Donaldson L 1999 Performance-Drien Organizational Change:The Organizational Portfolio. Sage, Thousand Oaks, CA

Donaldson L 2001 The Contingency Theory of Organizations.Sage, Thousand Oaks, CA

Drazin R, Van de Ven A H 1985 Alternative forms of fit incontingency theory. Administratie Science Quarterly 30:514–39

Jones G R 1998 Organizational Theory: Text and Cases, 2ndedn. Addison-Wesley Publishing Company, Reading, MA

Keller R T 1994 Technology–information processing fit and theperformance of R&D project groups: a test of contingencytheory. Academy of Management Journal 37: 167–79

Kraft K L, Puia G M, Hage J 1995 Structural contingencytheory revisited: main effects versus interactions in Child’sNational Study of Manufacturing and Service Sectors. ReueCanadienne des Sciences de l’Administration—CanadianJournal of Administratie Sciences 12: 182–94

Pennings J M 1987 Structural contingency theory: a multivariatetest. Organization Studies 8: 223–40

VandeVenA H,DrazinR1985 The concept of fit in contingencytheory. In: Staw B M, Cummings L L (eds.) Research inOrganizational Behaiour. JAI Press, Greenwich, CT, Vol. 7,pp. 333–65

L. Donaldson

Structural Equation Modeling

In 1980, Peter Bentler (1980, p. 420) stated thatstructural equation modeling held ‘the greatest prom-ise for furthering psychological science.’ Since then,there have been many important theoretical andpractical advances in the field. So much so, in fact, thatMuthen (2001) announced a ‘second generation’ ofstructural equation modeling. The purpose of thisarticle is to provide a description of the ‘first gen-eration’ of structural equation modeling and to outlinebriefly the most recent developments that constitutethe ‘second generation’ of structural equation model-ing. This article will not devote space to the com-parison of specific software packages. For the mostpart, existing software packages differ in terms of theuser interface. The competitive market in structuralequation modeling software means that new develop-ments available in one software package will soon beavailable in others.

1. The First Generation: Definition and BriefHistory of Structural Equation Modeling

Structural equation modeling can be defined as a classof methodologies that seeks to represent hypothesesabout the means, variances, and covariances ofobserved data in terms of a smaller number of‘structural’ parameters defined by a hypothesizedunderlying conceptual or theoretical model. Histori-

cally, structural equation modeling derived from thehybrid of two separate statistical traditions. The firsttradition is factor analysis developed in the disciplinesof psychology and psychometrics. The second tra-dition is simultaneous equation modeling developedmainly in econometrics, but having an early history inthe field of genetics and introduced to the field ofsociology under the name path analysis.

The subject of this article concerns the combinationof the latent variablefactor analytic approach withsimultaneous equation modeling. The combination ofthese methodologies into a coherent analytic frame-work was based on the work of Jo reskog (1973),Keesling (1972) and Wiley (1973).

2. The General Model

The general structural equation model as outlined byJo reskog (1973) consists of two parts: (a) the structuralpart linking latent variables to each other via systemsof simultaneous equations, and (b) the measurementpart which links latent variables to observed variablesvia a restricted (confirmatory) factor model. Thestructural part of the model can be written as

ηBηΓξζ (1)

where η is a vector of endogenous (criterion) latentvariables, ξ is a vector of exogenous (predictor) latentvariables, B is a matrix of regression coefficientsrelating the latent endogenous variables to each other,Γ is a matrix of regression coefficients relating en-dogenous variables to exogenous variables, and ζ is avector of disturbance terms.

The latent variables are linked to observable vari-ables via measurement equations for the endogenousvariables and exogenous variables. These equationsare defined as

yΛyηε (2)

and

xΛxξδ (3)

where Λy

and Λx

are matrices of factor loadings,respectively, and ε and δ are vectors of uniqueness,respectively. In addition, the general model specifiesvariances and covariances for ξ, ζ, ε, and δ, denoted Φ,Ψ, Θε, and Θδ, respectively.

Structural equation modeling is a methodologydesigned primarily to test substantive theories. Assuch, a theory might be sufficiently developed tosuggest that certain constructs do not affect otherconstructs, that certain variables do not load oncertain factors, and that certain disturbances andmeasurement errors do not covary. This implies thatsome of the elements of B, Γ, Λ

y, Λ

x, Φ, Ψ, Θε, and Θδ

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are fixed to zero by hypothesis. The remaining par-ameters are free to be estimated. The pattern of fixedand free parameters implies a specific structure for thecovariance matrix of the observed variables. In thisway, structural equation modeling can be seen as aspecial case of a more general coariance structuremodel defined as ΣΣ(Ω), where Σ is the populationcovariancematrix andΣ(Ω) is amatrix valued functionof the parameter ector Ω containing all of theparameters of the model.

3. Identification and Estimation of StructuralEquation Models

A prerequisite for the estimation of the parameters ofthe model is to establish whether the parameters areidentified. We begin with a definition of identificationby considering the problem from the covariancestructure modeling perspective. The population co-variance matrix Σ contains population variances andcovariances that are assumed to follow a modelcharacterized by a set of parameters contained in Ω.We know that the variances and covariances in Σ canbe estimated by their sample counterparts in thesample covariance matrix S using straightforwardequations for the calculation of sample variances andcovariances. Thus, the parameters in Σ are identified.What needs to be established prior to estimation iswhether the unknown parameters in Ω are identifiedfrom the elements of Σ.

We say that the elements in Ω are identified if theycan be expressed uniquely in terms of the elements ofthe covariance matrix Σ. If all elements in Ω areidentified then the model is identified. A variety ofnecessary and sufficient conditions for establishing theidentification of the model exist and are described inKaplan (2000).

Major advances in structural equation modelinghave, arguably, been made in the area of parameterestimation. The problem of parameter estimationconcerns obtaining estimates of the free parameters ofthe model, subject to the constraints imposed by thefixed parameters of the model. In other words, the goalis to apply a model to the sample covariance matrix S,which is an estimate of the population covariancematrix Σ that is assumed to follow a structuralequation model. Estimates are obtained such that thediscrepancy between the estimated covariance matrixΣ Σ(Ω ) and the sample covariance matrix S, is assmall as possible. Therefore, parameter estimationrequires choosing among a variety of fitting functions.Fitting functions have the properties that (a) the valueof the function is greater than or equal to zero, (b) thevalue of the function equals zero if, and only if, themodel fits the data perfectly, and (c) the function iscontinuous. We first consider estimators that assumemultivariate normality of the data. We will discussestimators for non-normal data in Sect. 5.

3.1 Normal Theory-based Estimation

The most common estimation method that rests on theassumption of multivariate normality is the method ofmaximum likelihood. If the assumption of multivariatenormality of the observed data holds, the estimatesobtained from maximum likelihood estimation pos-sess certain desirable properties. Specifically, maxi-mum likelihood estimation will yield unbiased andefficient estimates as well as a large sample goodness-of-fit test.

Another fitting function that enjoys the sameproperties under the same assumption is the general-ized least squares (GLS) estimator developed for thestructural equation modeling context by Jo reskog andGoldberger (1972).

4. Testing

Once the parameters of themodel have been estimated,the model can be tested for global fit to the data. Inaddition, hypotheses concerning individual par-ameters can also be evaluated. We consider modeltesting in this section.

4.1 Testing Global Model Fit

A feature of maximum likelihood estimation is thatone can explicitly test the hypothesis that the modelfits data. The null hypothesis states that the specifiedmodel holds in the population. The correspondingalternative hypothesis is that the population covari-ance matrix is any symmetric (and positive definite)matrix. Under maximum likelihood estimation, thestatistic for testing the null hypothesis that the modelfits in the population is referred to as the likelihoodratio (LR) test. The large sample distribution of thelikelihood ratio test is chi-squared with degrees offreedom given by the difference in the number ofnonredundant elements in the population covariancematrix and the number of free parameters in themodel.

4.2 Testing Hypotheses Regarding IndiidualParameters

In addition to a global test of whether the model holdsexactly in the population, one can also test hypothesesregarding the individual fixed and freed parameters inthe model. Specifically, we can consider three alterna-tive ways to evaluate the fixed and freed elements ofthe model vis-a -vis overall fit.

The first method rests on the difference between thelikelihood ratio chi-squared statistics comparing agiven model against a less restrictive model. Recallthat the initial hypothesis, say H

, is tested against the

alternative hypothesis Hathat Σ is a symmetric positive

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definite matrix. Consider a second hypothesis, say H

that differs from H

in that a single parameter thatwas restricted to zero is now relaxed. Note that thealternative hypothesis is the same in both cases.Therefore, the change in the chi-squared value betweenthe two models can be used to test the improvement infit due to the relaxation of the restriction. Thedistribution of the difference between the two chi-squared tests is distributed as chi-squared with degreesof freedom equaling the difference in degrees-of-freedom between the model under H

and the less

restrictive model under H

. In the case of a singlerestriction described here, the chi-squared differencetest is evaluated with one degree of freedom.

There are other ways of assessing the change in themodel without estimating a second nested model. TheLagrange multiplier (LM) test can be used to assesswhether freeing a fixed parameter would result in asignificant improvement in the overall fit of the model.The LM test is asymptotically distributed as chi-squared with degrees of freedom equaling the differ-ence between the degrees of freedom of the morerestrictive model and the less restrictive model. Again,if one restriction is being evaluated, then the LM testwill be evaluated with one degree of freedom. The LMtest is also referred to as the modification index.

Finally, we can consider evaluating whether fixing afree parameter would result in a significant decrementin the fit of the model. This is referred to as the Waldtest.

The Wald test is asymptotically distributed as chi-squared with degrees of freedom equaling the numberimposed restrictions. When interest is in evaluatingone restriction the Wald test is distributed as chi-squared with one degree of freedom. The square rootof the Wald statistic in this case is a z statistic. Thereason why the LM and Wald tests can be used asalternative approaches for testing model parameters isthat they are asymptotically equivalent to each otherand to the chi-squared difference test.

4.3 Alternatie Methods of Model Fit

Since the early 1980s, attention has focused on thedevelopment of alternative indices that provide rela-tively different perspectives on the fit of structuralequation models. The development of these indiceshas been motivated, in part, by the known sensitivityof the likelihood ratio chi-squared statistic to largesample sizes. Other classes of indices have beenmotivated by a need to rethink the notion of testingexact fit in the population—an idea that is deemed bysome to be unrealistic in most practical situations.Finally, another class of alternative indices has beendeveloped that focuses on the cross-validation ad-equacy of the model.

In this section we will divide our discussion ofalternative fit indices into three categories: (a)

measures based on comparative fit to a baseline model,including those that add a penalty function for modelcomplexity, (b) measures based on population errorsof approximation, and (c) cross-validation measures.

4.3.1 Measures based on comparatie fit to a base-line model. Arguably, the most active work in thearea of alternative fit indices has been the develop-ment of what can be broadly referred to as measuresof comparatie fit. The basic idea behind these indicesis that the fit of the model is compared to the fit ofsome baseline model that usually specifies completeindependence among the observed variables. The fitof the baseline model will usually be fairly large andthus the issue is whether one’s model of interest is animprovement relative to the baseline model. The in-dex is typically scaled to lie between zero and one,with one representing perfect fit relative to this base-line model. The sheer number of comparative fit in-dices precludes a detailed discussion of each one. Wewill consider a subset of indices here that serve toillustrate the basic ideas.

The quintessential example of a comparative fitindex is the normed-fit index (NFI) proposed byBentler and Bonett (1980). This index can be written as

NFIχbχ

t

χb

(4)

where χt

bis the chi-squared for the model of complete

independence (the so-called baseline model) and χtis

the chi-squared for the target model of interest. Thisindex, which ranges from zero to one, provides ameasure of the extent to which the target model is animprovement over the baseline model.

The NFI assumes a true null hypothesis andtherefore a central chi-squared distribution of the teststatistic. However, an argument could be made thatthe null hypothesis is never exactly true and that thedistribution of the test statistic can be better approxi-mated by a non-central chi-squared with noncentral-ity parameter λ. An estimate of the noncentralityparameter can be obtained as the difference betweenthe statistic and its associated degrees of freedom.Thus, for models that are not extremely misspecified,an index developed by McDonald and Marsh (1990)and referred to as the relatie noncentrality index(RNI) can be defined as

RNI[(χ

bdf

b)(χ

tdf

t)]

χbdf

b

(5)

The RNI can lie outside the 0–1 range. To remedy this,Bentler (1990) adjusted the RNI so that it would lie inthe range of 0–1. This adjusted version is referred to asthe comparatie fit index (CFI).

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Finally, there are classes of comparative fit indicesthat adjust existing fit indexes for the number ofparameters that are estimated. These are so calledparsimony-based comparative fit indices. The rationalebehind these indices is that a model can be made to fitthe data by simply estimating more and more par-ameters. Indeed, a model that is just identified fits thedata perfectly. Therefore, an appeal to parsimonywould require that these indices be adjusted for thenumber of parameters that are estimated.

It is important to note that use of comparativeindices has not been without controversy. In par-ticular, Sobel and Borhnstedt (1985) argued early onthat these indices are designed to compare one’shypothesized model against a scientifically question-able baseline hypothesis that the observed variablesare completely uncorrelated with each other. Yet, asSobel and Borhnstedt (1985) argue, one would neverseriously entertain such a hypothesis, and that perhapsthese indices should be compared with a differentbaseline hypothesis. Unfortunately, the conventionalpractice of structural equation modeling as repre-sented in scholarly journals suggests that these indiceshave always been compared with the baseline model ofcomplete independence.

4.3.2 Measures based on errors of approximation. Itwas noted earlier that the likelihood ratio chi-squared test assesses an exact null hypothesis that themodel fits perfectly in the population. In reality, it isfairly unlikely that the model will fit the data per-fectly. Not only is it unreasonable to expect ourmodels to hold perfectly, but trivial misspecificationscan have detrimental effects on the likelihood ratiotest when applied to large samples. Therefore, a moresensible approach is to assess whether the model fitsapproximately well in the population. The difficultyarises when trying to quantify what is meant by‘approximately.’

To motivate this work it is useful to differentiateamong different types of discrepancies. First, there isthe discrepancy due to approximation, which measuresthe lack of fit of the model to the populationcovariance matrix. Second, there is the discrepancy dueto estimation, which measures the difference betweenthe model fit to the sample covariance matrix and themodel fit to the population covariance matrix. Finally,there is the discrepancy due to overall error thatmeasures the difference between the elements of thepopulation covariance matrix and the model fit to thesample covariance matrix.

Measures of approximate fit are concerned with thediscrepancy due to approximation. Based on the workof Steiger and Lind (1980) it is possible to assessapproximate fit of a model in the population. Themethod of Steiger and Lind (1980) utilizes the rootmean square error of approximation (RMSEA) formeasuring approximate fit. In utilizing the RMSEA

for assessing approximate fit, a formal hypothesistesting framework is employed. Typically an RMSEAvalue 0.05 is considered a ‘close fit.’ This valueserves to define a formal null hypothesis of close fit. Inaddition, a 90% confidence interval around theRMSEA can be formed, enabling an assessment of theprecision of the estimate (Steiger and Lind 1980).Practical guidelines recommended suggest that valuesof the RMSEA between 0.05 and 0.08 are indicativeof fair fit, while values between 0.08 and 0.10 areindicative of mediocre fit.

4.3.3 Measures that assess cross-alidationadequacy. Another important consideration in evalu-ating a structural model is whether the model iscapable of cross-validating well in a future sample ofthe same size, from the same population, and sampledin the same fashion. In some cases, an investigatormay have a sufficiently large sample to allow it to berandomly split in half with the model estimated andmodified on the calibration sample and then cross-validated on the alidation sample. When this ispossible, then thefinalfittedmodel fromthe calibrationsample is applied to the validation sample covariancematrix with parameter estimates fixed to the esti-mated values obtained from the calibration sample.

In many instances, investigators may not havesamples large enough to allow separation into cali-bration and validation samples. However, the cross-validation adequacy of the model remains a desirablepiece of information in model evaluation. Two com-parable methods for assessing cross-validation ad-equacy are the Akaike information criterion (AIC)and the expected cross-validation index (ECVI). Inboth cases, it is assumed that the investigator is in aposition to choose among a set of competing models.The model with the lowest AIC or ECVI value will bethe model that will have the best cross-validationcapability.

5. Statistical Assumptions Underlying StructuralEquation Modeling

As with all statistical methodologies, structuralequation modeling requires that certain underlyingassumptions be satisfied in order to ensure accurateinferences. The major assumptions associated withstructural equation modeling include: multivariatenormality, no systematic missing data, sufficientlylarge sample size, and correct model specification.

5.1 Sampling Assumptions

For the purposes of proper estimation and inference, avery important question concerns the sampling mech-anism. In the absence of explicit information to the

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contrary, estimation methods such as maximum like-lihood assume that data are generated according tosimple random sampling. Perhaps more often thannot, however, structural equation models are appliedto data that have been obtained through somethingother than simple random sampling. For example,when data are obtained through multistage sampling,the usual assumption of independent observations isviolated resulting in predictable biases. Structuralequation modeling expanded to consider multistagesampling will be discussed in Sect. 6.

5.2 Non-normality

A basic assumption underlying the standard use ofstructural equation modeling is that observations aredrawn from a continuous and multivariate normalpopulation. This assumption is particularly importantfor maximum likelihood estimation because the maxi-mum likelihood estimator is derived directly from theexpression for the multivariate normal distribution. Ifthe data follow a continuous and multivariate normaldistribution, then maximum likelihood attains optimalasymptotic properties, viz., that the estimates arenormal, unbiased, and efficient.

The effects of non-normality on estimates, standarderrors, and tests of model fit are well known. Theextant literature suggests that non-normality does notaffect parameter estimates. In contrast, standarderrors appear to be underestimated relative to theempirical standard deviation of the estimates. Withregard to goodness-of-fit, research indicates that non-normality can lead to substantial overestimation oflikelihood ratio chi-squared statistic, and this over-estimation appears to be related to the number ofdegrees of freedom of the model.

5.2.1 Estimators for non-normal data. The estima-tion methods discussed in Sect. 3 assumed multi-variate normal data. In the case where the data arenot multivariate normal, two general estimators existthat differ depending on whether the observed vari-ables are continuous or categorical.

One of the major breakthroughs in structuralequation modeling has been the development ofestimation methods that are not based on the as-sumption of continuous multivariate normal data.Browne (1984) developed an asymptotic distributionfree (ADF) estimator based on weighted least-squarestheory, in which the weight matrix takes on a specialform. Specifically, when the data are not multivariatenormal, the weight matrix can be expanded to in-corporate information about the skewness and kur-tosis of the data. Simulation studies have shownreasonably good performance of the ADF estimationcompared with ML and GLS.

A problem with the ADF estimator is the compu-tational difficulties encountered for models of mod-erate size. Specifically, with p variables there are u12p(p1) elements in the sample covariance matrixS. The weight matrix is of order uu. It can be seenthat the size of the weight matrix grows rapidly withthe number of variables. So, if a model was estimatedwith 20 variables, the weight matrix would contain22,155 distinct elements. Moreover, ADF estimationrequired that the sample size (for each group ifrelevant) exceed p12p(p1) to ensure that theweight matrix is non-singular. These constraints havelimited the utility of the ADF estimator in appliedsettings.

Another difficulty with the ADF estimator is thatsocial and behavioral science data are rarely con-tinuous. Rather, we tend to encounter categorical dataand often, we find mixtures of scale types within aspecific analysis. When this is the case, the assumptionsof continuity as well as multivariate normality areviolated. Muthen (1984) made a major contribution tothe estimation of structural equation models underthese more realistic conditions by providing a unifiedapproach for estimating models containing mixturesof measurement scales.

The basic idea of Muthen’s (1984) method is thatunderlying each of the categorical variables is a latentcontinuous and normally distributed variable. Thestatistics for estimating the model are correlationsamong these latent variables. For example, thePearson product–moment correlations between ob-served dichotomous variables are called phi coef-ficients. Phi coefficients can underestimate the ‘true’ orlatent correlations between the variables. Thus, whenphi coefficients are used in structural equation models,the estimates may be attenuated. In contrast, the latentcorrelations between dichotomous variables are re-ferred to as tetrachoric correlations, and their usewould result in disattenuated correlation coefficients.Other underlying correlations can also be defined.

The procedure developed by Muthen (1984) handlesmixtures of scale types by calculating the appropriatecorrelations and the appropriate weight matrix cor-responding to those correlations. These are enteredinto a weighted least-squares analysis yielding correctgoodness-of-fit tests, estimates, and standard errors.Unfortunately, aswith theADFestimator, themethodof Muthen is computationally intensive. However,research has shown how such estimation methods canbe developed that are not as computationally in-tensive.

5.3 Missing Data

Generally, statistical procedures such as structuralequation modeling assume that each unit of analysishas complete data. However, for many reasons, unitsmay be missing values on one or more of the variables

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under investigation. Numerous ad hoc approachesexist for handling missing data, including listwise andpairwise deletion. However, these approaches assumethat the missing data are unrelated to the variablesthat have missing data, or any other variable in themodel, i.e., missing completely at random (MCAR)—a heroic assumption at best. More sophisticatedapproaches for dealing with missing data derive fromthe seminal ideas of Little and Rubin (1987).

Utilizing the Rubin and Little framework, a majorbreakthrough in model-based approaches to missingdata in the structural equation modeling context wasmade simultaneously by Allison (1987) and Muthen etal. (1987). They showed how structural equationmodeling could be usedwhenmissing data aremissing-at-random (MAR), an assumption that ismore relaxedthan missing-completely-at-random.

The approach advocated by Muthen et al. (1987)was compared with standard listwise deletion andpairwise deletion approaches (which assume MCAR)in an extensive simulation study. The general findingswere that Muthen et al.’s approach was superior tolistwise and pairwise deletion approaches for handlingmissing data even under conditions where it was notappropriate to ignore the mechanism that generatedthe missing data.

The major problem associated with Muthen et al.’sapproach is that it is restricted to modeling undera relatively small number of distinct missing datapatterns. Small numbers for distinct missing data pat-terns will be rare in social and behavioral scienceapplications. However, it has proved possible to applyMAR-based approaches (Little and Rubin 1987) tomodeling missing data within standard structuralequation modeling software in a way that mitigates theproblems with Muthen et al.’s approach. Specifically,imputation approaches have become available formaximum likelihood estimation of structural modelparameters under incomplete data assuming MAR.Simulation studies have demonstrated the effective-ness of these methods in comparison to listwisedeletion and pairwise deletion under MCAR andMAR.

5.4 Specification Error

Structural equation modeling assumes that the modelisproperly specified.Specificationerror isdefinedas theomission of relevant variables in any equation of thesystem of equations defined by the structural equationmodel. This includes the measurement model equa-tions as well as the structural model equations. As withthe problems of non-normality and missing data, thequestion of concern is the extent to which omittedvariables affect inferences.

The mid-1980s saw a proliferation of studies on theproblem of specification error in structural equationmodels. On the basis of work by Kaplan and others

(summarized in Kaplan 2000) the general finding isthat specification errors in the form of omittedvariables can result in substantial parameter estimatebias. In the context of sampling variability, specifica-tion errors have been found to be relatively robust tosmall specification errors. However, tests of freeparameters in the model are affected in such a way thatspecification error in one parameter can propagate toaffect the power of the test in another parameter in themodel. Sample size also interacts with the size and typeof the specification error.

6. Modern Deelopments Constituting the‘Second Generation’

Structural equation modeling is, arguably, one of themost popular statistical methodologies available toquantitative social scientists. The popularity of struc-tural equation modeling has led to the creation of ascholarly journal devoted specifically to structuralequation modeling as well as the existence ofSEMNET, a very popular and active electronic dis-cussion list that focuses on structural equation model-ing and related issues. In addition, there are a largenumber of software programs that allow for sophis-ticated and highly flexible modeling.

In addition to ‘first-generation’ applications ofstructural equation modeling, later developments haveallowed traditionally different approaches to statisticalmodeling to be specified as special cases of structuralequation modeling. Space constraints make it im-possible to highlight all of the modern developments instructural equation modeling. However, the moreimportant modern developments have resulted fromspecifying the general model in a way that allows a‘structural modeling’ approach to other types ofanalytic strategies. The most recent example of thisdevelopment is the use of structural equation modelingto estimate multilevel data as well as specifyingstructural equation models to estimate growth par-ameters from longitudinal data.

6.1 Multileel Structural Equation Modeling

Attempts have been made to integrate multilevelmodeling with structural equation modeling so as toprovide a general methodology that can account forissues of measurement error and simultaneity as wellas multistage sampling.

Multilevel structural equation modeling assumesthat the levels of the within-group endogenous andexogenous variables vary over between-group units.Moreover, it is possible to specify a model which isassumed to hold at the between-group level and thatexplains between-group variation of the within-groupvariables. An overview and application of multilevelstructural equation modeling can be found in Kaplan(2000).

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6.2 Growth Cure Modeling from the StructuralEquation Modeling Perspectie

Growth curve modeling has been advocated for manyyears by numerous researchers for the study ofinterindividual differences in change. The specificationof growth models can be viewed as falling within theclass of multilevel linear models (Bryk andRaudenbush 1992), where level 1 represents intra-individual differences in initial status and growth andlevel 2 models individual initial status and growthparameters as a function of interindividual differences.

Research has also shown how growth curve modelscould also be incorporated into the structural equationmodeling framework utilizing existing structural equa-tion modeling software. The advantages of the struc-tural equation modeling framework to growth curvemodeling is its tremendous flexibility in specifyingmodels of substantive interest. Developments in thisarea have allowed the merging of growth curve modelsand latent class models for the purpose of isolatingunique populations defined by unique patterns ofgrowth.

7. Pros and Cons of Structural EquationModeling

Structural equation modeling is, without question,one of the most popular methodologies in the quan-titative social sciences. Its popularity can be attributedto the sophistication of the underlying statisticaltheory, the potential for addressing important sub-stantive questions, and the availability and simplicityof software dedicated to structural equation modeling.However, despite the popularity of the method, it canbe argued that ‘first-generation’ use of the method isembedded in a conventional practice that precludesfurther statistical as well as substantive advances.

The primary problem with the ‘first-generation’practice of structural equation modeling lies inattempting to attain a ‘well-fitting’ model. That is, inconventional practice, if a model does not fit from thestandpoint of one statistical criterion (e.g., the like-lihood ratio chi-squared test), then other conceptuallycontradictory measures are usually reported (e.g., theNNFI). In addition, it is not uncommon to findnumerous modifications made to an ill-fitting model tobring it in line with the data, usually supplemented bypost hoc justification for how the modification fit intothe original theoretical framework.

Perhaps the problem lies in an obsession with nullhypothesis testing—certainly an issue that has receivedconsiderable attention. Or perhaps the practice of‘first-generation’ structural equation modeling is em-bedded in the view that only a well-fitting model isworthy of being interpreted. Regardless, it has beenargued by Kaplan (2000) that this conventionalpractice precludes learning valuable informationabout the phenomena under study that could other-

wise be attained if the focus was on the predictiveability of a model. Such a focus on the predictiveability of the model combined with a change of viewtoward strict hypothesis testing might lead to furthersubstantive and statistical developments. Opportun-ities for substantive development and model improve-ment emerge when the model does not yield accurateor admissible predictions derived from theory. Oppor-tunities for statistical developments emerge when newmethods are developed for engaging in predictionstudies and evaluating predictive performance.

That is not to say that there are no bright spots inthe field of structural equation modeling. Indeed,developments in multilevel structural equation model-ing, growth curve modeling, and latent class appli-cations suggest a promising future with respect tostatistical and substantive developments. However,these ‘second-generation’ methodologies will have tobe combined with a ‘second-generation’ epistemologyso as to realize the true potential of structural equationmodeling in the array of quantitative social sciences.

See also: Causation (Theories and Models): Con-ceptions in the Social Sciences; Factor Analysis andLatent Structure, Confirmatory; Factor Analysis andLatent Structure: Overview; Growth Curve Analysis;Indicator: Methodology; Latent Structure and CausalVariables; Linear Hypothesis: Regression (Basics);Statistical Analysis: Multilevel Methods

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D. Kaplan

Structuralism

For expository reasons, a specific sense of struc-turalism will be distinguished from a loose sense. Thespecific sense is the program which Claude Le vi-Strauss, borrowing from structural linguistics, tried tointroduce into anthropology. It is defined by thevarious, not always compatible, statements by Le vi-Strauss. The loose sense is reserved for those authorswho deserve to be discussed when speaking about astyle of approach which has manifested itself indifferent ways in different times and places, but whichin the history of ideas can generally be traced back tocertain writings of Emile Durkheim and MarcelMauss. Included in this groups are figures who wrotewell before Le vi-Strauss, such as Dutch authors whomade many of the analytic discoveries found in Le vi-Strauss’s The Elementary Structures of Kinship(1969a), but who did not speak of structuralism, aswell as contemporaries of Le vi-Strauss who thoughknowledgeable of Le vi-Strauss’s writings did notregard themselves necessarily as structuralists in hissense, or did so only part of the time.

The specific sense of structuralism poses its ownproblems. Which of Le vi-Strauss’s many doctrinesand which of the many intellectual influences on Le vi-Strauss are to be regarded as structuralist? Is it notarbitrary to regard some as structuralist and to

disregard the others? Yet taken as a whole, Le vi-Strauss’s style of thought is so personal as to beimpossible to imitate in its entirety. Given the elusivenature of his language and his many self-contra-dictions, many would not want to. Le vi-Strauss’sthought and prose, for example, have been said to beinfluenced by the French symbolist poets and thesurrealists (e.g., Leach 1970, p. 16); and according tohis own testimony he has been influenced by suchcomposers as Debussy, Stravinsky, and Wagner (Le vi-Strauss 1969b, p. 15). Elsewhere (Le vi-Strauss 1955,pp. 48–50) he describes the influence on his thought ofgeology, psychoanalysis, and Marxism. ‘All threeshowed that understanding consists in the reduction ofone type of reality to another; that true reality is neverthe most obvious of realities … .’ In response Leach(1970, p. 13) has written, ‘the relevance of Marxistideology for an understanding of Le vi-Strauss is dif-ficult to determine. Le vi-Strauss’ use of dialectic, withthe formal sequence of thesis–antithesis–synthesis,is Hegelian rather than Marxist and his attitude tohistory seems to be quite contrary to Marxist dogma.’Le vi-Strauss dedicated The Saage Mind (1966) to thephenomenologist Merleau-Ponty. How are we pre-cisely to relate the influence on him of Americananthropologists such as Boas, Kroeber, and Lowie tohis structuralism?

Perhaps we are on surer ground when it comes tothe debts which he admits to owing Marcel Granet,Georges Dume zil, and Henri Gre goire (Le vi-Strauss1969b, p. 15, n. 5). Also there is less difficulty in seeingthe similarity between his structuralism and Gestaltpsychology with its emphasis or the perception ofpatterns (Le vi-Strauss 1963a, pp. 324–41). The rel-evance of Gestalt was recognized by another of hisinfluences, namely Wiener (1948, p. 18, ff.), one of thefounders of communications theory. Indeed gamestheory, the mathematical theory of communicationand cybernetics are obvious and acknowledgedsources of some of his more unquestionably structural-ist ideas (Le vi-Strauss 1963a, p. 283; for a discussionsee Ardener 1971, liv ff.). The most direct source ofcourse is the linguistic methods of the structuralistlinguists Troubetzkoy, Jakobson, and Halle (cf.Benveniste 1962). In an anticipatory response to astatement of Wiener (1948, p. 24) that sociology andanthropology lacked the long runs of statistics underconstant conditions necessary for the applications ofthe mathematics of communications theory, Le vi-Strauss wrote in a paper of 1945 on Structural Analysisin Linguistics and in Anthropology that, ‘It [linguistics]is probably the only one [social science] which cantruly claim to be a science.’ Le vi-Strauss’s scientism (acharacteristic of his thought), like that of A. R.Radcliffe-Brown, is thus programmatic rather thanrealized. However, it is the methods which haveallegedly turned linguistics into a scientific socialscience that Le vi-Strauss wants to use to achieve thesame transformation in social anthropology.

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Structural Equation Modeling

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International Encyclopedia of the Social & Behavioral Sciences ISBN: 0-08-043076-7