Anderson & Gerbing SEM 2steps PB1988 HARD

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Psychological Bulletin1988, Vol. 103, No. 3,411-423 Copyright 1988 by the American Psychological Association, Inc.0033-2909/88/$00.75

Structural Equation Modelingin Practice: AReviewand Recommended Two-Step Approach

JamesC.AndersonJ. L. Kellogg Graduate School of ManagementNorthwesternUniversity

David W.GerbingDepartment of Management

Portland State University

In this article, we provide guidance for substantive researchers on the use of structural equationmodeling in practice for theory testing and development. Wepresent a comprehensive, two-stepmodeling approach that employs a series of nested models and sequential chi-squaredifference tests.Wediscuss the comparative advantages of this approach over aone-step approach. Considerationsin specification, assessment of fit, and respecification of measurement models using confirmatoryfactor analysis are reviewed. As background to the two-step approach, the distinction between ex-ploratory and confirmatory analysis, the distinction between complementary approaches for theorytesting versus predictive application, and some developments in estimation methods also are dis-cussed.

Substantive use of structural equation modeling has beengrowing in psychologyand the social sciences. One reason forthis is that these confirmatory methods (e.g., Bentler, 1983;Browne, 1984; Joreskog, 1978)provideresearcherswithacom-prehensive means for assessing and modifying theoreticalmodels. Assuch, theyoffer great potential for furthering theorydevelopment. Because of their relative sophistication, however,a number of problems andpitfalls in their application canhin-der this potential from being realized. The purpose of this arti-cle is to provide some guidance for substantive researchers onthe use of structural equation modeling in practice for theorytesting and development. We present a comprehensive, two-stepmodelingapproach that provides a basis for makingmeaningfulinferencesabout theoretical constructs and their interrelations,aswell asavoiding some specious inferences.

The model-building task can be thought of as the analysis oftwo conceptually distinct models (Anderson &Gerbing, 1982;Joreskog &Sorbom, 1984). A confirmatory measurement, orfactor analysis, model specifies the relations of the observedmeasures to their posited underlyingconstructs, with thecon-structs allowed to intercorrelate freely. A confirmatory struc-tural model then specifies the causal relations of the constructsto one another, as posited by some theory. With full-informa-tion estimation methods, such as those provided in the EQS(Bentler, 1985)or LISREL (Joreskog&Sorbom, 1984)programs,the measurement and structural submodels can be estimatedsimultaneously. The ability to do this in a one-step analysis ap-

This work was supported in part by the McManus Research Profes-sorshipawarded toJamesC.Anderson.

Wegratefully acknowledge the comments and suggestions of JeanneBrett, Claes Fornell, David Larcker, William Perreault, Jr.,and JamesSteiger.

Correspondence concerning this article should be addressed to JamesC. Anderson, Department of Marketing, J. L. Kellogg Graduate Schoolof Management, Northwestern University, Evanston, Illinois 60208.

proach, however, does not necessarily mean that it is the pre-ferred way to accomplish the model-building task.

In this article, we contend that there is much to gain in theorytesting and the assessment of construct validity from separateestimation (and respecification) of the measurement modelprior to the simultaneous estimation of the measurement andstructural submodels. The measurement model in conjunctionwith the structural model enables a comprehensive, confirma-toryassessment of construct validity (Bentler, 1978). Themea-surement model provides aconfirmatory assessment of conver-gent validity and discriminant validity (Campbell & Fiske,1959). Given acceptable convergent and discriminant validi-ties, the test of the structural model then constitutes a confir-matory assessment of nomological validity (Campbell, 1960;Cronbach &Meehl, 1955).

The organizationof the article is as follows: As backgroundto the two-step approach, webegin with a section in whichwediscuss the distinction between exploratory and confirmatoryanalysis, the distinction between complementary modeling ap-proaches for theory testing versus predictive application, andsome developments in estimation methods. Following this, wepresent theconfirmatorymeasurement model; discuss the needfor unidimensional measurement; and then consider the areasof specification, assessment of fit, and respecification in turn.In the next section, after briefly reviewing the confirmatorystructural model, we present a two-step modeling approachand, in doing so, discuss the comparative advantages of this two-step approach over a one-step approach.

BackgroundExploratory Versus Confirmatory Analyses

Although it is convenient to distinguish between exploratoryand confirmatory research, in practice this distinction is not asclear-cut. As Joreskog(1974) noted, "Many investigations are tosome extent both exploratoryandconfirmatory, since they involvesome variablesof known and other variablesof unknowncompc-

411


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412 JAMES C. ANDERSON AND DAVID W. GERBINGsition" (p. 2). Rather than as a strict dichotomy, then, the distinc-tion in practice between exploratory and confirmatory analysiscan be thoughtofas that ofanordered progression. Factor analysiscanbe used to illustrate this progression.

An exploratory factor analysis in which there is noprior speci-ficationof the numberof factors isexclusively exploratory. Usinga maximum likelihood (ML) or generalized least squares (GLS)exploratory program represents the next step in the progression,in that ahypothesized number of underlyingfactors can be speci-fied and the goodness of fit of the resulting solution can be tested.At this point, there is a demarcation where one moves from anexploratory programtoaconfirmatoryprogram.Now,ameasure-ment model needs to be specified apriori, although the parametervalues themselves are freelyestimated. Although this has histori-callybeen referred to asconfirmatory analysis, amore descriptiveterm might be restricted analysis, in that the valuesfor manyofthe parameters havebeen restricted a priori, typically to zero.

Because initially specified measurement models almost invari-ably fail toprovide acceptable fit, thenecessaryrespecification andreestimation using the same data mean that the analysis is notexclusively confirmatory. After acceptable fit has been achievedwith a series of respecincations, the next step in the progressionwould be to cross-validate the final model on another sampledrawn from the population towhichthe results are to begeneral-ized. This cross-validation would be accomplished by specifyingthesame model withfreelyestimated parameters or, inwhat repre-sents the quintessential confirmatory analysis, the same modelwith the parameter estimates constrained to the previously esti-mated values.Complementary Approaches for Theory Testing VersusPredictive Application

A fundamental distinction can be made between the use ofstructural equation modeling for theory testing and develop-ment versus predictive application (Fornell & Bookstein, 1982;Joreskog &Wold, 1982). This distinction and its implicationsconcern a basic choice of estimation method and underlyingmodel. For clarity, we can characterize this choice as one be-tween a full-information (ML or GLS) estimation approach(e.g., Bentler, 1983; Joreskog, 1978) in conjunction with thecommon factor model (Harman, 1976) and a partial leastsquares (PLS) estimation approach (e.g., Wold, 1982) in con-junctionwiththe principal-component model (Harman, 1976).

For theory testing and development, the ML or GLS ap-proach has several relative strengths. Underthe common factormodel, observed measures are assumed to have random errorvariance and measure-specific variance components (referredto together as uniqueness in the factor analytic literature, e.g.,Harman, 1976) that are not of theoretical interest. This un-wanted part of the observed measures is excluded from thedefinition of the latent constructs and is modeled separately.Consistent with this, the covariances among the latent con-structs are adjusted to reflect the attenuation in the observedcovariances due to these unwanted variance components. Be-cause of this assumption, the amount of variance explained inthe set of observed measures is not of primary concern. Re-flecting this, full-information methods provide parameter esti-mates that best explain the observed covariances. Two further

relative strengths of full-information approaches are that theyprovide the most efficient parameter estimates (Joreskog &Wold, 1982)and an overall test of model fit. Becauseof the un-derlying assumption of random error and measure specificity,however, there is inherent indeterminacy in the estimation offactor scores (cf. Lawley &Maxwell, 1971; McDonald & Mu-laik, 1979;Steiger, 1979). This isnot a concern in theory testing,whereas in predictive applications this will likely result in someloss ofpredictive accuracy.

For application and prediction, a PLS approach has relativestrength. Under this approach, one can assume that all observedmeasure variance is useful variance to be explained. That is,under a principal-component model, no random error varianceor measure-specific variance(i.e., unique variance) is assumed.Parameters are estimated so as to maximize the variance ex-plained ineither the set of observed measures (reflectivemode)or the set of latent variables (formative mode; Fomell &Bookstein, 1982). Fit isevaluatedon the basis ofthe percentageof variance explained in the specified regressions. Because aPLS approach estimates the latent variables as exact linearcombinations of the observed measures, it offers the advantageof exact definition of component scores. This exact definitionin conjunction with explaining a large percentage of the vari-ance in the observed measures isuseful inaccurately predictingindividuals' standings on the components.

Some shortcomings of the PLS approach also need to bementioned. Neitheran assumption of nor an assessment ofuni-dimensional measurement (discussed in the next section) ismade under a PLS approach. Therefore, the theoretical mean-ing imputed to the latent variables can be problematic. Further-more, because it is a limited-information estimation method,PLSparameter estimates are not asefficient as full-informationestimates (Fornell &Bookstein, 1982; Joreskog &Wold, 1982),andjackknife orbootstrap procedures (cf.Efron &Gong, 1983)are required to obtain estimates of the standard errors of theparameter estimates (Dijkstra, 1983). And no overall test ofmodel fit is available. Finally, PLS estimates will be asymptoti-cally correct only under the joint conditions of consistency(sample size becomes large)and consistencyat large (the num-ber of indicators per latent variable becomes large; Joreskog&Wold, 1982). In practice, the correlations between the latentvariables will tend to be underestimated, whereas the corre-lations of the observed measures with their respective latentvariableswill tend to be overestimated (Dijkstra, 1983).

These two approaches to structural equation modeling, then,canbe thought of as a complementary choice that depends onthe purpose of the research: ML or GLS for theory testing anddevelopment and PLS for application and prediction. As Jore-skog and Wold (1982) concluded, "ML is theory-oriented, andemphasizes the transition from exploratory to confirmatoryanalysis. PLS is primarily intended for causal-predictive analy-sisinsituationsofhighcomplexitybut lowtheoretical informa-tion" (p. 270). Drawingon this distinction, weconsider, in theremainder of this article, a confirmatory two-step approach totheory testinganddevelopment usingML or GLSmethods.Estimation Methods

Since the inception of contemporary structural equationmethodology in the middle 1960s (Bock & Bargmann, 1966;


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STRUCTURAL EQUATION MODELING IN PRACTICE 413Joreskog, 1966, 1967), maximum likelihood has been the pre-dominant estimation method. Under the assumption ofa multi-variate normal distribution of the observed variables, maxi-mum likelihood estimators have the desirable asymptotic, orlarge-sample, properties of being unbiased, consistent, andefficient (Kmenta, 1971). Moreover, significance testing of theindividual parameters is possible because estimates of the as-ymptotic standard errors of the parameter estimates can be ob-tained. Significance testing of overall model fit also is possiblebecause the fit function is asymptotically distributed as chi-square, adjustedby aconstant multiplier.

Although maximum likelihood parameter estimates in atleast moderately sized samples appear to be robust against amoderate violation of multivariate normality (Browne, 1984;Tanaka, 1984), the problem is that the asymptotic standard er-rors and overall chi-square test statistic appear not to be. Re-lated to this, using normal theory estimation methodswhen thedata have an underlying leptokurtic (peaked) distribution ap-pearstolead torejectionofthe null hypothesisforoverall modelfit more often than would be expected. Conversely, when theunderlying distribution isplatykurtic (flat), the opposite resultwould be expected to occur (Browne, 1984). Toaddress thesepotential problems, recent developments in estimation proce-dures, particularlyby Bentler (1983) and Browne(1982, 1984),have focused on relaxing the assumption of multivariate nor-mality.

In addition to providing more general estimation methods,thesedevelopments haveled to a more unified approach toesti-mation. The traditionalmaximum likelihoodfit function (Law-ley, 1940), based on the likelihood ratio, is

F(0) =ln| 2(9)1- ln|S| +tr[SZ(r']~ U)for p observed variables, withap Xp sample covariance matrixS, and p X p predicted covariance matrix 2(fl), where0 is thevector of specified model parameters to be estimated. The spe-cific maximum likelihood fit function in Equation 1 can be re-placed by a more general fit function, which is implemented inthe EQS program (Bentler, 1985) and in the LISREL program,beginning with Version7(Joreskog&Sorbom, 1987):

)=[s- < (2)where s is a p* X 1vector (such that p' = p(p + 1)12) of thenonduplicated elementsofthe full covariance matrix S (includ-ing the diagonal elements),a(6) is the correspondingp* X 1 vec-tor ofpredicted covariancesfrom X(0), and U isap*Xp*weightmatrix. Fit functions that can be expressed in this quadraticform define a family of estimation methods called generalizedleast squares (GLS). As can be seen directly from Equation 2,minimizing the fit function F(0) is the minimization of aweighted function of theresiduals, definedby s - a(6). The like-lihood ratio fit function of Equation 1 and the quadratic fitfunction of Equation 2 are minimized through iterative algo-rithms (cf.Rentier, 1986b).

The specific GLS method of estimation is specified by thevalue of U in Equation 2. Specifying U as I implies that mini-mizing F is the minimization of the sum of squared residuals,that is, ordinary, or "unweighted," least squares estimation. Al-ternately, when it is updated as a function of the most recent

parameter estimates obtained at each iteration during the esti-mation process, U can be chosen so that minimizing Equation2 isasymptotically equivalent to minimizing the likelihood fitfunction of Equation 1 (Browne, 1974;Lee &Jennrich, 1979).

Other choices of U result in estimation procedures that donot assume multivariate normality. The most general proce-dure, provided by Browne (1984), yields asymptotically distri-bution-free (ADF) "best" generalized least squares estimates,with corresponding statistical tests that are"asymptotically in-sensitive to the distribution of the observations" (p. 62). Theseestimators are provided by the EQS program and the LISREL 7program. TheEQSprogram refers tothese ADFGLS estimatorsas arbitrary distribution theory generalized least squares(AGLS; Bentler, 1985), whereas the LISREL 7 program refersto them as weighted least squares (WLS; Joreskog & Sorbom,1987).

The value of U for ADFestimation isnoteworthy in at leasttwo respects. First, the elements of U involve not only the sec-ond-order product moments about the respective means (vari-ances and covariances) of the observed variables but also thefourth-order product moments about the respective means.Therefore, as seen from Equation 2, although covariances arestill being fitted by the estimation process, as in traditionalmax-imum likelihood estimation, U now becomes the asymptoticcovariance matrix of the sample variances and covariances.Second, in ML or GLS estimation under multivariate normaltheory, Equation 2simplifies to a more computationally tracta-bleexpression, suchas in Equation 1. Bycontrast, in ADF esti-mation, onemust employthe full Umatrix. Forexample, whenthere are only 20observed variables, U has 22,155unique ele-ments (Browne, 1984). Thus, the computational requirementsofADF estimation canquickly surpass the capability of presentcomputers as the number of observed variables becomes mod-eratelylarge.

To address this problemof computational infeasibility whenthe number of variables is moderately large, both EQS and LIS-REL 7 useapproximations of the full ADF method. Bentler andDijkstra (1985) developed what they called linearized estima-tors, which involvea single iteration beginning from appropri-ate initial estimates, such as those provided by normal theoryML. This linearized (L) estimation procedure is referred to asLAGLSin EQS. The approximation approach implemented inLISREL 7(Joreskog&Sorbom, 1987) uses an option for ignoringthe off-diagonal elementsin U, providing whatare called diago-nally weighted least squares(DWLS)estimates.

Bentler (1985) also implemented in the EQSprogram an esti-mationapproach that assumes a somewhat more general under-lying distribution than the multivariate normal assumed forML estimation: elliptical estimation. The multivariate normaldistribution assumes that each variable has zero skewness(third-order moments) and zero kurtosis (fourth-order mo-ments). The multivariate elliptical distribution is a generaliza-tion of the multivariate normal in that the variables maysharea common, nonzero kurtosis parameter (Bentler, 1983; Beran,1979;Browne, 1984).Aswiththe multivariate normal, iso-den-sity contoursareellipsoids, but they mayreflect more platykur-tic or leptokurtic distributions, depending on the magnitudeand direction of the kurtosis parameter. The elliptical distribu-tion with regard to Equation 2 is a generalization of the multi-


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414 JAMES C. ANDERSON AND DAVID W. GERBINGvariate normal and, thus, provides moreflexibility in thetypesof data analyzed. Another advantageof this distribution is thatthe fourth-order moments can be expressed as a function of thesecond-order moments withonlythe additionof a singlekurto-sisparameter, greatlysimplifying the structure of U.

Bentler (1983) and Mooijaart and Bentler (1985) have out-lined an estimation procedure even more ambitious than anyof those presently implemented inEQSor LISREL 7. This proce-dure, called asymptotically distribution-free reweighted leastsquares (ARLS), generalizesonBrowne's(1984) ADF method.In an ADF method (or AGLSin EQS notation), U is denned asa constant before the minimization of Equation 2 begins. Bycontrast, inARLS,U isupdatedat each iterationof theminimi-zation algorithm. This updating is based on Bentler's (1983)expression of higher order moment structures, specified as afunction of the current estimates of the model parameters,thereby representing a generalization of presently estimatedsecond-order moment structures.

Inadditionto the relaxationof multivariatenormality, recentdevelopments in estimation procedures have addressed at leasttwo other issues. One problem is that when the data are stan-dardized, the covariances are not rescaled by known constantsbut by data-dependent values (i.e., standard deviations) thatwill randomly vary across samples. Becauseof this, when theobservedvariable covariances are expressed as correlations, theasymptotic standard errorsand overall chi-squaregoodness-of-fit tests are not correct without adjustments to the estimationprocedure (Bentler&Lee, 1983). Acompanion program toLIS-REL 7, PRELIS (Joreskog &Sorbom, 1987), can provide suchadjustments. Asecond problem is the use of product-momentcorrelations when the observed variables cannot beregardedascontinuous (cf.Babakus, Ferguson,&Joreskog, 1987). PRELISalso canaccount for this potential shortcomingofcurrent usagebycalculating the correct polychoric and polyserialcoefficients(Muthen, 1984)and then adjusting the estimation procedureaccordingly.

Insummary, thesenewestimation methods represent impor-tant theoretical advances. The degree, however, to which esti-mation methods that do not assume multivariate normalitywillsupplant normal theory estimation methods inpractice has yetto be determined. Many data sets may be adequately character-izedby themultivariate normal, muchas the univariate normaloften adequately describes univariate distributions of data.And, as Bentler (1983) noted, referring to the weight matrixU, "an estimated optimal weight matrix should be adjusted toreflect the strongest assumptions about the variables that maybe possible" (p. 504). Related to this, the limited number ofexistingMonte Carlo investigations of normal theory ML esti-mators applied to nonnormal data (Browne, 1984; Harlow,1985;Tanaka, 1984) hasprovided support for the robustnessofML estimation for the recoveryofparameter estimates, thoughtheir associated standard errors may be biased. Because assess-ments of the multivariate normality assumption now can bereadily made by using the EQS and PRELIS programs, a re-searcher can makean informed choice on estimation methodsinpractice, weighingthe trade-offs between the reasonablenessof anunderlying normal theory assumption and the limitationsof arbitrary theory methods (e.g., constraintsonmodel sizeand

the need for larger sample sizes, which wediscuss later in thenext section).

Confirmatory Measurement ModelsAconfirmatory factor analysis model, or confirmatory mea-

surement model, specifies the posited relations of the observedvariables to the underlying constructs, with the constructs al-lowed to intercorrelate freely. Usingthe LISREL program nota-tion, this model can be given directly from Joreskog and Sor-bom (1984, pp. 1.9-10)as

x = A + S, (3)where x is a vector of q observed measures, $ is a vector of nunderlyingfactorssuchthat n


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STRUCTURAL EQUATION MODELING IN PRACTICE 415

Pad (7) Specificationwherea is any indicator of construct anddisany indicator ofanother construct, *. Note that when =f,Equation 7 re-duces to Equation 6; that is, internal consistency represents aspecial case of external consistency. Because it often occurs inpractice that there are less than four indicators of a construct,external consistency then becomes the sole criterion for assess-ingunidimensionality.Theproduct rulesforinternalandexter-nal consistency,whichare usedinconfirmatory factor analysis,can be used to generate a predicted covariance matrix for anyspecified model and set of parameter estimates.

In building measurement models, multiple-indicator mea-surement models (Anderson& Gerbing, 1982; Hunter&Gerb-ing, 1982)arepreferred becausetheyallowthe most unambigu-ous assignment of meaning to the estimated constructs. Thereason for this is that with multiple-indicator measurementmodels, each estimated construct isdennedby at least two mea-sures, and each measureis intendedas an estimateof only oneconstruct. Unidimensional measuresof this type have been re-ferred to ascongeneric measurements (Joreskog, 1971). Bycon-trast, measurement models that contain correlated measure-ment errors or that have indicators that load on more than oneestimated construct do not represent unidimensional constructmeasurement (Gerbing & Anderson, 1984). As a result, assign-ment of meaning to such estimated constructs can be problem-atic (cf. Bagozzi, 1983;Fomell, 1983; Gerbing & Anderson,1984).

Some dissent, however, exists about the application of theconfirmatory factoranalysis model forassessing unidimension-ality. Cattell (1973, 1978) has argued that individual measuresor items, like real-life behaviors, tend to be factoriallycomplex."In other words, to show that agiven matrix is rank one is notto prove that the items are measuring a pure unitarytrait factorin common: it may be a mixture of unitary traits" (Cattell,1973,p. 382). AccordingtoCattell (1973),although these itemsare unidimensional with respect to each other, theysimply mayrepresenta"bloated specific" in the context of the true (sourcetrait) factor space. That is, the items represent a "psychologicalconcept ofsomethingthat is behaviorallyverynarrow" (Cattell,1973, p. 359).

We agreewithCattell (1973, 1978) that estimated first-orderfactors may not correspond to the constructs of interest (cf.Gerbing & Anderson, 1984). The measurement approach thatwe haveadvocated isnot, however, necessarily inconsistent withCattell's (1973, 1978) approach. The two approaches can be-come compatible whenthe levelof analysis shifts from the indi-vidual items to a corresponding set of composites denned bythese items. Further analyses of these composites could then beundertaken to isolate the constructs of interest, whichwouldbeconceptualized as higher order factors (Gerbing & Anderson,1984). One possibility is a second-order confirmatory factoranalysis as outlined by, for example, Joreskog (1971) or Weeks(1980).Another possibility is to interpret the resultingcompos-ites within an existing "reference factor system," such as the16 personality dimensions provided by Cattell (1973) for thepersonality domain.

Setting the metric of thefactors. Foridentification of themea-surement model, one must set the metric (variances) of the fac-tors. Apreferred way of doing this is to fix the diagonal of thephi matrix at 1.0, givingall factors unit variances, rather thanto arbitrarily fix the pattern coefficient for one indicator ofeachfactor at 1.0 (Gerbing &Hunter, 1982). Setting the metric inthis way allows a researcher to test the significance of each pat-tern coefficient, which is of interest, rather than to forgo thisand test whetherthe factor variances are significantly differentfrom zero, which typicallyis not of interest.

Single indicators. Although having multiple indicators foreach construct is strongly advocated, sometimes in practiceonly a single indicator of some construct isavailable. And, asmost often is the case, this indicator seemsunlikely to perfectlyestimate the construct (i.e., has no random measurement erroror measure-specificitycomponent). The question then becomes"At what values shouldthe theta-delta andlambda parametersbeset?"Toanswerthis, ideally,a researcher would like to haveanindependent estimate forthe error varianceofthe single indi-cator, perhaps drawn from prior research, but often this is notavailable.

In the absence of an independent estimate, the choiceofval-ues becomes arbitrary. In the past, a conservative value for t,,such as.1 sj, has been chosen, and its associated X has been setat .95s, (e.g., Sorbom &Joreskog, 1982).Another conservativealternative to consider is to set 9, for the single indicator at thesmallest value found for the other, estimated error variances(9(). Although this value isstill arbitrary, it has theadvantageof beingbasedoninformation specific to thegivenresearch con-text. That is, this indicator shares a respondent sample and sur-veyinstrument with the other indicators.Sample size needed. Because full-information estimationmethods depend on large-sample properties, a natural concernis the sample size needed to obtain meaningful parameter esti-mates. In a recent Monte Carlo study, Andersonand Gerbing(1984) and Gerbing and Anderson(1985)have investigated MLestimation for a number of sample sizes and a variety of con-firmatory factor models in which the normal theory assump-tion was fully met. The results of this studywere that althoughthe bias in parameter estimates is of no practical significancefor sample sizes as low as 50, for a givensample, the deviationsof the parameter estimates from their respective population val-uescan bequite large. Whereas this doesnot present a problemin statistical inference, because the standard errors computedby the LISREL program are adjusted accordingly, a sample sizeof 150 or more typically will be needed to obtain parameterestimates thathavestandard errors small enoughtobe ofpracti-cal use.

Related to this, two problems in the estimation of the mea-surement model that are morelikely to occur with small samplesizes are nonconvergence and improper solutions. (Wediscusspotential causes of these problems within the Respecificationsubsection.) Solutions are nonconvergent when an estimationmethod'scomputational algorithm, within a set number of iter-ations, is unable to arrive at values that meet prescribed, termi-nation criteria (cf. Joreskog, 1966, 1967). Solutions are im-proper when the values for one or more parameter estimates


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416 JAMES C. ANDERSON AND DAVID W. GERBINGare not feasible, suchas negative variance estimates (cf. Dillon,Kumar, & Mulani, 1987;Gerbing&Anderson, 1987;vanDriel,1978).AndersonandGerbing(1984) found thatasample sizeof150 will usuallybe sufficient to obtain a convergedand propersolution for models with three or more indicators per factor.Measurement models in which factors are denned byonlytwoindicators per factor can be problematic, however, so largersamples may be needed to obtain a converged and proper solu-tion.

Unfortunately, a practical limitation of estimation methodsthat require information from higher order moments (e.g.,ADF) is that they correspondingly require larger sample sizes.The issue is not simply that larger samples are needed to pro-vide more stable estimates for consistent estimators. Perhapsof greater concern, the statistical properties of full-informationestimators are asymptotic; that is, they have proven to be trueonly for largesamples. Thus, acritical taskis to establish guide-lines regarding minimum sample sizes for which the asymp-totic properties of these more general, arbitrary distributiontheoryestimators can be reasonably approximated.Presently, such guidelines on minimum sample sizes havenotbeen determined. Initial studies by Tanaka (1984) and Harlow(1985)suggest that a sample sizeofat least 400 or 500 is needed.Furthermore, consider Browne's (1984) comments regardingthe choice of the best generalized least squares (BGLS) estima-tors:

Wenotethat the terra "best"is used in averyrestrictedsensewithrespect to a specific asymptoticpropertywhich possibly may notcarryover to finite samples. It is possible that otherestimatorsmayhaveother properties which render them superior to BGLS estima-tors for practical applications of samples of moderatesize. (p. 68)

Related to these comments, in a small, preliminary MonteCarlostudy, Browne(1984) found theBGLS estimates providedby the asymptotic distribution-free procedure to have "unac-ceptable bias" (p. 81) for someof the parameters witha samplesizeof 500.

Assessment of FitAfter estimating a measurement model, given a converged

and proper solution, a researcher would assess how well thespecified model accounted for the data with one or more overallgoodness-of-fit indices.The LISREL program providesthe prob-abilityvalue associated with the chi-square likelihood ratio test,the goodness-of-fit index, and the root-mean-square residual(cf. Joreskog & Sorbom, 1984, pp. 1.38-42). Anderson andGerbing (1984) gave estimates of the expected values of theseindices, and their 5th-or 95th-percentile values,for avarietyofconfirmatory factor models and sample sizes. The chi-squareprobability value and the normed and nonnormed fit indices(Bentler &Bonett, 1980) are obtained from the EQS program(Bentler, 1985,p. 94).'

Convergent validity can be assessed from the measurementmodel bydetermining whether each indicator's estimated pat-tern coefficient on its posited underlying construct factor issig-nificant (greater than twice its standard error). Discriminantvalidity can be assessed for two estimated constructs by con-straining the estimated correlation parameter (0S) between

them to 1.0 and then performing a chi-square difference teston the values obtained for the constrained and unconstrainedmodels (Joreskog, 1971). "Asignificantly lowerx2 value for themodel in which the trait correlations are not constrained tounity would indicate that the traitsare not perfectly correlatedand that discriminant validity is achieved" (Bagozzi & Phillips,1982,p.476). Although thisis anecessarycondition fordemon-strating discriminant validity, the practical significance of thisdifference will depend on the research setting. This test shouldbe performed for one pair of factors at a time, rather than as asimultaneous test of all pairs of interest.2 The reason for this isthat a nonsignificant valuefor one pair of factors can be obfus-cated bybeing tested with several pairs that havesignificant val-ues. A complementary assessment of discriminant validity isto determine whether the confidence interval (two standarderrors) around the correlation estimate between the two factorsincludes 1.0.

RespecificationBecause the emphasisof this article is on structural equation

modeling inpractice, werecognize that mostoften some respec-ification of the measurement model will be required. It must bestressed, however, that respecification decisions should not bebased on statistical considerations alone but rather in conjunc-tion with theory and content considerations. Considerationoftheoryand content both greatly reduces the number of alternatemodelsto investigate (cf.Young, 1977)and reducesthe possibil-ity of taking advantage of sampling error to attain goodnessof fit.

Sometimes, the first respecification necessary is in responseto nonconvergence or an improper solution. Nonconvergencecan occur because of a fundamentally incongruent pattern ofsample covariances that is caused either by sampling error inconjunction witha properly specified model or by a misspecifi-cation. Relyingon content, one can obtain convergence for themodel by respecifying one or more problematic indicators todifferent constructsor by excludingthemfrom further analysis.

Considering improper solutions, van Driel (1978) presentedthree potential causes; sampling variations in conjunction withtrue parameter values close to zero, a fundamentally misspeci-fied model, and indefiniteness (underidentification) of themodel. VanDriel showedthat it is possible to distinguish which

1 The normed fit index (Bentler &Bonett, 1980) can also becalcu-lated by using the LISREL program. This isaccomplished by specifyingeach indicator as aseparatefactor andthenfixing lambda as anidentitymatrix, theta delta as a null matrix, and phi as a diagonal matrix withfreely estimatedvariances. Using the obtained chi-square value for thisoverall null model (xl>)> inconjunction with the chi-squarevalue (xi)from the measurement model, one can calculate the normed fit indexvalueas(xl>-Xm)/X0.2 When anumber ofchi-square difference testsareperformed for as-sessments of discriminant validity, the significance level for each testshould be adjusted tomaintain the "true" overall significance level forthe family of tests (cf.Finn, 1974). This adjustment can be given asa0 = 1 - (1- 01)', wherea0is theoverall significance level, typicallysetat .05; a, is the significance levelthat should be used foreach individualhypothesis test of discriminant validity; and / is the number of testsperformed.


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STRUCTURAL EQUATION MODELING IN PRACTICE 417of these causes is the likely one by examining the confidenceinterval constructed around the negative estimate. When posi-tive values fall within this confidence interval and the size of theinterval is comparable to that for proper estimates, the likelycause of the improper estimate is sampling error. Building onthis work, Oerbing and Anderson (1987) recently found thatfor improper estimates due to sampling error, respecifying themodel with the problematic parameter fixed at zero has no ap-preciable effect on the parameter estimates of other factors oron the overall goodness-of-fit indices. Alternately, this parame-ter can be fixed at some arbitrarily small, positive number (e.g.,.005) to preserve the confirmatory factor model (cf. Bentler,1976).

Given a converged and proper solution but unacceptableoverall fit, there are four basic ways to respecify indicators thathave not "worked out as planned": Relate the indicator to adifferent factor, delete the indicator from the model, relate theindicator to multiple factors, or use correlated measurementerrors. The first two ways preserve the potential to have unidi-mensional measurement and are preferred because of this,whereas the last twowaysdo not, therebyobfuscating the mean-ing of the estimated underlying constructs. The use of corre-lated measurement errors can be justified only when they arespecified a priori. As an example, correlated measurementer-rors may be expected in longitudinal research when the sameindicators aremeasured at multiple points intime. Bycontrast,correlated measurement errors should not be used as respecifi-cations because they take advantage of chance, at a cost of onlya single degree of freedom, with a consequent loss of interpret-ability and theoretical meaningfulness (Bagozzi, 1983; Fornell,1983). Gerbing and Anderson(1984)demonstrated how the un-critical use of correlated measurement errors for respecifica-tion, although improving goodness of fit, can mask a true un-derlying structure.

In our experience, the patterning of the residuals has beenthe most useful for locating the source of misspecification inmultiple-indicator measurement models. The LISREL programprovides normalized residuals (Joreskog & Sorbom, 1984, p.1.42), whereasthe EQSprogram (Bentler, 1985,pp. 92-93) pro-vides standardized residuals. Although Bentler and Dijkstra(1985) recently pointed out that the normalized residualsmaynot be strictly interpretable as standard normal variates (i.e.,normalized residuals greater than 1.96 in magnitude may notbe strictly interpretable as statistically significant), nonetheless,the pattern of large normalized residuals (e.g., greater than 2 inmagnitude) is still informative for respecification. For example,an indicator assigned to the wrongfactor will likely have a pat-tern of large negative normalized residuals with the other indi-catorsof the factor to which it was assigned (representing over-fitting), and when another factor on which it should belong ex-ists, an obverse pattern of large positive residuals will beobserved with the indicators of this factor (representing under-fitting). As another example, indicators that are multidimen-sional tend to have large normalized residuals (the result of ei-ther underfittingoroverfilling)wilh indicators ofmorelhanonefactor, which often represents the only large normalized resid-ual for each of these other indicators.

Useful adjuncts to the pattern of residuals are similarity (orproportionality) coefficients (Anderson & Gerbing, 1982;

Hunter, 1973) and multiple-groups analysis (cf. Anderson &Gerbing, 1982; Nunnally, 1978), each of which can readily becomputed wilh Ihe ITANprogram (Gerbing &Hunler, 1987).Asimilarity coefficient, u,j, for any two indicators, x, and Xj , canbedefined for a set of qindicators as

42i1'2 (8)

The value of this index ranges from -10to +1.0, with valuesgreater in magnilude indicating greater internal and externalconsistency for Ihe Iwo indicators. Thus, similarity coefficientsare useful because they efficiently summarize the internal andexternal consistency of the indicators with one another. Alter-nate indicators of the same underlyingfactor, therefore, shouldhave similaritycoefficients that are typically .8 or greater.

Multiple-groups analysis is a confirmatory estimationmethod thai is complementary to full-information estimationof multiple-indicator measuremenl models. Wilh multiple-groups analysis, each conslruct factor is defined as simply theunit-weighted sum of its posited indicators. The factor loadingsare simply the correlation of each indicator withIhe composite(construct factor), and the factor correlations are oblained bycorrelating Ihe composites. Communalities are computedwithin each group of indicators by iteration. By usingcommu-nalities, the resultanl indicator-factor and factor-factor corre-lations are corrected for attenuation due to measuremenl error.Because multiple-groupsanalysis estimates arecomputed fromonly those covariances of the variables in Ihe equation on whichIhe estimates are based, these estimates more clearly localizemisspecification, makingil easier to detect (Anderson & Gerb-ing, 1982). For example, if an indicator is specified as beingrelated to the wrong factor, then the multiple-groups analysisshows this by producing a higher factor loading for this indica-tor on the correct factor. Full-information melhods, byconlrasl,draw on all indicator covariances to produce estimates thatminimize the fit function (Joreskog, 1978).

In summary, a researcher should use these sources of infor-mation about respecification in an integrative manner, alongwithcontent considerations, in making decisionsabout respeci-fication. In practice, the measurement model may sometimesbe judged to provide acceptable fil even Ihough Ihe chi-squarevalue is still slatistically significant. This judgment should besupported by Ihe values of the normed fit index and the otherfit indices, particularly Ihe rool-mean-square residual index inconjunction wilh the number of large normalized or standard-ized residuals (and the absolute values of Ihe largest ones).

One-StepVersusTwo-StepModelingApproachesThe primary contention of this article is thai much is to be

gained from separate estimation and respecification of the mea-suremenl model prior to the simultaneous estimation of themeasurement andstruclural submodels. In putting forth a spe-cific two-step approach, we use the concepls of nested models,pseudo chi-square tests, and sequential chi-square differencetests (SCDTs) and draw on some recent work from quantitative


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418 JAMES C. ANDERSON AND DAVID W. GERBINGpsychology (Steiger, Shapiro, &Browne, 1985). These tests en-able a separate assessment of the adequacy of the substantivemodel of interest, apart from that of the measurement model.We firstpresent the structural modeland discuss the concept ofinterpretational confounding (Burt, 1973,1976).

A confirmatory structural model that specifies the positedcausal relations of the estimated constructs to one another canbe given directly from Joreskogand Sorbom (1984, p. 1.5).Thismodel can be expressed as

(9)where i? is a vector of m endogenous constructs, {is a vector ofn exogenous constructs, B is an m Xm matrix of coefficientsrepresenting the effects of the endogenous constructs on one an-other, r is an m X n matrix of coefficients representing theeffects of the exogenous constructs on the endogenous con-structs, and f is avector of mresiduals (errors inequations andrandom disturbance terms).

The definitional distinction between endogenous and exoge-nous constructs is simply that endogenous constructs have theircausal antecedents specified within the model under consider-ation, whereas the causes of exogenous constructs are outsidethe model and not of present interest. Note that this distinctionwas not germane in the specification of confirmatory measure-ment models, giveninEquation 3. Because of this, all observedmeasures were denoted simply as x. In contrast, when struc-tural models are specified, only observed measures of exoge-nousconstructsare denoted as x,whereas observed measuresofendogenousconstructsare denoted asy. Separate measurementsubmodels are specified for x and y (cf. Joreskog & Sorbom,1984, pp. 1.5-6), which then are simultaneously estimated withthe structural submodel.

In the presence of misspecification, the usual situation inpractice, a one-step approach in which the measurement andstructural submodels are estimated simultaneously will sufferfrom interpretation^ confounding (cf, Burt, 1973,1976). Inter-pretationalconfounding "occursas the assignment of empiricalmeaning to an unobserved variable which is other than themeaning assigned to it by an individual a priori to estimatingunknown parameters" (Burt, 1976, p. 4). Furthermore, thisempirically dennedmeaningmaychange considerably, depend-ing on the specification of free and constrained parameters forthe structural submodel. Interpretational confounding is re-flected by marked changes in the estimates of the pattern co-efficients when alternate structural modelsare estimated.

The potential for interpretational confounding isminimizedby prior separate estimation of the measurement model be-cause no constraints are placed on the structural parametersthat relate the estimated constructs to one another. Given ac-ceptable unidimensional measurement, the pattern coefficientsfrom the measurement model should change only trivially,if atall, when the measurement submodel and alternate structuralsubmodels are simultaneously estimated. With a one-step ap-proach, the presence of interpretational confounding may notbe detected, resultingin fit being maximized at the expenseofmeaningful interpretabilityof the constructs.

Recommended Two-Step Modeling ApproachFor assessing the structural model under a two-step ap-

proach, we recommend estimating a series of five nested struc-tural models. Amodel, Mj, issaid to be nested withinanothermodel, M! ,when its set of freely estimated parameters is a sub-set of those estimated in M,, and this can be denoted as M2 notethat it will not, however, be independent from these tests. Ina similar way,the SCDT comparison of M0 - MS will not beindependent from the earlier sequenceof three SCDT compari-sons (or from M, - M,). Nevertheless, the additional SCDTcomparisons ofM, - M, andMc - M, can beusefully inter-spersed with the earlier sequence of SCDT comparisons to pro-vide a decision-tree framework that enables a better under-standing of which, if any, of the three alternative theoreticalmodels should be accepted. We present one decision-treeframework for this set ofSCDTcomparisons inFigure 1.

A decision-tree framework. As can be seen from Figure 1,under this decision-tree framework, a researcher would first per-form theSCDT ofM,- M,.This SCDT providesanasymptoti-cally independent assessment of the theoretical model's expla-nation of the relations of the estimated constructs to one an-other. In other words, one can make an asymptoticallyindependent test of nomological validity. Note that becauseM, - MS isasymptotically independent ofMs, aresearchercanbuild a measurement model that has the best fit from a contentand statistical standpoint, where respecification mayhave beenemployed to accomplish this, and still provide a statistical as-sessment of the adequacy of the theoretical model of interest.

Before continuingwith this decision tree, weshould mentionanother comparative strength of the two-step approach. Notonly does theSCDT comparison ofM, - M5provideanassess-ment of fit forthe substantive modelof interest to the estimatedconstruct covariances, but it also requires the researcher tocon-sider the strength of explanation of this theoretical model overthat ofaconfirmatory measurement model. Comparing the de-grees of freedom associated with this SCDT withthe total num-ber available, [(m+n)(m+ n - l)]/2, indicates this inferentialstrength. That is, the ability to make any causal inferencesabout construct relations from correlational data depends di-rectly on the available degrees of freedom. Thus, for example,a researcher who specifies a substantive model in which eachconstruct is related by direct causal paths to all others wouldrealizefrom this test the inabilitytomakeanycausal inferences.This is because no degrees of freedom would exist for theSCDT; the theoretical "causal"model isindistinguishable fromaconfirmatory measurement model, and any causal interpreta-tion should be carefully avoided. To the extent, however, that a"considerable" proportion of possible direct causal paths arespecified as zero and there is acceptable fit, one can advancequalified causal interpretations.

The SCDT comparison of Mc Mt provides further under-standing of the explanatory ability afforded by the theoreticalmodel of interest and, irrespective of the outcome of theMt - Mscomparison, wouldbeconsidered next.Bagozzi (1984)recently noted the need to consider rival hypotheses in theoryconstruction and stressed that whenever possible, these rivalex-planations should be tested within the same study. Apart fromthis but again stressing the need to assess alternative models,MacCallum (1986) concluded from his research on specifica-tion searches that "investigators should not interpret a nonsig-nificant chi-square as a signal to stop a specification search"(p. 118). SCDTs are particularly well-suited for accomplishingthese comparisons between alternative theoretical models.

Consider first theupper branch of the decision tree in Figure1, that is, thenull hypothesis that Mt - M, = 0 is not rejected.Given this, when both the Mc M, and the Mc Ms compari-sons also are not significant, Mc would be accepted because itisthe most parsimonious structural modelofthe three hypothe-sized, theoretical alternativesand because it provides adequateexplanation of the estimated construct covariances. When ei-

3 Steiger et al. (1985)developed these analytic resultswithin thecon-text of exploratory maximum likelihood factor analysis, in which thequestionof interest is the number of factors that best representsa givencovariance matrix. However, their derivationsweredeveloped for a gen-eral discrepancy function, of which the fit function used in confirma-tory analyses of covariance structures (cf. Browne, 1984;Joreskog,1978) is a special case. Their results even extend to situations in whichthe null hypothesisneed not be true. In suchsituations, the SCDTswillstill be asymptotically independent but asymptotically distributed asnoncentral chi-square variates.

4 A recent development in the EQS program (Bentler, 1986a) is theprovision of Wald tests and Lagrange multiplier tests (cf. Buse, 1982),each ofwhichisasymptoticallyequivalent tochi-squaredifference tests.Thisallowsa researcher, withinasinglecomputer run, toobtain overallgoodness-of-fit information that isasymptotically equivalent to whatwould be obtained from separate SCDT comparisons of Mc and M0with the specified model, Mt.


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420 JAMES C. ANDERSON AND DAVID W. GERBING

M - Ms

M M,MC MsIsgn

M - Mu

Respecify Muasalternate model,Mu';then M, - Mu. a

-Accept Mu

'Relax constrant inMu thats"next-most-likely." model Mu ;thenMu2 - MS"

Figure 1. A decision-tree framework for the set of sequential chi-square difference tests (SCDTs). Mt =theoretical model of interest; M, = measurement (or "saturated")model; Mc and Mu = next most likelyconstrained and unconstrained structural models, respectively;ns andsign indicate that the null hypothesisfor each SCDTis not or is rejected, respectively, at the specified probability level {e.g., 05)."Themodelingapproach shifts from beingconfirmatory tobeing increasinglyexploratory.

ther Mc-Mt orMc-Ms issignificant, the M,-Mu comparisonwould be assessed next. If this SCDT is not significant, it indi-cates that relaxingthe next most likely constraint or constraintsfrom a theoretical perspective in M, does not significantly addto its explanation of the construct covariances, and with parsi-mony preferred when given no difference in explanation, Mtwould be accepted. Obversely, a significant result would indi-cate that the additional estimated parameter or parameters in-crementally contribute to the explanation given by M, andwould lead to the acceptance of Mu. Note that because of theadditive property of chi-square values and their associated de-grees offreedom, oneneed not performtheSCDT of Mu - MS,which must be nonsignificant given the earlier pattern of SCDTresults.

Consider now the lower branch of the decision tree, that is,the null hypothesis that Mt M5 = 0 is rejected. As with theupper branch, when both theMc - M, and Mc - Ms compari-sons are not significant, a researcher would accept Mc. The ex-planation in this situation, however, would be that one or moreparameters that were being estimated in Mt weresuperfluous inthat they were not significantly contributing to the explanationof the construct covariancesbut were"costing" their associateddegrees of freedom. Constraining these irrelevant parameters inMcgains their associated degrees of freedom, with no apprecia-ble loss of fit. As a result, although Mt - MS wassignificant,Mc MS, which has essentially the same SCDT value, is notbecause of these additional degreesof freedom.

When either the Mc - Mt orMe - Ms comparison issignifi-cant, the SCDT of M, - Mu is considered next. Given thatMt Ms has already been found significant, a nonsignificantvalue for MI Mu would indicate the need to respecify Mn as

some alternative structural model, Mu-, such that M, , and the modeling ap-proach would shift from being confirmatory to being increas-ingly exploratory. In practice, a researcher might at this pointalso constrain to zero, or "trim" any parameters from Mt thathave nonsignificant estimates (Mr).A respecification search forMU' would continue until both a significant value of Mt MU'and a nonsignificant value of MU' Ms are obtained or untilthere are no further constrained parameters that wouldbe theo-retically meaningful to relax.

Before moving on to consider the Mu Msbranch, weshouldnote that even though Mt Ms is significant and M, Mn isnot, it ispossible toobtain a SCDT value forMu - Ms that isnot significant. Althoughthis may not occur often in practice,a researcher would still not accept M0 in this situation. The ra-tionale underlying this isthat, given that M, - Ms issignificant,there must be one or more constrained parameters in M, that,when allowed to be unconstrained (as Mu"), would provide asignificant increment in the explanation of the estimated con-struct covariancesover M,; that is, theSCDT ofM,- Mtfwouldbe significant. Given this andthat Mu - M,was notsignificant,Mu. - M, must also not be significant. Therefore, Mu. wouldprovide a significant increment in explanation over M, andwould provide adequate explanation of the estimated constructcovariances.

ThefinalSCDT comparison ofMu - Ms isperformed whenM, - Mn is significant (asis M, - Ms). WhenthisSCDT value issignificant, a researcher wouldaccept Mu. The next most likely


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STRUCTURAL EQUATION MODELING IN PRACTICE 421unconstrained theoretical alternative, though less parsimoniousthan Mt, isrequired foracceptable explanationof the estimatedconstruct covariances. Finally, when Mu - Ms issignificant, aresearcher would relax one or more parameters in Mu that is"next most likely" from a theoretical perspective, yielding amodel M , such that Mu < MU2. Then, a researcher would per-form theSCOTofMU2 - Ms, and aswith M, - M0.,the model-ing approach would shift from beingconfirmatory to being in-creasingly exploratory. A respecification search for M02 wouldcontinue until a nonsignificant value of MUJ Ms is obtainedor until no further constrained parameters are theoreticallymeaningful to relax. Note that the critical distinction betweenMU2 and Mrf isthat withMU2, the respecification search contin-ues along the same theoretical direction, whereas with Mu., therespecification search calls for a change in theoretical tack. Thisis reflected by the fact that Mu


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422 JAMES C. ANDERSON AND DAVID W. GERBINGeling approach that draws on past research and experience, aswell assome recent analytic developments. Wehavealso offeredguidance regarding the specification, assessment, and respeci-fication ofconfirmatory measurement models.

As we have advocated, there is much to be gained from atwo-step approach, compared with a one-step approach, to themodel-buildingprocess. Atwo-step approach has a number ofcomparative strengths that allow meaningful inferences to bemade. First, it allows tests of the significance for all pattern co-efficients. Second, the two-step approach allows an assessmentof whether any structural model would give acceptable fit.Third, one can makeanasymptotically independent test of thesubstantive or theoretical model of interest. Related to this,because a measurement model serves as the baseline model inSCDTs, the significance of the fit for it is asymptotically inde-pendent from the SCDTs of interest. As a result, respecificationcan be made to achieve acceptable unidimensional constructmeasurement. Finally, the two-stepapproach provides apartic-ularly useful framework for formal comparisons of the substan-tive model of interest with next most likely theoretical alterna-tives.

Structural equation modeling, properly employed, offersgreat potential for theory development and construct validationin psychologyand the social sciences. If substantive researchersemploy the two-stepapproach recommended in this article andremain cognizant of the basic principles of scientific inferencethat we have reviewed, the potential of these confirmatorymethods can be better realized in practice.

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Received September 26,1986Revision received August 15, 1987

Accepted August 25,1987

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Anderson & Gerbing SEM 2steps PB1988 HARD