Structural equation modeling with Lisrel application in tourism.pdf

Tourism Management 20 (1999) 71—88

Structural equation modeling with Lisrel: application in tourism

Yvette Reisinger!,*, Lindsay Turner"

! Tourism Program, Faculty of Business and Economics, Monash University, Melbourne, Australia" Department of Applied Economics, Victoria University of Technology, Melbourne, Australia

Abstract

Structural equation modeling (SEM) is widely used in various disciplines. In the tourism discipline SEM has not been frequentlyapplied. This paper explains the concept of SEM using the Lisrel (Linear Structural Equations) approach: its major purpose,application, types of models, steps involved in formulation and testing of models, and major SEM computer software packages andtheir advantages and limitations. ( 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Structural equation modeling; Lisrel; Tourism

1. Introduction

During the last decade structural equation modeling(SEM) has increasingly been applied in the marketingdiscipline, particularly in the US. In the tourism disci-pline this technique has not been applied widely. Theconcept of SEM is difficult to understand due to itsstatistical complexity and non-user-friendly computermanuals.

This paper introduces the basic concepts associatedwith SEM using Lisrel in a comprehensive and non-technical manner. A number of technical terms are de-fined to make the paper accessible to readers not familiarwith Lisrel. The purpose of this paper is to (1) explain theconcept of SEM modeling, its major objectives and ad-vantages; (2) show how useful structural models are insolving research problems within the tourism discipline;(3) present the major steps involved in the formulationand testing of a Lisrel model through an application ofLisrel modeling to test a hypothesis about the relation-ship between the tourist perceptions of a beach resortand their satisfaction with the resort; (4) draw attentionto potential limitations associated with the Lisrel ap-proach; and (5) introduce major SEM computer softwaredescribing benefits and limitations.

*Corresponding author. Tel.: (03) 990 47028; fax: (03) 990 47050;e-mail: [email protected].

2. What is LISREL?

Lisrel stands for LInear Structural RELationships andis a computer program for covariance structure analysis.It is a multivariate technique which combines (confirma-tory) factor analysis modeling from psychometric theoryand structural equations modeling associated witheconometrics. It was originally introduced by Joreskogand Van Thillo in 1972. Currently, its 7th and 8th re-leases are available on the market.

3. Objectives of SEM

The primary aim of SEM is to explain the pattern ofa series of inter-related dependence relationships simulta-neously between a set of latent (unobserved) constructs,each measured by one or more manifest (observed)variables.

The measured (observed) variables in SEM havea finite number of values. Examples of measured vari-ables are distance, cost, size, weight or height. The meas-ured (manifest) variables are gathered from respondentsthrough data collection methods, or collected as second-ary data from a published source. They are representedby the numeric responses to a rating scale item on a ques-tionnaire. Measured variables in SEM are usuallycontinuous.

On the other hand, latent (unobserved) variables arenot directly observed, have an infinite number of values,

0261-5177/99/$ — see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 2 6 1 - 5 1 7 7 ( 9 8 ) 0 0 1 0 4 - 6

and are usually continuous. Examples of latent con-structs are attitudes, customer satisfaction, perception ofvalue or quality. Latent variables are theoretical con-structs which can only be determined to exist as a combi-nation of other measurable variables. As such they aresimilar to Principal Components and are sometimes the-oretically justified from a previous factor analysis.

In this primary form of analysis, SEM is similar tocombining multiple regression and factor analysis. Assuch the SEM expresses the linear causal relationshipbetween two separate sets of latent constructs (whichmay have been derived by two separate factor analyses).When using SEM these latent constructs are termed‘‘exogenous’’ (independent) constructs and ‘‘endogenous’’(dependent) constructs. Fig. 6 shows that endogenouslatent constructs such as repeat visitation and satisfac-tion depend on independent exogenous latent constructssuch as culture and perception.

The SEMs include one or more linear regression equa-tions that describe how the endogenous constructs de-pend upon the exogenous constructs. Their coefficientsare called path coefficients, or sometimes regressionweights.

However, there is an important difference betweenfactor analysis modeling and SEM modeling. In factoranalysis the observed variables can load on any and allfactors (constructs). The number of factors is constrained.When using SEM, confirmatory factor analysis is usedand the observed variables are loaded onto particularconstructs. The loadings are free or fixed at particularvalues, and the independence or covariance of variablesis specified.

Although the primary purpose of SEM is the analysisof latent constructs and in particular the analysis ofcausal links between latent constructs, SEM is also ca-pable of other forms of analysis. SEM can be used toestimate variance and covariance, test hypotheses, con-ventional linear regression, and factor analysis. In com-plex analysis frameworks SEM may be preferable toconventional statistical methods, for example, where it isrequired to test whether factor analysis on data fromseveral populations yields the same factor model simulta-neously. Another example is where a multiple regressionis required to test for several dependent variables fromthe same set of independent variables simultaneously,particularly if it is possible for one dependent variable tosimultaneously cause another. The SEM is a powerfulmethod for effectively dealing with multicollinearity(when two or more variables are highly correlated) whichis one of the benefit of SEM over multiple regression andfactor analysis.

All aspects of SEM modeling must be directed bytheory, which is critical for model development andmodification. A clear misuse of SEM can occur whendata are simply fitted to a suitable SEM and theory isthen expanded from the analytic result.

4. Application of SEM modeling

SEM modeling has been widely used in a numberof disciplines, including psychology (Agho et al., 1992;Shen et al., 1995), sociology (Kenny, 1996), economics(Huang, 1991), criminology (Junger, 1992), cross-national(Mullen, 1995; Singh, 1995), cross-cultural research(Riordan & Vandenberg, 1994), healthcare (Babakus& Mangold, 1992; Taylor, 1994a; Taylor & Cronin,1994), gerontology (Russell, 1990), human resourcesmanagement (Medsker et al., 1994), environmentalstudies (Nevitte & Kanji, 1995), family studies(Fu & Heaton, 1995), religious studies (Legge, 1995),migration studies (Sandu & DeJong, 1995), marketingand many others. In the marketing discipline, Lisrelhas been used in a variety of applications, includingconsumer behavior (Oliver & Swan, 1989; Singh, 1990;Fornell, 1992; Heide & Miner, 1992; Lichtensteinet al., 1993; McCarty & Shrum, 1993; Taylor & Baker,1994; Spreng et al., 1996), organizational buying behavior(Michaels et al., 1987), channel management (Schul& Babakus, 1988), product policy (DeBrentani & Droge,1988), pricing strategy (Walters & MacKenzie, 1988),advertising (MacKenzie & Lutz, 1989), salesforcemanagement (Dubinsky et al., 1986), retailing (Goodet al., 1988), international marketing (Han, 1988),services marketing (Arora & Cavusgil, 1985; Crosbyet al., 1990; Hui & Bateson, 1991; Francese, 1991; Cronin& Taylor, 1992; Brown et al., 1993; Price et al., 1995;Taylor, 1994b), and service satisfaction (Jayanti &Jackson, 1991).

The reason why SEM modeling has been applied in somany disciplines is its ability to solve research problemsrelated to causal relationships between latent constructswhich are measured by observed variables. For example,such causal relationships can be found in educationalresearch where the causes of educational achievementsand failure are analyzed or in consumer behavioral re-search where the reasons for purchasing various productsand services are analyzed. Many important marketing,psychological or cultural concepts are latent constructs,with unknown reliability, measured by multiple observedvariables. The lower the measurement reliability, themore difficult is to observe relationships between thelatent constructs and other variables. By using SEMsone can model important latent constructs while takinginto account the unreliability of the indicators.Also, many latent constructs such as perceptions,evaluation, satisfaction or behavior measures havelow reliability. By using regression one may get coeffi-cients with unexpected signs: the predictors one wouldexpect to be positively related to the dependent variableend up with negative coefficients, or vice versa. Regres-sion analysis as opposed to SEM analysis does noteliminate the difficulties caused by unreliable measures.The SEMs consider unknown reliability of the measures

72 Y. Reisinger, L. Turner / Tourism Management 20 (1999) 71—88

and rank the measures in terms of their importance(Bacon et al., 1998).

5. Application in tourism

SEM modeling has not been widely used in thetourism discipline, outside the US. However, the applica-tion of SEM in tourism is important as a tool for promo-ting better quality research. Tourism researchers areoften faced with a set of interrelated questions. Whatvariables determine tourist arrivals to a particular desti-nation? How does demand as a latent variable combinewith supply variables to affect tourist arrivals? How dodemand and supply variables simultaneously affect tour-ist purchasing decisions and holiday satisfaction? Howdoes tourist holiday satisfaction result in repeat visitationand loyalty to a destination? Many of the same indepen-dent variables affect different dependent variables withdifferent effects. Other multivariate techniques do notaddress these questions within a single comprehensivemethod.

In the tourism discipline, Lisrel modeling has recentlybeen used to assess traveler types (Nickerson & Ellis,1991), hotel guest satisfaction (Gundersen et al., 1996),service quality and satisfaction in the hotel/motel indus-try (Getty & Thompson, 1994; Thomson & Getty, 1994),tourists’ and retailers’ perceptions of service levels ina tourism destination (Vogt & Fesenmaier, 1994), cul-tural differences between Asian tourists and Australianhosts (Reisinger & Turner, 1998a) and cultural differ-ences between Korean tourists and Australian serviceproviders (Reisinger & Turner, 1998b).

Gundersen et al. (1996) identified important factors forhotel guest satisfaction among business travelers. Theanalysis covered three departments of hotel operations(receptions, food and beverage, and housekeeping) andtwo dimensions of satisfaction (tangible and intangible).The structural model showed the relationships amongthe three departments and overall satisfaction with thehotel. The highest loadings were noted on the serviceaspects of all departments. The major explanatory vari-ables for overall hotel guest satisfaction were tangibleaspects of housekeeping and intangible aspects of recep-tion service, suggesting that by focusing on these factorshigh levels of satisfaction among business travelers canbe achieved.

Getty and Thompson (1994) tested customers’ percep-tions of the lodging sector performance on multipledimensions and the perceived level of overall qualitypossessed by the lodging property, satisfaction with thelodging experience, and the customers’ willingness torecommend the property. The customers’ intentions torecommend the property were determined by their per-ceptions of the overall quality of the property, rather

than their satisfaction with the stay. The specific qualityand satisfaction dimensions, most responsible for willing-ness to recommend the property, included the generalappearance of the property, perceived value associatedwith the stay, willingness of employees to listen, and thedegree to which the property provided a safe environ-ment. Thompson and Getty (1994) suggested that cus-tomers’ intentions to provide positive word of mouth area function of their perceptions of the overall quality of thelodging property, rather than their expressed level ofsatisfaction with stay.

Vogt and Fesenmaier (1994) used four service qualitydimensions (reliability, responsiveness, assurance, acces-sibility), measured by 13 items, to evaluate service experi-ences as perceived by tourists and retailers. The resultsshowed that tourists evaluated services differently toretailers.

Reisinger and Turner (1998a) determined which cul-tural differences between Asian tourists and Australianhosts are predictors of tourist satisfaction. The resultsindicated that differences in cultural values and rules ofsocial behavior between Asian tourists and Australianhosts directly influence tourist satisfaction, and differ-ences in perceptions of service directly influence touristsocial interaction. Perceptions of service providersonly indirectly determine satisfaction, through themediating effect of interaction. Reisinger and Turner(1998b) also determined that differences in rules ofsocial behavior and perceptions of service determineKorean tourist satisfaction while differences in rules ofsocial behavior determine social interaction with Koreantourists.

6. Types of LISREL modeling

The general Lisrel model has many submodels asspecial cases. Firstly, the geometric symbols and math-ematical notations are presented below, followed bya presentation of the different submodels.

Abbreviationsx — measured independent variabley — measured dependent variablem — latent exogenous construct explained by x-variablesg — latent endogenous construct explained by y-variablesd — error for x-variablee — error for y-variablej — correlation between measured variables and all latentconstructsc — correlation between latent constructs m (exogenous)and g (endogenous)/ — correlation between exogenous latent constructs mb — correlations between endogenous latent constructs g.

Y. Reisinger, L. Turner / Tourism Management 20 (1999) 71—88 73

sch

Fig. 1. Path diagram of a hypothetical model — Submodel 1.

Submodel 1 is the Lisrel model which is designed tomeasure observed variables. The model has only x, m, andd-error variables. There are no y- and g-variables (seeprior abbreviations). This type of model is presented inthis paper as an example. The data used measure only thecorrelation between the constructs and not cause andeffect (see Fig. 1).

Submodel 2 is the Lisrel model which is designed toestimate ‘‘causal’’ relationships among directly measured‘‘causal’’ and ‘‘caused’’ variables. The model has no latentvariables but there are two kinds of directly measured

variables: x, y, and f-error variables. There are four typesof Lisrel Submodel 2: a single regression model, bivariatemodel, recursive model, and non-recursive model. Theexamples of these models are graphically presented inFigs. 2—5.

Full LISREL combines features of Submodel 1 andSubmodel 2. It involves x, y, g, m, and e, d, f-variables (seeFig. 6).

Submodel 3 is the Lisrel model with no x-variables:Submodel 3A involves only y, g, m, and e, f-error vari-ables; Submodel 3B involves only y and g-variables with


Fig. 2. A single regression model.

Fig. 3. Bivariate model.

Fig. 4. Recursive model.

no m-variables. Models 3A and 3B are not presentedvisually as they are parts of the Full Lisrel model. Sincethe Full Lisrel model has a large number of parameters,the advantage of using Submodel 3, rather than the fullLisrel, is that it has fewer parameter matrices, althougheach one is large, and it can handle models in whicha d correlates with an e (Joreskog & Sortom, 1989b).

7. The stages involved in Lisrel modeling

There are eight stages involved in the process of Lisrelmodeling and testing. These are presented in Fig. 7.

Stage 1. The first part of stage 1 focuses on the devel-opment of a theoretical model with the linkages (definedcausal relationships) between latent constructs and their

measurable variables, reflecting proposed hypotheses.This part represents the development of a structural model.

For example, the hypothesized model consists of twodimensions (latent constructs): tourist perceptions ofa beach resort and tourist satisfaction with the resort.The perception of a beach resort is measured by percep-tions of: (1) a hotel; (2) service providers; (3) leisureactivities; and (4) accessibility to the resort. The satisfac-tion with the resort is measured by satisfaction with (1)a tangible product; (2) an intangible product (service);and (3) degree of relaxation. It is hypothesized that thereis a strong relationship (correlation) between the percep-tion of a beach resort and tourist satisfaction with theresort (Submodel 1) (see Fig. 1). It can also be hy-pothesized that tourist satisfaction is determined by tour-ist perceptions of this resort (Full Lisrel) (see Fig. 6).


Fig. 5. Non-recursive model.

Fig. 6. Full Lisrel model.

The second part of stage 1 involves the operationaliz-ation of the latent constructs via the measured variablesand describing the way in which they are represented byempirical indicators (manifest variables). This part rep-resents the development of a measurement model. Thehypothetical measurement model is represented simplyby a two-construct model (perception and satisfaction) asshown in Table 1.

The first part of stage 1 also provides informationabout the validities and reliabilities of the variables. Theanalysis is predominantly confirmatory because it deter-mines the extent to which the proposed model is consis-tent with the empirical data.

The most critical point at this stage is to include all keypredictive variables (multiple indicators of the latentvariables) to avoid a specification error. The justificationfor inclusion of the specific latent constructs and theirindicators in a model can be provided by factor analysis.

However, this inclusion together with causation must betheoretically sound and be weighted against the limita-tions of SEM and computer programs. A model shouldcontain at most 20 variables (5—6 constructs each meas-ured by 3—4 indicators) (Bentler & Chou, 1987). Theinterpretation of the results and their statistical signifi-cance becomes difficult as the number of concepts be-comes large (exceeding 20).

Stage 2 involves the construction of a path diagram. Inorder to understand the geometric symbols representedin models and correctly draw a path diagram, Lisrel v.7requires familiarity with the Greek letters describingLisrel matrices. The symbols used were presentedpreviously.

In a path diagram all causal relationships betweenconstructs and their indicators are graphically presentedwith arrows. They form a visual presentation of thehypotheses and the measurement scheme. A curved line


Fig. 7. Stages involved in the application of structural equation modeling. Source: Hair et al. (1995) Multivariate Data Analysis with Readings.p. 628—629.

indicates a correlation/covariance between constructs,e.g. between perceptions and satisfaction (Submodel 1)(see Fig. 1). A straight arrow indicates a direct causalrelationship from a construct to its indicators, and directcausal—effect relationship between constructs. The directarrow from perceptions to satisfaction changes the Sub-model 1 into a Full Lisrel analysis (with y-variables) bystating that perception causes satisfaction (see Fig. 6).

All constructs fall into two categories: exogenous andendogenous. Exogenous constructs are independent vari-

ables and are not caused/predicted by any other variablein a model (there are no straight arrows pointing tothese constructs, e.g. perception in Full Lisrel). Endogen-ous constructs are predicted by other constructs andrelationships contained in the model (there are arrowspointing to these constructs, e.g. satisfaction in FullLisrel). They can also predict other endogenous con-structs. In order to avoid specification error attention hasto be paid not to omit any exogenous or endogenousconstructs.


Table 1Two-construct measurement model

Variables/indicators Indicators Loadings on constructs

Perception Satisfaction

Hotel (x1) L1

Service providers (x2) L2

Leisure activities (x3) L3

Accessibility (x4) L4

Tangible product (x5) L5

Intangible product (x6) L6

Relaxation (x7) L7

Table 2Lisrel notation for the measurement model

Exogenous Exogenous Errorindicator constructs

x1

" jx11

m1

# d1

x2

" jx21

m1

# d2

x3

" jx31

m1

# d3

x4

" jx41

m1

# d4

x5

" jx52

m2

# d5

x6

" jx62

m2

# d6

x7

" jx72

m2

# d7

Correlation among exogenous constructs (/)

m1

m2

m1

—m2

/21

—

j — correlation between manifest variables and latent constructs m.

A path diagram should show all causal relationships.The number of causal paths should be theoretically justi-fied. All relationships are to be linear.

Stage 3 involves the formal mathematical specificationof the model by describing the nature and number ofparameters to be estimated (which variables measurewhich constructs), translating the path diagram intoa series of linear equations which link constructs, andtranslating the specified model into Lisrel language in theform of matrices, indicating hypothesized correlationsamong constructs or variables. The coefficient matricesrepresent the paths in a model.

Because both constructs in the hypothetical path dia-gram are exogenous (Submodel 1), the measurementmodel and associated correlation matrices for exogenousconstructs and indicators are needed. The appropriateLisrel notation is shown in Table 2.

At this stage a distinction must be made between fixed,constrained and free parameters. Fixed parameters spec-ify values a priori and they are not estimated as part ofthe model. An example of a fixed parameter would be toassign j

11"1.00 so that j

21, j

31, and j

41would be

compared against correlations when all parameters arefree. Restricted (constrained) parameters are unknownand are estimated by the model. For example, when twoindependent variables (m

1and m

2) have the same impact

on a dependent variable (g1) one can specify that

c11"c

12(c"correlation between latent constructs). In

this case it is only necessary to estimate one parameter todetermine the value of the other parameter. Free para-meters have unknown values, are not constrained to beequal to any other parameter, and need to be estimatedby the program (Diamantopoulos, 1994).

The Lisrel v.7 analysis run is based on control com-mands which consist of several lines. The Lisrel controllines and the input are explained in the Lisrel SPSSmanuals. Since there are 5 Lisrel submodels, differentinput files need to be requested for each model. Thecontrol commands for each submodel should be de-veloped according to the instructions given in the Lisrelv.7 manual. Also, the correct specification of each linedepends on the research phenomena under study. Once

the Lisrel program is run it reproduces all command files.The matrix and the parameter specifications should beimmediately inspected to make sure that no errors havebeen made.

Stage 4 considers whether the variance/covariance orcorrelation matrix is to be used as the input data, and thisinvolves an assessment of the sample size. The covariancematrix is used when the objective is to test a theory,provide comparisons between different populations orsamples, or to explain the total variance of constructsneeded to test the theory. However, because the diagonalof the matrix is not one, interpretation of the results ismore difficult, because the coefficients must be inter-preted in terms of the units of measure for the constructs.The correlation matrix allows for direct comparisons ofthe coefficients within a model. Therefore, it is morewidely used. The correlation matrix is also used to under-stand the patterns of relationships between the con-structs. It is not used to explain the total variance ofa construct as needed in theory testing. Thus, interpreta-tion of the results and their generalizability to differentsituations should be made with caution, when the cor-relation matrix is used (Hair et al., 1995). The correlationmatrix for the hypothesized model is presented inTable 3.

The most widely used method for computing the cor-relations or covariances between manifest variables isPearson product—moment correlation and the correla-tion matrix is computed using Prelis (Joreskog &Sorbom, 1988).

Sample size plays an important role in estimating andinterpreting SEM results as well as estimating samplingerrors. Although there is no correct rule for estimatingsample size for SEM, recommendations are for a sizeranging between 100 and 200 (Hair et al., 1995). A sampleof 200 is called a ‘‘critical sample size’’. The sample size


Table 3Correlation matrix for the hypothesized model

Variables Hotel Service Leisure Access Tangible Intangible Relaxation(x

1) providers activities (x

4) product product (x

7)

(x2) (x

3) (x

5) (x

6)

x1

Hotel 1.000x2

Providers !0.349 1.000x3

Leisure 0.562 0.786 1.000x4

Access 0.612 0.677 0.432 1.000x5

Tangible 0.899 0.231 0.521 0.421 1.000x6

Intangible 0.123 0.899 0.789 0.513 0.222 1.000x7

Relaxation 0.433 0.335 0.788 0.188 0.111 0.654 1.000

should also be large enough when compared with thenumber of estimated parameters (as a rule of thumb atleast 5 times the number of parameters), but with anabsolute minimum of 50 respondents. The sample sizedepends on methods of model estimation which are dis-cussed later.

After the structural and measurement models are spe-cified and the input data type is selected, the computerprogram for model estimation should be chosen. Thereare many various programs available on the market.Although some offer different advantages, the Lisrel com-puting program has been the most widely used program.Other computer programs are discussed later.

Stage 5 addresses the issue of model identification, thatis, the extent to which the information provided by thedata is sufficient to enable parameter estimation. Ifa model is not identified, then it is not possible to deter-mine the model parameters. A necessary condition forthe identification is that the number of independent para-meters is less than or equal to the number of elements ofthe sample matrix of covariances among the observedvariables.

For example, if t parameters are to be estimated, theminimum condition for identification is

t)s,

where s"1/2(p#q)(p#q#1), p is the number of y-variables and q the number of x-variables.

z If t"s the set of parameters is just identified (there isonly one and only one estimate for each parameter).

z If t(s, the model is overidentified (it is possible toobtain several estimates of the same parameter).

z If t's, the model is unidentified (an infinite number ofvalues of the parameters could be obtained).

In a just-identified model, all the information availableis used to estimate parameters and there is no informa-tion left to test the model (df"0). In an overidentifiedmodel there are positive degrees of freedom (equal tos!t), thus, one set of estimates can be used to test themodel. In the unidentified model, one must either (1) addmore manifest variables; (2) set certain parameters to

zero; or (3) set parameters equal to each other (Aaker& Bagozzi, 1979) to make all the parameters identified.However, all three steps can be applied if they are justi-fied by theory.

The condition t)s is necessary, but not sufficient, forthe identification of a Lisrel model. In fact, there are nosufficient conditions for the full structure model. TheLisrel program provides warnings about identificationproblems (Diamantopoulos, 1994).

The symptoms of potential identification problems are:(1) very large standard errors for coefficients; (2) theinability of the program to invert the information matrix;(3) impossible estimates (e.g. negative and non-significanterror variances for any construct); and (4) high correla-tions ($0.80 or above) among observed variables. Thesesymptoms must be searched out and eliminated (Hairet al., 1995).

There are several sources of identification problems: (1)a large number of coefficients relative to the number ofcorrelations or covariances, indicated by a small numberof degrees of freedom — similar to the problems of overfit-ting, that is, insufficient sample size; (2) the use of reci-procal effects (two-way causal arrows between theconstructs); (3) failure to fix the scale of a construct, thatis, incorrect assignment of parameters as fixed or free(Hair et al., 1995); (4) skewness; (5) nonlinearity; (6) het-eroscedasticity; (7) multicollinearity; (8) singularity; and(9) autocorrelation.

It should be noted that heteroscedasticity, causedeither by nonnormality of the variables or the lack ofa direct relationship between variables, is not fatal to ananalysis. The linear relationship between variables iscaptured by the analysis but there is even more predicta-bility if the heteroscedasticity is accounted for as well. If itis not, the analysis is weakened, but not invalidated.

Multicollinearity and singularity are problems witha correlation matrix that occur when variables are toohighly correlated. For multicollinearity, the variables arevery highly correlated (0.8 and above); and for singular-ity, the variables are perfectly correlated and one of thevariables is a combination of one or more of the othervariables. When variables are multicollinear or singular,


Table 4Initial estimates (TSLS)

Variables Perception Satisfaction

Hotel 0.866 0.000Service providers 0.847 0.000Leisure activities 0.801 0.000Accessibility 0.702 0.000Tangible product 0.000 0.780Intangible product (service) 0.000 0.923Relaxation 0.000 0.930


Perception 1.000Satisfaction 0.664 1.000

Table 5Lisrel estimates (Maximum Likelihood)




Perception 1.000Satisfaction 0.666 1.000

they contain redundant information and they are not allneeded in the analysis.

The potential solutions for identification problems are:(1) to eliminate some of the estimated coefficients (delet-ing paths from the path diagram); (2) to fix the measure-ment error variances of constructs if possible, if negativechange to 0.005; (3) to fix any structural coefficients thatwere reliably known, that is, eliminate correlations overone because of multi-collinearity of variables; (4) to re-move multicollinearity by using data reduction methodslike Principal Components Analysis; (5) to eliminatetroublesome variables, e.g. highly correlated variables,redundant variables; (6) to check univariate descriptivestatistics for accuracy (e.g. out-of-range values, plausiblestandard deviations, coefficients of variation); (7) to checkfor missing values; (8) to identify nonnormal variables,e.g. check for skewness and kurtosis; (9) to check foroutliers; (10) to check for nonlinearity and heteroscedas-ticity; and (11) to reformulate the theoretical model toprovide more constructs relative to the number of rela-tionships examined.

The Lisrel v.7 program offers seven different kinds ofparameter estimation methods: instrumental variables(IV), two-stage least squares (TSLS), unweighted-leastsquares (ULS), generalized least squares (GLS), max-imum likelihood (ML), generally weighted least squares(WLS) and diagonally weighted least squares (DWLS).The most widely used are the TSLS and MLE methods.The two-stage least squares (TSLS) method computes theinitial estimates, and the maximum likelihood estimation(MLE) method computes the final solution. The TSLSmethod (as well as the IV method) of model estimation isnon-iterative and fast. The MLE method (as well as ULS,GLS, WLS, and DWLS methods) is an iterative proced-ure and it successively improves initial parameter esti-mates. The MLE method may be used to estimateparameters under the assumption of multivariate nor-mality and is robust against departures from normality.When using the MLE method the standard errors (SE)and Chi-square goodness-of-fit measures may be used ifinterpreted with caution. The MLE method is also moreprecise in large samples. The minimum sample size toensure appropriate use of MLE is 100. As the sampleincreases the sensitivity of the method to detect differ-ences among the data also increases. However, as thesample exceeds 400—500 the method becomes ‘‘too sensi-tive’’ and almost any difference is detected, making all fitmeasures poor (Hair et al., 1995). The initial and finalestimates computed by TSLS and MLE methods for thehypothetical model are presented in Tables 4 and 5.

A comparison of the TSLS with those of the final MLestimates reveals that they are very accurate. No differ-ence is larger than 0.02.

The Lisrel program also offers three types of solution:non-standardized, standardized and completely stand-ardized. In the non-standardized solution all latent and

manifest variables are non-standardized. The non-stand-ardized parameter estimates show the resulting change ina dependent variable from a unit change in an indepen-dent variable, all other variables being held constant.Non-standardized coefficients are computed with allvariables in their original metric form and describe theeffect that variables have in an absolute sense. Thus, theycan be used to compare similar models in other popula-tions. However, they are tied to the measurement units ofthe variables they represent. Any change in the measure-ment unit for an independent or dependent variablechanges the value and comparability of parametersacross populations (Bagozzi, 1977).

In the standardized solution only latent variables (con-structs) are standardized and the manifest variables (xand y) are left in their original metric. In the completelystandardized solution both the latent and the manifestvariables are standardized. The standardized parametersreflect the resulting change in a dependent variablefrom a standard deviation change in an independentvariable. The standardized parameters are appropriate tocompare the relative contributions of a number of


Table 7Measurement error for indicators

Variables Hotel Service Leisure Accessibility Tangible Intangible Relaxationproviders activities product product

Hotel 0.585Providers 0.000 0.572Leisure activities 0.000 0.000 0.453Accessibility 0.000 0.000 0.000 0.321Tangible product 0.000 0.000 0.000 0.000 !0.325Intangible product 0.000 0.000 0.000 0.000 0.000 0.422Relaxation 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Table 6Initial results of the measurement model



independent variables on the same dependent variableand for the same sample of observations. They are notappropriate to compare across populations or samples(Bagozzi, 1980).

Step 6 involves the assessment of the model fit usinga variety of fit measures for the measurement and struc-tural model (and supporting/rejecting the proposed hy-potheses). However, before evaluating the goodness-of-fitbetween the data and model several assumptions of SEMmust be met. These are: (1) independence of variables; (2)random sampling of respondents; (3) linearity of all rela-tionships; (4) multivariate normality of distribution (im-portant in the use of Lisrel); (5) no kurtosis and noskewness; (6) appropriate data measured on interval orratio scale; (7) sample size 100—400; and (8) exploratorypurpose of the study.

The above assumptions can be tested through pro-gram Prelis. The generalized least squares (GLS), whichis an alternative estimation method, can adjust for theviolations of these assumptions. However, as the modelsbecome large and complex, the use of this method be-comes more limited.

Additionally, if the use of SEM is associated withtime-series data, considerable care is required to test forautocorrelation and stationarity, and where required thedata transformed into a non-stationary series.

Once the assumptions are met, the results must first beexamined for offending estimates which are coefficientsthat exceed acceptable limits. The common examples are:

(1) negative error variances or non-significant error vari-ances for any construct; (2) standardized coefficients ex-ceeding or very close to 1.0; and (3) very large standarderrors associated with any estimated coefficient.

These offending estimates must be resolved beforeevaluating the model results. In the case of negative errorvariances (Heywood case) the offending error variancescan be changed to a very small positive value (0.005). Ifcorrelations in the standardized solution exceed 1.0, ortwo estimates are highly correlated, one of the constructsshould be removed (Hair et al., 1995). Tables 6 and7 present the Heywood case.

Tables 6 and 7 show that a loading for tangible prod-uct is greater than 1.0 (known as a Heywood case).A corresponding negative error measurement value forthe same variable is derived (!0.325). Such estimates areinappropriate and must be corrected before the modelcan be interpreted and the goodness-of-fit assessed. Inthis case, the variable will be retained and the corres-ponding negative error variance will be set to a smallvalue of 0.005 to ensure that the loading will be less than1.0. The model is then reestimated. Since in examiningthe new results, no offending estimates are found (a newloading on a tangible product is 0.996 and error varianceis 0.005), the model can be assessed for its goodness-of-fit.

When assessing model fit, attention must be paid bothto the measurement and the structural models. Fornell(1987) suggested simultaneous evaluation of both mod-els. However, Anderson and Gerbing (1982) reported thatproper evaluation of the measurement model (latent vari-ables) is a pre-requisite to the evaluation of the structuralmodel (the analysis of the causal relations among thelatent variables). The Lisrel program runs the assessmentof both models simultaneously.

There are three types of goodness-of-fit measurement:(1) absolute fit measures (assess the overall model fit,both structural and measurement together, with no ad-justment for overfitting); (2) incremental fit measures(compare the proposed model to a comparison model);and (3) parsimonious fit measures (adjust the measures offit to compare models with different numbers of coeffi-cients and determine the fit achieved by each coefficient).


In order to achieve a better understanding of the accepta-bility of the proposed model multiple measures should beapplied (Hair et al., 1995).

The absolute fit measures provide information on theextent to which the model as a whole provides an accept-able fit to the data. They are evaluated by:

(a) Likelihood ratio of Chi-square to the degrees offreedom (the acceptable range is between 0.05 and0.10—0.20). A large value of Chi-square indicates a poorfit of the model to the data, a small value of Chi-squareindicates a good fit. The degrees of freedom judgewhether the Chi-square is large or small. The number ofdegrees of freedom is calculated as

df"1/2[(p#q) (p#q#1)]!t

where p is the number of endogenous indicators, q thenumber of exogenous indicators, p#q the number ofmanifest variables and t the number of independent para-meters to be estimated.

(b) Goodness-of-fit index (GFI) which is an indicatorof the relative amount of variances and covariances joint-ly accounted for by the model; shows how closely theproposed model comes to a perfect one (takes valuesbetween 0 and 1 and the closer to unity, the better themodel fit). A marginal acceptance level is 0.90.

(c) Root-mean-square residuals (RMSR) reflect theaverage amount of variances and covariances not ac-counted for by the model. The closer to zero the betterthe fit. A marginal acceptance level is 0.08. RMSR mustbe interpreted in relation to the sizes of the observedvariances and covariances.

(d) Root-mean-square error of approximation(RMSEA).

(e) Non-centrality parameter (NCP).(f ) Scaled non-centrality parameter (SNCP).(g) Expected cross-validation index (ECVI).The NCP, SNCP and ECVI are used in comparison

among alternative models.Results for the hypothetical model:

Revised modelChi-square 15.87Degrees of freedom 10Significance level 0.08Goodness-of-fit index (GFI) 0.949Adjusted goodness-of-fit index (AGFI) 0.889Root mean square residual (RMSR) 0.056

The Chi-square value (15.87 with 10 df) has a statisticalsignificance level of 0.08, above the minimum level of0.05, but not above the more conservative levels of 0.10or 0.20. This statistic shows some support for a notionthat the differences between the predicted and actualmatrices are non-significant and it indicates an accept-able model fit. The GFI of 0.949 is quite high, but not

when adjusted for model parsimony (different number ofcoefficients). The root-mean-square residual (RMSR) in-dicates that the average residual correlation is 0.056,acceptable given strong correlations in the original cor-relation matrix.

Since these statistics provide overall measures of fitthey do not express the quality of the model. It has beenargued that the Chi-square measure-of-fit should not beregarded as the best indicator of the model fit, parti-cularly when there is data departure from normality.Lack of normality can inflate the Chi-square statisticsand create upward bias for determining significance ofthe coefficients. Also, the use of Chi-square is not valid inmost applications (Joreskog & Sorbom, 1989a). Al-though the Chi-square measure may be treated as a testof the hypothesis, the statistical problem is not one oftesting an hypothesis (which a priori might be consideredfalse), but one of fitting the model to the data, anddeciding whether the fit is adequate or not. Joreskog andSorbom (1989b) explain that in most empirical work,models are only experimental and only regarded as anapproximation to reality, and for this reason the Chi-square should not be used. Other reasons for not usinga Chi-square, as a criteria for judging the adequacy ofthe model, include sample size and problems related tothe power of the test. Large samples tend to increasethe Chi-square due to specification error in the model.Thus, Chi-square statistics should be interpreted withcaution.

Measures such as GFI, AGFI and RMSR also do notexpress perfectly the quality of the models. According toJoreskog and Sorbom (1989b), quality should be judgedby other internal and external criteria such as theoreticalgrounding. For instance, these measures can show poorfit because of one relationship only being poorly deter-mined. Thus, a fit of 0.5—0.6 does not precisely statewhether the model is or is not supported by the data. Inpractice, it can suggest that some of the poorly describedrelationship paths should be eliminated from the modelto make it more sound, and the model should be runagain to see if an improved fit can be obtained. Thesemeasures also do not indicate what is wrong with themodel (Joreskog & Sorbom, 1989b). As to the RMSR,they work best if all observed variables are standardized.

While all the absolute measures might fall within ac-ceptable levels, the incremental fit and parsimonious fitindices are needed to ensure acceptability of the modelfrom other perspectives. The incremental fit measuresassess the incremental fit of the model compared to a nullmodel (the most simple model that can be theoreticallyjustified, e.g. a single construct model related to allindicators with no measurement error). These are: (1)Tucker-Lewis measure (TL); and (2) normed fix index(NFI). All these incremental fit measures should exceedthe recommended level of minimum 0.90 to supportacceptance of the proposed model (Hair et al., 1995).


Squared multiple correlations for x-variablesHotel Providers Leisure Access Tangible Intangible Relaxation0.745 0.720 0.648 0.483 0.600 0.863 0.867

Results in the hypothetical model:

Null modelChi-square 210.876Degrees of freedom 15Significance level 0.000Tucker Lewis (TL) 0.9414Normed Fix Index (NFI) 0.9239

Both incremental fit measures exceed the recommen-ded level of 0.90 and support acceptance of the proposedmodel.

The drawback of the overall fit measures is that theydo improve as estimated coefficients are added. Thus,parsimonious fit measures should be applied. They deter-mine the model fit in comparison to models of differingcomplexity. The fit is compared versus the number ofestimated coefficients (or degrees of freedom) needed toachieve the level of fit (fit per coefficient). The two mostappropriate parsimonious fit measures are:

(1) normed Chi-square (Chi-square/df) (the recom-mended level is between 1.0 and 2.0); and

(2) adjusted for the degrees of freedom goodness-of-fitindex (AGFI) (takes values between 0 and 1; and thecloser to unity, the better the model fit). If there is a dropin AGFI as compared to GFI, the overall fit of the modelcan be questioned (Hair et al., 1995).

Results in the hypothetical model: the normed Chi-square (Chi-square divided by degrees of freedom) is1.587 (15.87/10). This falls within the recommended levelsof 1.0—2.0. The AGFI value of 0.889 is close to therecommended level of 0.90. These results show supportfor model parsimony.

In summary, the various measures of overall modelgoodness-of-fit gave support to the results of an accept-able representation of the hypothesized constructs.

The other parsimonious fit measures are: (3) parsi-mony normed fit index (PNFI); (4) parsimonious good-ness-of-fit index (PGFI); (5) comparative fit index (CFI);(6) incremental fit index (IFI); (7) relative fit index (RFI);and (8) critical N (CN) (Hair et al., 1995).

The fit of the measurement model is assessed by examin-ing squared multiple correlation coefficients (SMC) forthe y- and x-variables which indicate how well the y- andx-variables measure the latent construct, the largestamount of variance accounted for by the constructs, andthe extent to which the individual variables are free frommeasurement error. They also represent the reliabilities(convergent validities) of these measures. These coeffi-cients lie between 0 and 1 (the closer to 1, the better thevariable acts as an indicator of the latent construct).

Results in the hypothetical model:

The results show that all x-variables are goodmeasurements of both constructs.

The total coefficients of determination (TCD) (R2) forall y- and x-variables provide measures of how well the y-and x-variables as a group measure the latent constructs.The closer to 1, the better.

Results in the hypothetical model: total coefficient ofdetermination for x-variables is 0.981. All x-variables asa group measure the latent constructs very well.

The fit of the structural model is assessed by thesquared multiple correlations (SMC) for structural equa-tions which indicates the amount of variance in eachendogenous latent variable accounted for by the indepen-dent variables in the relevant structural equation, and thetotal coefficient of determination (TCD) (R2) for struc-tural equations which shows the strength of the relation-ships for all structural relationships together.Results in the hypothetical model: the R2 for the totalstructural equations is 0.788.

The results of structural equation modeling can beaffected by multicollinearity as in regression. If largevalues of correlation for the observed variables appearthe deletion of one variable or reformulation of the causalrelationships should be considered. Although there isno limit on what defines ‘‘high’’ correlation, values ex-ceeding 0.90 and even 0.80 can indicate problems, valuesbelow 0.8 can be compensated for by declaringcovariance paths between independent variables. That is,it is possible to model multicollinearity within the SEManalysis.

Each of the constructs can also be evaluated separatelyby:(1) Examining the indicators’ loadings (t-values for thepaths) for statistical significance. If the t-values asso-ciated with each of the loadings for the path coefficients,are larger than 2 the parameters are significant andvariables are significantly related to their specified con-structs, thus verifying the relationships among indicatorsand constructs.

The examination of the t-values associated with eachof the loadings in the hypothetical model indicates thatfor each variable they exceed the critical values for the0.05 significance level (critical value"1.96) and the 0.01significance level as well (critical value"2.576). Thus, allvariables are significantly related to their specified con-structs, verifying posited relationships among indicatorsand constructs (see Table 8).

(2) Examining the correlation between the latent con-structs (/ values and t-value); and Correlation amonglatent constructs (t-value in parentheses)


Table 8Construct loadings (t-values in parentheses)


Hotel 0.644 0.000(6.543)

Service providers 0.788 0.000(7.899)

Leisure activities 0.655 0.000(8.999)

Accessibility 0.566 0.000(6.888)

Tangible product 0.000 0.877(12.435)

Intangible product (service) 0.000 0.899(11.456)

Relaxation 0.000 0.901(12.345)

Fig. 8. The Q-plot for the hypothetical model.

Perception SatisfactionPerception 1.000Satisfaction 0.899 1.000

(14.567)

The examination of the correlation between the latentconstructs and the t-value show that the correlation isvery high and the t-value exceeds the critical value of1.96, indicating that the latent constructs are significantlycorrelated with each other.

(3) Assessing the standard errors (SE) for each coeffic-ient and construct. Standard errors show how accuratelythe values of the parameters are estimated. The smallerthe standard errors, the better the estimation. However,what is small or large depends on the units of measure-ment in latent constructs and the magnitude of the para-meter estimate itself. The standard errors are correctunder assumptions of multivariate normality. Theyshould be interpreted with caution if the condition ofnormality does not hold. Therefore, t-values are better tobe used as independent units of measurement.

The model fit can also be examined by assessing thefitted residuals (FR) which represent the differences be-tween the observed and the fitted correlations calculatedfrom the model. They should be relatively small to thesize of the elements of the correlation matrices, to indi-cate that the fit of the models is acceptable. However,since the fitted residuals depend on the metric of theobserved variables (the unit of measurement), they canvary from variable to variable and are difficult to use inthe assessment of fit. The problem is avoided by evaluat-ing the standardized residuals (SR) (fitted residualsdivided by their standard errors) which are independentof the metric of the observed variables and can be inter-preted as standard normal deviations. An SR whichexceeds the value of 2.58 in absolute terms indicatessubstantial specification and prediction error, for a pairof indicators (Hair et al., 1995).

However, the best picture of fit is obtained by lookingat the Q-plot which plots the standardized residuals(horizontal axis) against the quartiles of the normal dis-tribution (vertical axis). The best possible fit is obtainedwhen all residuals lie in a straight vertical line, the worstis when the residuals lie in a horizontal straight line. Anacceptable fit is indicated when the residuals lie approx-imately along the diagonal, with steeper plots (greaterthan 45°) representing better fits (see Fig. 8). If the pat-tern of residuals is non-linear this indicates departurefrom normality, linearity and/or specification errors inthe model (Joreskog & Sorbom, 1989b).

Stage 7 considers whether modifications to the modelhave to be made in the light of the results obtained at theprevious stage. At this stage the analysis becomes ex-ploratory in nature and results from previous analysis areused to develop a better fitting model. The aim is toidentify specification errors and produce a new modelwhich fits the data better. This new model has to beverified on a second independent sample.

The first modification to the model may be donethrough examination of the standardized residuals andthe modification indices. The standardized residuals(normalized) are provided by the program and representthe differences between the observed correlation/covariance and the estimated correlation/covariancematrix. Residuals values greater than $2.58 are con-sidered statistically significant at 0.05 level. Significantresiduals indicate substantial error for a pair of indi-cators. The acceptable range is one in 20 residuals ex-ceeding 2.58 by chance. In the hypothetical model onlyone value exceeded 2.58. Thus, only one correlation fromthe original input matrix has a statistically significantresidual. This falls within the acceptable range.

The other two ways in which modification to a modelcan be made is by deleting or adding parameters. In bothcases, deleting or adding parameters should be guided bytheory. Non-significant t-values can give insight as towhich parameters should be eliminated. However, ifa theory suggests that particular parameters should beincluded in the model, even non-significant parametersshould be retained because the sample size maybe too


small to detect their real significance (Joreskog &Sorbom, 1989b).

The effect of the deletion on the model fit can beassessed by comparing the Chi-square values of the twomodels, particularly, the differences in Chi-squares (D2).

The modification indices (MI) can be used to decidewhich parameters should be added to the model. The MIare measures of the predicted decrease in the Chi-squarethat results if a single parameter (fixed or constrained) isfreed (relaxed) and the model reestimated, with all otherparameters maintaining their present values. The im-provement in fit from relaxing the parameters is accept-able under the condition that only one parameter isrelaxed at a time. The model modification should neverbe based solely on the MI. Modification of the modelsmust be theoretically justified. Consideration should begiven to whether some of the parameters are not neces-sary to measure the latent constructs. According toDarden (1983), achieving a good fit at all costs is notalways recommended because a good fit for a model maytheoretically be inappropriate. There are many modelsthat could fit the data better. In fact, a poor fit tells more,that is, the degree to which the model is not supported bythe data.

If the best fitting model cannot be found using theconfirmatory strategy (where the researcher specifiesa single model and SEM to assess its significance), thenthe best fitting model can also be determined by compari-son of competing or nested models. In a competingstrategy a number of alternative models are compared tofind a model which comes closest to a theory. Differencesbetween models are indicated by the differences in theChi-square values for the different models. These differ-ences can be tested for statistical significance with theappropriate degrees of freedom. The objective is to findthe ‘best’ fit from among the set of models. In the devel-opment strategy, an initial model goes through a series ofmodel respecifications in order to improve the model fit(Hair et al., 1995).

Stage 8 involves the cross-validation of the model witha new data set. This is done by dividing the sample intotwo parts to conduct a validation test. The Lisrel abilityto analyze multi-sample analysis can be used for thispurpose. The cross-validation test is important whenmodification indices are used and the model did notprovide an acceptable fit. This test also should be usedwhen the model shows an acceptable fit in the firstanalysis. Cross-validation can also be used to comparecompeting models in terms of predictive validity andfacilitate the selection of a model; to compare the differ-ence between samples belonging to different populations,and to assess the impact of moderating variables (Dia-mantopoulos, 1994; Sharma et al., 1981).

However, cross-validation has its limitations: (1) itimplies access to the raw data; (2) the sample must belarge enough to divide it into sub-samples and generate

reliable estimates (minimum sample size should be be-tween 300 and 500 observations); and (3) bias may occurif sample splitting is done randomly (Diamantopoulos,1994).

8. Comparison of computer programs for modelestimation

The last decade has seen improvement of SEM soft-ware packages. There are several computer packagesdesigned for the analysis of SEM, e.g. Calis, Cosan,EzPath, Liscomp, and Streams. However, there are onlythree programs, Lisrel, EQS and Amos, which are popu-lar and widely used. These programs address the sameissues. However, they do vary slightly in their functions,methods of parameter estimation, number and quality offit indices, and/or efficiency. For example, Lisrel andAmos produce different values for Hoelter’s N (the lar-gest sample size for which the model is not rejected).Amos uses a 0.05 significance level for this test, and Lisrela 0.01 level. Using the 0.05 level leads to a lower criticalN than using 0.01. This means that Lisrel indicatesa good model more often than Amos does (Hox, 1995).

Lisrel is considered by most researchers as the flagshipstructural equation modeling (SEM) program. Lisrel v8with Prelis 2 provides estimates of the observed variableerror variances; demands sample size requirements; pro-vides for univariate statistics, tests for skewness and kur-tosis, multivariate normality Lisrel v.8 can imposelinear/nonlinear constraints, includes every fit statisticknown (31), including the Bentler—Bonett normal andnon-normal fit indices, computes modification indices,offers automatic path diagrams, and includes an exten-sive collection of sample programs and data sets. Thisversion allows for the analysis of categorical data (byconverting raw data into matrices of tetrachoric, poly-choric and mixed correlations prior analysis). Modelscan be easily specified with the Simplis command lan-guage and modified interactively by adding or deletingpaths on screen. Lisrel now reads SPSS system files.

Lisrel has memory allocation problems, design flaws(absence of the standardized GUI built-in editor), theability to make interactive changes to the font size andstyle for the screen and the printer, copying and pastinginformation clipboards, specifying where to save files.Lisrel permits unreasonable estimates to be printed (cor-relations greater 1.00, standard errors abnormally largeor small, negative variances). Simplis may be used tospecify simple to moderately sophisticated models only.Although Simplis minimizes the possibility of makingmistakes, the more complex procedures require the use oftraditional Lisrel syntax programs. In addition, Lisrel isalso unable to release or free up memory used duringa run, and has no support in setting up the model andanalysis options (beyond the Windows Notebook).


Amos v3.6 specifies models using a path diagram or bysyntax. Amos displays parameter estimates graphically,provides a variety of methods of parameter estimation,including MLE, GLS, ULS, a ‘‘scale-free’’ least squares,and ADF. Amos calculates numerous fit indices, includingboth absolute and comparative indices, it also computesmodification indices and estimates parameter change stat-istics as the effect of constraints on model fit. Amos can fitmultiple models into a single analysis. A multiple-groupanalysis is also possible, even with different models fordifferent groups. Amos efficiently handles missing data bycase-wise maximum likelihood. Bootstrapped standarderrors and confidence intervals are available for all esti-mates, as well as for sample means, variances, covariancesand correlations. Amos reads raw data from SPSS*.SAVand dBASE (3 and 4) files, in addition to ASCII files. AmosDraw includes the tools that produce publication-qualitypath diagrams. Although AMOS allows parameter equal-ity constraints, it does not allow for more complex con-straints such as the linear combination constraints allowedby EQS or non-linear constraints included in Lisrel v.8.Amos also does not allow the user to restrict parameterestimates to a certain range. EQS automatically prohibitsimpossible parameter values, such as negative variances.Instead, Amos, like Lisrel, allows for these impossiblevalues. Amos does not accommodate ordinal and nonmet-ric data, and it does not recognize exotic correlationcoefficients (polychoric).

EQS (EQuationS) v.5 assumes the data are multivari-ate normal and it takes non-normality into account. EQSuses the Bentler—Weeks model as a basis for testing thefull range of structural equation models. Estimation ofunreasonable values is prevented by constraining thevalue of the offending parameter to ‘‘0’’. A wide range ofgoodness-of-fit (GOF) indices are provided, includingstatistical, practical fit and model parsimony, chi-square,Normed and Non-normed indices and a revised versionof the NFI that overcomes the underestimation of fit insmall samples.

EQS 5 includes polyserial and polychoric correlationsfor treating categorical data, the Lagrange Multiplierand Wald test for model specification, and the ability toimport matrices. In addition, EQS includes theSatorra—Bentler scaled chi-square statistics, the most re-liable test statistic for evaluating covariance structuremodels under various distributions and sample sizes. Theprogram has a highly integrated visual interface withextensive data exploration and manipulation options.EQS 5 includes the model drawing tool Diagrammer,although this tool is less reliable than the Amos equiva-lent. The Amos drawing path diagram program hasa drawing tool and a number of drawing functions withthe ability to move, rotate and duplicate. The Amosdrawing tools are better than EQS, particularly in theircapacity to write the path diagram to a graphics file, andthe capacity to work through the Windows clipboard.

9. Conclusion

SEM modeling is a powerful tool enabling researchersto go beyond factor analysis into the arena of determin-ing whether one set of unobserved constructs (dimen-sions) can cause (be seen to be likely to determine)another set of dimensions. In tourism studies, it is oftenthe case that the variables under study cannot be directlyobserved or measured (for example, motivation, satisfac-tion, importance, perception) yet these unobserved vari-ables might be hypothesized to cause one another. SEManalysis is a methodology capable of handling this typeof analysis, along with more conventional regressionmodels, and simultaneous regression models, whilst ac-counting for multicollinearity and, with appropriate care,other assumptions of regression modeling.

In terms of the best software to use, there is a compro-mise to be made between ease of use, flexibility and theuser’s needs. In particular, the easier to use Amos pack-age might not meet the user’s expectations when morecomplex models are required to be estimated. Lisrel isa flagship SEM program, and probably most frequentlyused in the last five years. However, the link betweenAmos, SPSS and Windows will probably mean thatAmos will become the most widely and easily used pack-age, with more specialist use still requiring Lisrel. EQSappears at this time to stand alone and to be less flexible.For more detailed information readers should refer tospecialized literature such as the journal Structural Equa-tion Modeling.

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