nte Carlo evaluations o f goodn essoffi t indices

for structural equation models. In K. A. Bollen & J. S. Long (Eds.) . Testing structural

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Hayduk. L. A. (1987). Structural equation modeling with LISREL: Essentials and advances.

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Hoyle. R. H. (1995a). (Ed.) . Structural equation modeling: Concepts, issues, and applications.

Thousand Oaks. CA: Sage Publications.

Hoyle. R. H. (1995b). The structural equation modeling approach: Basic concepts and funda-

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