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Structural Equation Modeling

Yvette Reisingera; Felix Mavondob a The Fox School of Business Management, School of Hospitality and Tourism Management, Temple University, Philadelphia, PA, USA b Department of Marketing in the Faculty of Business and Economics, Monash University, Melbourne, Victoria, Australia

To cite this Article Reisinger, Yvette and Mavondo, Felix(2007) 'Structural Equation Modeling', Journal of Travel &

Tourism Marketing, 21: 4, 41 — 71 To link to this Article: DOI: 10.1300/J073v21n04_05 URL: http://dx.doi.org/10.1300/J073v21n04_05

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Structural Equation Modeling: Critical Issues and New Developments Yvette Reisinger Felix Mavondo

Downloaded By: [Monash University] At: 07:37 22 June 2011

ABSTRACT. This article focuses on a variety of research issues related to structural equation modelling. Methodological and analytical issues when applying SEM techniques are discussed. Guidelines on how to present and report SEM results in academic publications are provided, and the most common SEM computer software identified. The goal is to promote improved usage of SEM in tourism research. doi:10.1300/J073v21n04_05 [Article copies available for a fee from The Haworth Document Delivery Service: 1-800-HAWORTH. E-mail address: Website: © 2006 by The Haworth Press, Inc. All rights reserved.]

KEYWORDS. Structural Equation Modelling (SEM), tourism research

INTRODUCTION In the past 15 years structural equation modelling (SEM) has become an important tool in applied multivariate analysis for theory testing and causal modelling. It is a powerful statistical technique successfully used i n social and psychological and behavioural science research. However, SEM has not been widely applied in the tourism discipline. In the last decade the number of tourism studies that applied SEM has been increasing. The SEM techniques are statistically complex. The existing SEM software packages provide a large amount of information that creates uncertainty as to what should be examined, presented and reported to the reader. Investigators need to become well

acquainted with the critical issues and problems that occur in SEM application to enable a reader to obtain complete and accurate information, and to correctly evaluate the SEM findings. The authors assume the reader is familiar with the basic concepts and terminology of SEM. Readers who are not familiar with SEM concepts are advised to refer to the article “Structural equation modelling with LISREL: Application in tourism” by Y. Reisinger and L. Turner published in Tourism Management 20(1), 1999, pp.71-80, which gives a comprehensive introduction to the terminology. Readers should also consult more specialized literature such as the journal Structural Equation Modelling. SEM program manuals and textbooks should be, however, consulted first.

Dr. Yvette Reisinger is affliated with The Fox School of Business Management, School of Hospitality and Tourism Management, Temple University, 1700 North Broad Street, Philadelphia, PA 19122. Felix Mavondo (E-mail: [email protected]. edu.au) is Professor in the Department of Marketing in the Faculty of Business and Economics at Monash University, Clayton campus, Wellington Road, Melbourne, Victoria 3800, Australia. Journal of Travel & Tourism Marketing, Vol. 21(4) 2006 Available online at http://jttm.haworthpress.com Ó 2006 by The Haworth Press, Inc. All rights reserved. doi:10.1300/J073v21n04_05

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JOURNAL OF TRAVEL & TOURISM MARKETING


WHAT IS STRUCTURAL EQUATION MODELLING? Structural equation modelling (SEM) is an important multivariate technique. It simultaneously estimates and tests a series of hypothesized inter-related dependency relationships between a set of latent (unobserved) constructs, each measured by one or more manifest (observed) variables. The term ‘structural’ assumes that the parameters reveal a causal relationship. However, this is not always the case. At best, the technique shows whether the causal assumptions embedded in a model fit the data (Bollen, 1989). Even an appropriate use of this technique does not imply that correct causal inferences can be drawn from the results of an SEM-based analysis. Such inferences require additional assumptions concerning the context of a study and its data (Bentler, 1989). The SEM technique combines multiple regression and factor analysis. It expresses the linear relationship between latent constructs, which can be either exogenous (independent) or endogenous (dependent). In SEM linear regression equations describe how the endogenous constructs depend upon the exogenous constructs. The relationships between variables are expressed as coefficients termed path coefficients, or regression weights. The latent constructs may be derived from exploratory factor analyses. However, in exploratory factor analysis the observed variables can load on any and all factors, in SEM, confirmatory factor analysis is used and the observed variables can load on only one factor. SEM ASSUMPTIONS Several assumptions must be met to carry out the SEM analysis. These are: (1) linearity of all relationships; (2) homoscedasticity; (3) multivariate normality; (4) no kurtosis and no skewness; (5) no extreme cases such as outliers; (6) data measured on interval or ratio scale; (7) sample size 100-400 (or a minimum ratio of five times more cases than the number of independent variables; (8) discriminant validity of measures; (9) random sampling (except for longitudinal studies); and (10) independence of

error (not correlated to each other and to latent factors). SEM BENEFITS The primary purpose of SEM is to test and analyse interrelationships among latent constructs and their measured variables. The technique carries out the conventional linear regression and confirmatory factor analysis, estimates variances and covariances, and evaluates the adequacy of fit of a theoretical model to data. The technique allows for the development and contrasting of alternative models. In complex analysis SEM may be used to test associations among several dependent and independent variables simultaneously, or determine whether confirmatory factor analysis (CFA) on data from several populations yields the same factor structure. The SEM technique has considerable potential for theory testing and development as well as validation of constructs (Anderson, 1987; Anderson and Gerbing, 1988). SEM is a more powerful technique than multiple regressions and analysis of variance techniques. However, these techniques allow one relationship to be examined at a time (Hair et al., 2002). Structural equation modelling has the ability to accommodate multiple interrelated dependence relationship in a single model. Besides, structural equation modelling can be used to examine the nature and magnitude of postulated dependence relationships and at the same time assess the direct and indirect relationships of these variables (Schumacker and Lomax, 1996). Direct effects are measured by structure coefficient; a path coefficient. These beta (␤) values indicate the resultant change in a dependent variable as a consequence of a one unit change in an independent variable. Indirect effects are calculated where the independent variable influences the dependent variable through the path connecting each to two or more other variables. The beta value represents the resultant change in dependent variable as a consequence of a one unit change in an independent variable that is attributed to the indirect relationship. Total effects of an independent variable on a dependent variable are calculated by summing the direct and

Yvette Reisinger and Felix Mavondo

indirect effects (Schumacker and Lomax, 1996). Synonymous for SEM are covariance structure analysis, covariance structure modelling, and analysis of covariance structures.

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METHODOLOGICAL AND ANALYTICAL ISSUES IN SEM APPLICATION Model Conceptualization


APPLICATION OF SEM IN TOURISM RESEARCH SEM has been widely used in a number of disciplines, including psychology, sociology, economics, cross-cultural research, management, environmental studies, and marketing. During the last decade SEM has been used in a number of tourism studies because many tourism, marketing, psychological or cultural concepts are latent constructs measured by multiple observed variables (e.g., tourist satisfaction, perceptions, attitudes, values, loyalty). Tourism researchers frequently wish to identify the variables that determine tourist satisfaction with a particular destination, or how loyalty affects tourist purchasing decisions, or how tourist attitudes result in repeat visitation. SEM techniques address these questions within a single comprehensive approach. Recent examples of tourism studies that applied SEM are identified in Table 1. APPROACHES TO SEM SEM is usually viewed as a confirmatory rather than exploratory procedure, using one of three approaches: • Strictly confirmatory approach: a model is tested using SEM goodness of fit tests to determine if the pattern of variances and covariances in the data is consistent with a hypothesized model. • Model development approach: confirmatory and exploratory analysis is combined. A model is tested, and if found to be unacceptable, an alternative model is tested based on changes suggested by modification indexes. Cross-validation strategy is applied; the model is developed using a calibration data sample and then confirmed on an independent validation sample. • Alternative models approach: several causal models are tested to determine which has the best fit.

Structural equation modelling approach involves developing structural and measurement models. Structural Model Structural model is a theoretical model that shows structural relations among latent constructs (exogenous and endogenous) and their observed variables, together with the direct arcs connecting them, and the disturbance terms for these constructs. The linkages between the latent constructs reflect proposed hypotheses. The structural model represents the combined measurement model and path model. Measurement Model Figure 1 reflects a measurement model with two latent variables (perception and satisfaction) connected by a double headed curved line, which represents correlation. This provides a measure of their relationship which may arise from unmeasured influences not in the model. Each of the latent variables has several observed measures used to measure it. Each measured variable has an error associated with it (ei). These indicate the variance in the measured items not accounted for by the latent factor. Figure 2 reflects a measurement model with two latent variables (perception and satisfaction) connected by a one-headed arrow, which indicates direct relationship between constructs and their indicators. Measurement Model–Worked Example The measurement model specifies causal relations between latent constructs and their measures (one or more measured variables) and shows the way in which the latent constructs are operationalised via the measured variables (indicators) (see Table 2). The measurement model also provides overall model fit, which is

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TABLE 1. SEM Research Studies in Tourism Author (s)


Hwang, S., Lee Ch. & Chen, HJ.

Year

Journal

Theme

2005

Tourism Management

The relationships among tourists' involvement, place attachment and interpretation satisfaction in Taiwan's national parks

Kang, I., Jeon, SG., Lee, 2005 SJ. & Lee, Ch.

Tourism Management

The influence of interpersonal relationships between cockpit and cabin crews on the effectiveness of airline service operation

Barker, M., Page, S. & Meyer, D.

2003

Journal of Travel Research

Factors affecting crimes against tourists in urban areas

Lee, Ch & Back, K.J.

2003

Annals of Tourism Research

casino impact of residents' perception

Pritchard, M.

2003

Tourism Analysis

The attitudinal and behavioural consequences of destination performance

Ko, DW. $ Stewart, W.P. 2002

Tourism Management

Residents' perceptions of tourism impacts and attitudes toward tourism development

Seiler, V., Hsieh, S., Seiler, M. & Hsieh, C.

Journal of Travel & Tourism Marketing

Determinants of travel expenditures for Taiwanese travellers visiting the USA

Bigne, E., Sanchez, M. & 2001 Sanchez, J.

Tourism Management

The relationship between the image of a destination as perceived by tourists and their behavioural intentions and the post-purchase evaluation of the stay

Turner, L. & Reisinger, Y. 2001

Journal of Retailing and Determinants of shopping satisfaction Consumer Services

Yoon, YS., Gursoy, D. & Chen, J.

2001

Tourism Management

The effects of tourism impacts on local residents support for tourism development

Wober, K. & Gretzel, U.

2000


Factors affecting the success of an Internet-based marketing decision support systems

Reisinger, Y. & Turner, L. 1998

International Journal of Hospitality, Leisure and Tourism Management

Asian and Western cultural differences as predictors of tourist satisfaction

Reisinger, Y. & Turner, L. 1998

Journal of Travel and Tourism Marketing

Determinants of Asian and Western cultural differences

2002

FIGURE 1. Hypothetical Measurement Model with Correlated Latent Constructs

e1

hotel

e2

transport

accomm

Perceptions e3 e4

Satisfaction

access

e5 e6

food attractions

amenities

e7

Note: The curve between perception and satisfaction shows there is a relationship (correlation) between the latent constructs.

indicative of whether a set of items are unidimensional (Joreskog, 1993; Steenkamp and van Triip, 1991). Proper specification of the measurement model is required for a struc-

tural model to have meaning (Anderson and Gerbing, 1982). Latent constructs represent one-dimensional concepts in their purest form (Bollen,


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FIGURE 2. Hypothetical Measurement Model with Direct Relationship Connecting Latent Constructs

e1

hotel

e2

transport

accomm

Perceptions

access

e6

food

e3 e4

Satisfaction

e5

amenities

attractions

e7

Note: The straight arrow between perception and satisfaction shows direct effect (relationship) between the two latent constructs.


TABLE 2. Hypothetical Two Construct Measurement Model Latent constructs Perception

Satisfaction

Measured variables/indicators

Loadings on constructs

P1 Hotel

0.982

P2 Transport

0.971

P3 Food

0.891

P4 Attractions

0.789

S1 Accommodation

0.924

S2 Accessibility

0.853

S3 Amenities

0.791

1989). The latent construct is free of random error which is, however, accounted for in the indicators (measurement items). Covariance or correlation structure analysis should be performed to assess the measurement models for each construct. In the measurement model shown in Figure 3, the latent constructs are represented by eclipses and the observed variables (items) are represented by rectangles. Observed variables are connected to the latent constructs by an arrow, which indicates that the items are theoretically attributed to the construct. The loading coefficients are represented by the factor loadings adjoining the arrows. The response error (e) is linked to each of the items and connotes the portion of the observed variables not explained by the construct. The values immediately above the observed variables represent the variance of each item explained by the

latent construct (Schumacker and Lomax, 1996). In most applications the measurement model is a conventional confirmatory factor analysis (CFA) model confirming that the indicators assign themselves into factors corresponding to how the investigator has linked the indicators to the latent variables. The latent constructs/ variables are common factors measured by the variables that loaded on these constructs. The errors are uncorrelated, with the exception of longitudinal models that need correlated components at two or three more time periods (McDonald and Ho, 2002). In a pure measurement model (CFA) there is unmeasured covariance between each possible pair of latent variables, there are straight arrows from the latent variables to their respective indicators, from the error and disturbance terms to their respective variables, but there are no direct effects (straight arrows) connecting the latent variables. The examples of recent tourism studies that used CFA are identified in Table 3. Path Model Path model describes the dependency relations, usually treated as causal relations, between the latent constructs. The model is usually depicted in a circle-or rectangleand-arrow figure. Single arrows indicate causation between exogenous or intermediary variables and the dependent endogenous variables. Investigators should always specify linkages (existent and nonexistent) between latent variables. An arc represents a single direct connection between latent variable, path represents a sequence of arcs connecting several

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FIGURE 3. Measurement Model .85 e1 e2 e3

P1 P2 P3

.96 .94

.97 .79 .89 .62

e4

S1

e5

.92

.98

.72 Perceptions

Satisfaction

S2

.85 .79

.62

.79

P4

S3

e6

e7

TABLE 3. CFA Studies in Tourism, Hospitality and Leisure


Author (s)

Year

Journal

Theme

Bigne, J., Andreu, L & Gnoth, 2005 J.

Tourism Management

The influence of visitor emotions in a theme park environment on satisfaction and behavioural intentions

Gooroochurn, N & Sugiyarto, 2005 G.

Tourism Economics

Determinants of tourist destination competitiveness

Hwang, SN., Lee C. & Chen HJ

2005

Tourism Management

The relationships among tourists' involvement, place attachment and interpretation satisfaction in Taiwan's national parks

Karatepe, O., Avci, T. & Tenkinkus, M.

2005


The effects of service quality, interpersonal relationships, selling tasks, organization, internal communication and innovativeness on marketing culture in Turkish hotels

Douglas, A. & Mills, J.

2004


The extent to which the top ten Caribbean destinations market their tourism product using their national tourism organization websites

Gursoy, D., Kim, K & Uysal, M.

2004

Tourism Management

The impact of festivals and special events on local communities in Virginia, USA

Manning, M., Davidson, M. & 2004 Manning, R.

International Journal of Hospitality Management

Dimensions of tourism and hospitality organizational climate and their influence on employee turnover and customer satisfaction

Nyaupane, G., Morais, D. & Graefe, A.

2004


Leisure constraints in three nature-based tourism activities

Zalatan, A. Gursoy, D. & Gavcar, E.

2004 2003

Tourism Economics Annals of Tourism Research

The validity of tourist typology Dimensions of European leisure tourists' involvement profile

Kozak, N., Karatepe, O. & Avci, T.

2003

Tourism Analysis

The influence of the quality of airline services on customer satisfaction in Northern Cyprus

Pennington-Gray, L. & Kerstetter, D.

2002


Pleasure travellers' perceptions of the constraints to nature-based tourism

Yoon, YS., Gursoy, D. & Chen, J.

2001

Tourism Management

The effects of tourism impacts on local residents' support for tourism development, Virginia, USA

Morais, D., Backman, K. & Backman, S.

1999

Tourism Analysis

The antecedents of advertisements emotional appeal on intentions to purchase a leisure service

Kang, YS., Long, P. & Perdue, R.

1996


Resident attitudes toward legal gambling in the USA


variables. Investigators can choose between a path model (1) with latent variables or (2) with observed variables, or (3) with some latent and some observed variables (see Figure 4a,b,c). Recent examples of tourism studies that used structural modelling are identified in Table 4.


SEM Diagram Construction and Its Graphical Presentation The model should show paths connecting latent variables and identify the measurement model that relates latent constructs to observed variables. Latent or unobserved variables are shown as circles or ellipses; observed variables as squares. One-headed arrows indicate causal influence of one variable on the other one, and two-headed arrows indicate reciprocal influence of each variable on another. Curved lines represent non-directional correlation or covariation (see Figure 1); straight arrows indicate direct relationship between constructs and their indicators, and direct relationships between constructs (see Figure 2). There should be no straight arrows pointing to the exogenous constructs because they are independent and not determined by any other variable in a model. There should be arrows pointing to the endogenous constructs because they are determined by other constructs and relationships contained in the model. They can also determine other endogenous constructs. Free and fixed parameters can be shown using specific symbols defined in the text or in the figure. One must assure that (a) relationships in the model are linear; (b) measurement errors and disturbance terms are random and uncorrelated; (c) each latent variable has three or more observed measures; and (d) causal flow between the construct and the measure is unidirectional (recursive) (Edwards and Bagozzi, 2000). The most critical point at this stage is to include all key predictive variables (multiple indicators of the latent variables) to avoid a specification error. There must be justification for inclusion of the specific latent constructs. The paths must be theoretically sound to prevent capitalising on data.

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Validity Measurement SEM measures convergent and discriminant validity in one model. Construct validity refers to establishing what an instrument is actually measuring (Malhotra, 1999). It is “the extent to which the measure correlates with other measures designed to measure the same thing” (Churchill, 1979, p. 70). The correlation with other measures is measured by convergent validity, discriminant validity Convergent validity measures the extent to which an item in an instrument correlates with other measures of the same construct in the instrument. It is “the degree to which multiple methods of measuring a variable provide the same results” (O’Leary-Kelly and Vorkurka, 1998, p. 399). Confirmatory factor analysis is more appropriate than the multitrait-multimethod matrix (Cambell and Fiske, 1959) in testing convergent validity. Convergent validity can also be assessed by examining the regression coefficients in the measurement models (Steenkamp and van Trijp, 1991). For convergent validity to be established, the loadings between the measurement items and the latent construct should be greater than 0.7 and the overall fit of the model should be acceptable (Schumacker and Lomax, 1996). Discriminant validity assesses whether a construct does not correlate too highly with measures with which it is expected to be different to (Churchill, 1999). In order to establish discriminant validity one can use the values generated by SEM to determine both internal consistency (Fornell and Larcker, 1981) and also assess the difference between measures of different constructs within the same model (Sakar et al., 2001). Only constructs with measures that theoretically appear to be similar are assessed for discriminant validity. According to Fornell and Larcker (1981), two constructs are distinct if the average variance extracted (AVE) (i.e., the average variance shared between a construct and its measures) in a latent construct indicators exceeds the variance that the latent construct shares with another construct (i.e., the squared correlation between two constructs). If the construct has more in common with the latent construct than with other constructs one can conclude that the constructs are distinct from each other provid-

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FIGURE 4a. Path Model with Latent Variables and Direct Connections Between Variables

Repeat visit

causal path

Endogenous variable

Exogenous variable

Perception

Culture

causal path

causal path


Satisfaction

Endogenous variable

FIGURE 4b. Path Model with Observed Variables Intention to visit

Cultural norms

Perception of accommodation

Satisfaction with accommodation

FIGURE 4c. Path Model with Latent and Observed Variables

intention to come back Repeat visit

intention to go somewhere else

norms intention to stay at home Culture

Quality of experience

rules

accommodation

values

Satisfaction accessibility

amenities


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TABLE 4. Path Analysis Studies in Tourism, Hospitality and Leisure


Author (s)

Year

Journal

Theme

Jang, S., Cai, L., Morrison, A. 2005 and O'Leary, J.

International Journal of Tourism Research

The effects of travel activities and seasons on expenditure

Reisinger, Y & Mavondo, F.

2005


Implications of travel risk perception on travel anxiety and intentions to travel internationally

Kozak, N., Karatepe, O. & Avci, T.

2003

Tourism Analysis

Airline service quality dimensions as determinants of customer satisfaction

Yu, H. and Littrell, M.

2003


The relationships between travel activities, beliefs about a product authenticity and features, attitude toward shopping and intention to purchase a product

Subramaniam, N, McManus, 2002 L & Mia, L.


The impact of structure, need for achievement, participative budgeting on hotel managers' organisational commitment

Chen, J. & Gursoy, D.

International Journal of Contemporary Hospitality Management

The relationship between tourists' destination loyalty and their destination preferences

Reisinger, Y., Mavondo, F. & 2001 Weber, S.

Tourism, Culture & Communication

The relationships between lifestyle, preferences for activities, travel motivation, personality and cultural values

Uriely, N. & Reichel, A.

2000


Working tourists and their attitudes to hosts

Reisinger, Y. & Turner, L

1999

European Journal of Marketing Cultural analysis of Japanese tourists

Borchegrevink, C. & Boster, F.

1997


2001

ing discriminant validity (Hulland, 1999) (see Table 5). Model Specification Structural equation models must be justified from theoretical reasoning or literature (Hair et al., 2002; Schumaker and Lomax, 1996). If the relationships within the structural model lack theoretical basis, results may be misleading (Hair et al., 2002). Model specification refers to the formal mathematical specification of the relationships embedded in the model. Investigators should assert which parameters are fixed, constrained and free. The parameters of an SEM are the estimated loadings, error variances and covariances in the measurement model, and the estimated directed arc coefficients and disturbance variances and covariances in the path model. Fixed parameters specify values a priori and are not estimated as part of the model; constrained parameters are unknown, estimated by the model; free parameters have unknown values, are not constrained to be equal to any other

The antecedents of leader-member exchange development

parameter, and need to be estimated. Variable parameters correspond to arrows in the model, while null parameters correspond to an absence of an arrow. Relationships Between Latent Constructs and Their Measures The relationships between latent constructs and their measures should be accurately specified to ensure psychometrically correct measurement models, and all paths linking latent variables must be theoretically justified. Reflective measures reflect or manifest a latent construct or are caused by latent constructs (Fornell and Bookstein, 1982); latent constructs are causes of measures (Bollen, 1989) (see Figure 5). Latent constructs with reflective measures are often called latent variables (MacCallum and Browne, 1993). Formative measures are causes of latent constructs (Bagozzi and Fornell, 1982; MacCallum and Browne, 1993); they form latent constructs (Fornell and Bookstein, 1982; MacCallum and Browne, 1993) (see Figure 6).

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TABLE 5. Internal Consistency, Square Roots of Average Variance Extracted, and Correlation Matrix for the Hypothetical Constructs Construct

Internal consistency

1

2

Perception

.8536

.7153

Satisfaction

.8267

.436 (.36)

Repeat visitation

.7580

.446 (.49)

.428 (.41)

Culture

.7325

.159 (.14)

.374 (.46)

3

4

.6765 .6321 .283 (.37)

.7387


Note: Figures in italics and brackets represent covariance; the figures in bold represent the square root of average variance extracted; the figures in text (above the figures in italics) are correlations.

FIGURE 5. Reflective Measures: Observed Measures as the Result of a Latent Construct

accommodation

Satisfaction

transportation

ships among constructs cannot be meaningfully tested (MacCallum and Browne, 1993). For example, formative measures do not have error terms; they cannot be tested using CFA. Covariances among measures can be used to identify formative and reflective measures (Bollen and Lennox, 1991). Conditions for Establishing Causality

amenities

FIGURE 6. Formative Measures: Observed Measures as the Causes of a Latent Construct

satisfaction with accommodation

satisfaction with transportation

Satisfaction with a trip

satisfaction with amenities

Latent constructs with formative measures are often called composite variables. Because formative measures are used for constructs that are composites of specific variables, e.g., socio-economic status is a composite of occupation, education and income. If the relationships between constructs and measures are specified incorrectly, relation-

In order to establish causality four conditions must be met (1) the cause and the effect must be distinct entities; (2) the cause and the effect must be associated with each other; (3) cause must occur before the effect (temporal separation); and (4) competing explanations for the relationship between the cause and the effect must be eliminated (Bagozzi, 1980; Bollen, 1989; Cook and Campbell, 1979). The relationship between constructs can be decomposed into four components: (a) a direct effect (one variable directly affects another); (b) an indirect effect (the effect of one variable on another is mediated by one or more other variables); (c) a spurious component (that is due to common or correlated causes); and (d) an unanalyzed component (Edwards and Bagozzi, 2000) Missing Values and Outliers Information concerning missing values and outliers should be provided. Missing data is a common problem in SEM applications. Data can be missing completely at random and inde-



pendently of other observed variables (MCAR) and missing at random depending on other observed variables (MAR). Some argue that missing data can be ignored (Rubin, 1976). Others suggest deletion of missing data. However, listwise and pairwise deletion for missing data can lead to biased parameter estimates under MAR, but unbiased estimates under MCAR (McDonald and Ho, 2002). Thus, any SEM study should indicate the extent to which there is a missing data and describe the method used to deal with it. For example, if there are not too many missing data, multiple-group SEM may be used. This method requires a different group of missing data with equality constraints imposed across groups (see Allison, 1987; Muthen, Kaplan and Hollis, 1987). The expectation-maximization (EM) algorithm (Dempster, Laird and Rubin, 1977) or multiple imputations may also be used. In the later case multiple data sets are created with plausible values replacing the missing values and the analysis is applied to the complete data set (see Graham and Hofer, 2000). The computer software dealing with missing data include Mplus, AMOS, Mx and LISREL; all assume MAR. There is no correction available if the missing data cannot be ignored (not MAR) (McDonald and Ho, 2002). As to outliers, EQS permits the deletion from the analysis of outliers as judged by their contribution to Mardia’s coefficient of multivariate kurtosis (Bentler, 1989; Joreskog and Sorbom, 1986). Matrix Type and Sample Selection Matrix Type to Be Analyzed Different matrices (covariances, cross-products, covariances/means, correlations) can be analyzed. Investigators must decide whether to fit a model to a covariance matrix or a correlation matrix and provide some justification for their choice (Raykov, Tomer and Nesselroade, 1991). The covariance matrix is used when the objective is to test a theory, compare different populations or samples, or explain the total variance of constructs needed to test the theory. However, interpretation of the results is more difficult because the coefficients must be interpreted in terms of the units of measure for the constructs. The correlation matrix allows for

51

direct comparisons of the coefficients within a model and thus it is frequently used to understand the patterns of relationships between the constructs (it is not used to explain the total variance of a construct as needed in theory testing). Using correlation matrix is more advantageous in terms of results interpretation. If the latent variables are standardized and the model is fit to correlation matrix, then parameter estimates can be interpreted in terms of standardized variables. In research designs such as multisampling or repeated measures, it is necessary to use the covariance matrix to retain information about variances of variables (MacCallum and Austin, 2000), especially when cross-group equality constraints are imposed and the original metric is to be preserved in both groups. Standardizing covariance matrices separately for each sample discards important information about variability in each group (Raykov et al., 1991). If a scale is changed, a rescaling should be applied for the data from each sample when using scale-invariant models with each group. LISREL offers a common rescaling for multigroup solutions (Cudeck, 1989; Joreskog and Sorbom, 1988).The variable should be scale to similar variances before the SEM analyses. This can be achieved by moving the decimal point in some input variables (Bentler, 1989; Joreskog and Sorbom, 1988). In other research designs one may chose either covariance or correlation matrix. The most widely used method for computing the correlations or covariances between manifest variables is Pearson product-moment correlation. The correlation matrix in LISREL is computed using Prelis (Joreskog and Sorbom, 1988). Other programs that provide for correct estimation when analyzing a correlation matrix are RAMONA (Browne and Mels, 1998), SEPATH (Steiger, 1999), EQS (Bentler, 1989, 1995), LISREL (Joreskog and Sorbom, 1996) and Mx (Neale, 1997). The AMOS program (Arbuckle and Wothke, 1999) does not accept correlation matrices for analysis. It is desirable to provide the means and standard deviations of the observed variables in table or in an appendix. This information should also be available to the reader by other means such as electronic mail (Raykov et al., 1991).

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Sample Size Although MacCallum et al. (1996) provided a method for determining the minimum sample size to achieve a given level of power for tests of model fit, there is no correct rule for estimating sample size for SEM. Rather it is accepted that the absolute minimum sample size must be at least greater than the number of covariance or correlations in the input data matrix. The general rule of thumb is that there should be a minimum of five cases for every distinct variable that is used to measure a construct (Hair et al., 1995, 2002). This implies that as the model complexity increases so do the sample size requirements. Kline (1998) recommends 10 times as many cases as parameters (or ideally 20 times); 5 times or less is insufficient for significance testing of model effects. Recommendations are for a sample size ranging from 100 to 200 (Hair et al., 1995), with an absolute minimum of 50 respondents. A critical sample size is of 200. Unfortunately, too often small sample sizes (less than 100) are used in SEM studies with no discussion of whether the sample is sufficiently large enough to run SEM. SEM analysis of small samples lacks power and is questionable. The sample size has a significant influence on the complexity of a model. When sample size is small, simple models are preferred. When sample size is larger, more complex models are used. However, when one develops complex models with large number of parameters, one may obtain less precise estimates. Thus, sample size is critical for achieving acceptable fit measures. Only the fit measures that depend on sample size should be used, especially when comparing alternative models. For example, when comparing models with small sample size, the ECVI index (Browne and Cudeck, 1993), which is sensitive to the sample size (MacCallum and Austin, 2000) should be used. The sample size also depends on methods of model estimation, which are discussed later. Model Identification Model identification refers to the assessment of the extent to which the information provided by the data is sufficient to enable pa-

rameter estimation or allow a unique solution to be found for the equations constrained in the theoretical model. The model is identified if every parameter is identified and the model’s degree of freedom are greater than or equal zero. If some of the parameters are not identified then a model is not identified. A necessary condition for the identification is t ⱕ s, where t = number of independent parameters, s = number of elements of the sample matrix of covariances among the observed variables, s = 1/2(p + q)(p + q + 1), p = number of y-variables, q = number of x-variables. Structural models maybe just-identified, under-identified and over-identified. The model is just identified if t = s (there is only one estimate for each parameter; all the information available is used to estimate parameters and there is no information left to test the model. It is also not possible for two different sets of parameter values to produce the same variance-covariance matrix. Such a model has exactly zero degrees of freedom, which implies a ‘perfect fit’ between the data and the proposed model. Such a solution is of little interest, as it cannot be generalised. The model is overidentified if t < s (it is possible to obtain several estimates of the same parameter; there are positive degrees of freedom (equal to s-t), one set of estimates can be used to test the model. This means the sample matrix provides more information than needed that allows for a number of ways of estimating a parameter. The model is unidentified if t > s (an infinite number of values of the parameters could be obtained). A model has negative degrees of freedom, which means that it tries to estimate more parameters than there is information available. In this case, the results of the analysis lack robustness therefore additional parameters have to be fixed or constrained before the model can be identified (Hair et al., 2002). In order to make all the parameters identified one must either (1) add more observed variables; (2) fix some parameters to specific values; or (3) set parameters equal to each other (Aaker and Bagozzi, 1979). Direct reflective measures can also be added to the construct of interest. Many variables such as race, sex, occupation, religion, community, region, and even age influence behaviour and should be specified not as error-free formative measures


but as imperfect reflective measures of constructs that cause other constructs (Blalock (1971). The condition t ⱕ s is necessary, but not sufficient, for the identification of a model. One should also use technical methods for determining model identification (Bekker, Merckens and Wansbeek (1994) and examine three distinct problems: (a) identification of the measurement model, (b) identification of the path model; and (c) scaling of the latent variables (McDonald and Ho, 2002).


Identification of the Measurement Model The measurement model must be identified and one should show that this is the case. The parameters of the measurement model are identified when the factor loadings form independent clusters in which each observed variable loads on only one factor and it is a pure indicator of the factor. Also, each latent variable or common factor should have at least two pure indicators if the factors are correlated and at least three if they are not. This condition is called independent clusters basis (McDonald, 1999). Many measurement models can be identified by using Thurstonian conditions of simple structure which allows every variable to load on more than one factor. However, simple structures do not generally guarantee identified measurement models. It is important to show that the measurement model is identified by either identifying the presence of independent clusters, or at least an independent clusters basis (McDonald and Ho, 2002).

53

Identification of the Path Model The path model must be identified and it should be shown that this is the case. The path model identification relies on the selection of non-directed arcs. In the path diagram, an exogenous variable has no directed arc ending on it; its variance is not explained by variables in the model. An endogenous variable has at least one directed arc ending on it, originating from one or more exogenous or endogenous variables. A recursive model is identified if a) every equation in a model is a regression (difficult to test) and all covariances of disturbances of causally ordered variables are zero (precedence rule) (McDonald, 1997). According to the orthogonality rule, all disturbances of endogenous variables must be uncorrelated (McDonald, 1997). If a model is fully ordered except for the exogenous variables, the last two rules coincide (McDonald and Ho, 2002). Most applications use a recursive model in which all arrows flow one way, with no feedback looping (no reciprocal directional effects or feedback effects) and the residual error (disturbance terms) for the endogenous variables are uncorrelated (see Figure 7). In non-recursive models there are directed paths consisting of a sequence of unidirectional arcs forming closed loops (see Figure 8). The orthogonality rule applies to the recursive model identifiability but not to a non-recursive model. In the case of a recursive model, investigators should include or omit a non-directed arc, depending on the existence or non-existence of unmeasured common causes. If the resulting recursive model has no non-directed arcs between causally ordered variables, it is identi-

FIGURE 7. Hypothetical Recursive Model

Hotel

Satisfaction

Transport Demand Food

Attractions

Visitation

54


FIGURE 8. Hypothetical Non-Recursive Model

Hotel

Transport

Satisfaction

Visitation Food


Attractions

fied. In the case of a non-recursive model there is no clear identifiability rule (McDonald and Ho, 2002). Identification Problems and Their Sources Model identification is an issue of concern in SEM studies. Identification problems and their sources must be searched out and eliminated. If possible investigators should avoid identification problems by applying numerous strategies (see Reisinger and Turner, 1999; Hair et al., 1995, 2002).

Demand

in an absolute sense. They are appropriate to compare across groups with different variances, measurement errors and disturbance terms. They can be used to compare similar models in other populations. However, any change in the measurement unit for an independent or dependent variable changes the value and comparability of parameters across populations (Bagozzi, 1977). Investigators should decide about standardizing the variables in relation to the purpose of the study and justify their choice. Model Estimation

Standardization In order to avoid identification problems McDonald and Ho (2002) recommend standardizing the variables either before or after estimation. Standardized parameters are metrically comparable, as opposed to unstandardized values. The standardized parameters can be used to compare the relative contributions of independent variables on the same dependent variable and for the same sample of observations. They are not appropriate to compare across populations or samples (Bagozzi, 1980). A standardized solution avoids the problem of underidentifiability, aids interpretation of the results, in particular, standardized factor loadings, standardized regression coefficients, and standardized path coefficients. On the other hand, non-standardized coefficients are computed with all variables in their original metric form; they show the effect that variables have

Model estimation refers to whether the parameter estimates are consistent with covariance/ correlation matrix of the observed variables. It involves choosing the estimation technique for the specified model. It depends on the variable scale and the distributional property of the variables used in the model. Parameter Estimation Various parameter estimation techniques are available including instrumental variables (IV), two-stage least squares (TSLS), unweighted-least squares (ULS), maximum likelihood (MLE), ordinary (unweighted) least squares (QL), generalized least squares (GLS), weighted least squares (WLS) and diagonally weighted least squares (DWLS). The popular methods are the TSLS and MLE methods. The two-stage least-squares (TSLS) method com-



putes the initial estimates, the maximum likelihood estimation (MLE) method computes the final solution. The TSLS and MLE estimates are very comparable and usually do not differ by more than 0.02. The TSLS and IV methods are non-iterative and fast, the MLE, ULS, GLS, WLS and DWLS methods are iterative procedures and successively improve initial parameter estimates. The generalized least squares (GLS) is often used to adjust for the violations of the SEM assumptions which can be tested in LISREL through the program PRELIS. However, as the models become large and complex, the use of this method becomes more limited. The MLE method is the most commonly used. However, it requires large samples as it is more precise in large samples. The minimum sample size to ensure appropriate use of MLE is 100. As the sample exceeds 400-500 the method becomes “too sensitive” and almost any difference is detected as significant, making all fit measures poor (Hair et al., 1995). Multivariate Normality When deciding about the estimation method the assessment of the multivariate normality distribution of the observed variables should be made. This can be done by checking the distribution of each variable or multivariate skewness and kurtosis coefficients. The univariate distribution of the variables and the use of factor scores can detect outliers (Bollen and Arminger, 1991). Explanation should be given of the way the variables distribution was studied. For example, PRELIS and EQS provide different ways of studying the distribution of variables. The MLE or GLS parameter estimation methods should be applied with continuous multivariate normally distributed data (Joreskog and Sorbom, 1988). These methods can also be used if there is slight or moderate departure from normality. However, the chi-square value and the standard errors should be interpreted with caution. They are sensitive to departures from normality (Bentler, 1989; Joreskog and Sorbom, 1988). If there is excessive skewness and kurtosis in the data, MLE and GLS estimation methods can give biased standard errors and incorrect test statistics, other methods for parameter estimation may be used (Bentler,

55

1987; 1989; Joreskog and Sorbom, 1986, 1988; Muthen, 1984). Weighted least squares (WLS) discrepancy estimation methods give good parameter estimates without requiring multivariate normality (McDonald and Ho, 2002). However, each of the methods has its own limitations. They require large sample sizes to satisfy the asymptotic statistical theory underlying the SEM approach (Tanaka, 1987). One should note that parameter estimates can remain valid, but not the standard errors, even when the data are non-normal (Anderson 1989). Since much social and behavioural science data are not normally distributed an asymptotically distribution-free (ADF) estimator (Browne, 1984) should be used. However, the use of the ADF estimator requires extremely large sample, far larger than are available in SEM applications. In the context of analysis of variance and multivariate regression problems transformation of variables should be performed by using univariate Box-Cox transformations (Mooijaart 1993) and a robust transformation method (Yuan, Chan and Bentler, 2000). Non-normality may be caused by the presence of categorical variables or indicators. A continuous/categorical variable methodology (CVM) estimator (Muthen, 1984) should be used for the analysis of dichotomous, ordered polytomous, and measured variables. However, the CVM estimator also requires a very large sample size to obtain reliable weight matrices. The CVM estimator outperforms the ADF estimator when the number of categories of the variables are few (< 5) (Muthen, 1989; Muthen and Kaplan, 1992). Model Testing and Interpretation The model should be tested using a variety of fit measures for the measurement and structural model (to support/reject the proposed hypotheses). Model Fit Model fit is determined by the degree to which the hypothesized measurement model fits the actual model derived from the sample data. The analysis is achieved by examining a

56


variety of fit indices (fit measures) that subjectively indicate whether the theoretical model fits the data. Different sets of these indices are available in different computer programs. For example, LISREL prints 15 and AMOS prints 25 different goodness of fit measures.


Fit Measures There are absolute, relative, parsimony and noncentrality fit indices (Maruyama, 1998; Tanaka, 1993). The absolute fit measures indicate the extent to which the model as a whole, both path and measurement together, provides an acceptable fit to the data with no adjustment for overfitting; they do not use an alternative model as a base for comparison (e.g., Ù², GFI, AGFI, Hoelter’s CN, AIC, BIC, RMR, SRMR). Relative fit indices compare the incremental fit of the model tested to a null model (also called a baseline or independence model) (e.g., IFI, TLI, and NFI). Most of these fit indices are computed by using ratios of the model chi-square and the null model chi-square and degrees of freedom for the models. All of them have values that range between approximately 0 and 1.0. Some are ‘normed’ and their values cannot be below 0 or above 1 (e.g., NFI, CFI). Others are ‘nonnormed’ because they may be larger than 1 or slightly below 0 (e.g., TLI, IFI). The conventional cutoff for these indices is 0.90 for good fitting models; there is an argument this value should be increased to 0.95. The parsimonious fit indices are adjustments to most of the ones above. They adjust the fit to compare models with different complexity (different numbers of coefficients). They penalise less parsimonious models, so that simpler models are preferred over more complex ones. The more complex the model, the lower the fit index. Parsimonious fit indices include PRATION, PCLOSE, PGFI, PNFI, PNF2, PCFI. Noncentrality-based indices (RMSEA, CFI, RNI, CI) test the degree of rejection of an incorrect model. A summary of the commonly used model fit indices is displayed in Table 6. Although there are rules of thumb for acceptance of model fit (see Table 6), Bollen (1989) reports that these cutoffs are arbitrary. For example, Carmines and McIver (1981) state that CMIN should be in the 2:1 or 3:1 range for an acceptable

model. Kline (1998) says 3 or less is acceptable, and others allow values as high as 5 to consider a model adequate fit, while others insist on the value of 2 or less. As to RMSEA, a good fit is when its value is less than or equal to 0.05, adequate fit if less than or equal to 0.08. However, Hu and Bentler (1999) also suggested RMSEA < = 0.6 as the cutoff for a good model fit. As to NNFI, some investigators suggest values of .80, .90 and even .95 (Hu and Bentler, 1999) as the cutoff for a good model fit. It seems that it is better to compare the fit of one’s model to the fit of other. For example, a CFI of 0.85 may represent a better fit when the best prior model had a fit of 0.70. Problems with Fit Indices and Their Choice There is a little agreement as to the choice of fit indexes and criteria of model evaluation. The Chi-square index is the most common used by all computer programs. The rule is that the chi-square value should not be significant if there is a good model fit. If chi-square is < 0.05, the model is rejected. However, the chi-square index is not a very good fit index. The chi-square test depends on (1) the size of a model; models with more variables have larger chi-square; (2) the distribution of variables; highly skewed and kurtotic variables increase chi-square values (when multivariate normality is violated one should use Satorra-Bentler scaled chi-square, which adjusts model chisquare for non-normality; (3) the sample size: larger samples produce larger chi-squares that are likely to be significant, creating an error that is rejecting true model even when it is minimally false (Bentler and Bonnet, 1980). In very large samples, even tiny differences between the observed model and the perfect-fit-model may be significant. Small samples may be too likely to accept poor model. Hoelter’s critical N should be used to determine the sample size that must be reached to accept the model by chi-square, at the .05 or .01 levels. Hoelter’s N should be greater than 200. AMOS and LISREL computer Hoelter’s N. Problems associated with sample size and the statistical power have been extensively discussed in Saris and Satorra (1993). Moreover, the chi-square tests the hypothesis (the difference between the proposed and alternative model) (Bentler and Bonnet, 1980), not the model fit. Thus, the use of


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TABLE 6. Model Fit Indices and Their Acceptable Levels Goodness of Fit Criterion Absolute fit measures

Acceptable Level

Interpretation

Low c² value (relative to degrees of freedom) with significance level < .05 Ratio 2:1 or 3:

Value greater than .05 reflects acceptable fit Values between 0.05 and 0.20 indicate a good fit Non-significant & small values show good fit, significant and large values show poor fit

Chi-square (c²)

Chi-square/df or CMIN Goodness of Fit Index (GFI)

.90 or higher

Adjusted for the degrees of freedom GFI (AGFI)

.90 or higher


Akaike Information Criterion (AIC) Hoelter's Critical N (CN) Root Mean Square .08 or lower Residuals (RMSR) or RMS or RMR Model Comparison and Relative fit measures Tucker- Lewis Index (TLI) Value close to 1 or Bentler-Bonett Non-Normed Fit Index (BBNNFI) Normed Fit Index (NFI) or Value close to 1 Bentler-Bonett Normed Fit Index (BBNFI) or DELTA 1

Value close to 1 reflects good model fit, values < 3 reflects acceptable fit Values > .90 reflect good fit Shows the amount of variances explained by model. It is the equivalent of R2 in multiple regressions Values > .90 reflect a good fit Adjusts model fit for the degrees of freedom relative to the number of variables Small positive values relative to independence model indicate model parsimony Gives a sample size below which the model is acceptable and above which the model becomes poorly fitting Values close to 0 reflect good fit, Marginal acceptance level is 0.08 Reflects the average amount of variances and covariances not accounted for by the model

Bentler-Bonnet Index (BBI) Incremental Fit Index (IFI) or BL89 or DELTA 2 Relative Fit Index (RFI) or RH01 Model Parsimony and Parsimonious fit measures Parsimony Ratio (PRATIO)

.90 or higher .90 or higher

Values >.90 reflect a good fit, Values below .90 indicate the need to respecify the model Compares an absolutes null model with the theoretical model of interest, penalizes for model complexity Values below .90 indicate the need to respecify model Reflects the proportion by which model improves fit compared to the null model, provides a measure of the proportion of total covariance accounted for by the model Values > .90 reflect a good fit Values = or > .90 reflect a good fit

Values close to 1

Values > .90 reflect a good fit

.90 or higher

Parsimonious Fit Index (PCLOSE)

.90 or higher

Shows the extent to which good fit can be achieved by freeing constrained parameters Values close to 0 indicate no fit and values close to 1 indicate perfect fit; values > .90 reflect a good fit Takes into account the number of degrees of freedom; a high degree of fit with fewer degrees of freedom is desired

Noncentrality-based indices Root Mean Square Error of Approximation (RMSEA) (measure of misfit), or RMS, RMSE, discrepancy per degree of freedom Comparative Fit Index (CFI) or Bentler Comparative Fit Index Relative Non-Centrality Index (RNI) McDonald's Centrality Index (CI)

< .08

Values < .05 reflect a good fit Values between .05 and .08 reflect reasonable fit Estimates how well the fitted model approximates the population covariance matrix per degree of freedom

Value close to 1

Values > .90 reflect a good fit Penalizes for sample size, gives the best approximation of the population value for a single model Values > .90 reflect good fit Penalizes for sample size and model complexity Values > .90 reflect good fit

.90 or higher .90 or higher


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the chi-square test is not valid in most applications (Joreskog and Sorbom, 1989a). LISREL refers to chi-square fit index as chi-square, chi-square goodness of fit, and chi-square badness-of-fit. In small samples and loglinear analysis the chi-square index is called G2, the generalized likelihood ratio. The chi-square statistic divided by degrees of freedom (c²/df) or CMIN is used with a large sample size (Joreskog, 1969) and is also sensitive to sample size. In large samples even small differences may become significant, whereas in small samples even large differences may become non-significant. The absolute fit measures improve as estimated coefficients are added. The GFI, AGFI and RMSR indices can show poor fit because of one relationship only being poorly determined. They do not indicate whether the model is or is not supported by the data, what is wrong with the model, or which paths to eliminate to make it better fit (Joreskog and Sorbom (1989b). The RMSR index performs best if all observed variables are standardized. GFI and the Bentler-Bonnett indexes of fit depend on sample size (Bentler, 1990). AIC can only be used for non-hierarchical comparisons of non-nested models, e.g., model building and trimming. Some relative fit indices (and the noncentrality fit indices) are also affected by sample size; larger samples have a higher fit index, although Bollen (1990) showed that the TLI and IFI are relatively unaffected by the sample size. There is a wide disagreement on which fit indices to examine and report. Various fit indices were recommended by scholars. It was suggested to use and report chi-square value, degrees of freedom and corresponding p value. However, if only these statistics are considered almost every model can be rejected. Hu and Bentler (1998) recommended to use SRMSR, Bentler and Bonett (1980) recommended NNFI, and Bentler and Cudeck (1993) suggested RMSEA. RMSEA was proposed because this index yields appropriate conclusions about model quality and provides precise fit (Hu and Bentler, 1989, 1999). RMSEA and the associated confidence interval were also recommended by MacCallum and Austin (2000). It was also suggested to apply multiple measures in order to achieve a better acceptability

of the model. Kline (1998) recommended at least four tests such as chi-square, GFI, NFI or CFI; NNFI; and SRMR. Jaccard and Wan (1996) recommended using of at least three fit tests, one from each category mentioned above. Since there is no single index of fit that is superior to the others one should use a combination of the presented above fit indices and justify their choice. One should identify the value of the chi-square, degrees of freedom and corresponding p value as well as give information about the fit of the model provided by the GFI, AGFI, CFI and RMSEA indices; they have known distribution and are the most commonly used. One should avoid reporting all of the fit indices and the indices that are most optimistic about the fit of the proposed model. Also, not all fit indices can be computed by the SEM computer programs. For example, AMOS does not compute all fit indices if there are missing data. Acceptable Models Acceptable models are associated with low chi-square values for a given number of degrees of freedom, p values greater than the preset significance level, high descriptive goodness of fit indices, e.g., GFI, NFI, NNFI, or CFI (depending on the program), and a low root-mean square residual (in LISREL). It is also commonly accepted that an index greater than 0.09 and close to unity indicates a ‘perfect’ fit. It is also accepted that RMSEA less than 0.05 indicates a ‘good’ fit and RMSEA less than 0.08 indicates an ‘acceptable’ fit. However, these cutoff values might change. A more concrete basis for conventional cutoff values might be established in the future. Other Model Assessment Criteria The other criteria that might be used to assess model fit are offending estimates (coefficients that exceed acceptable limits) such as negative error variances or non-significant error variances for any construct, standardized coefficients exceeding or very close to 1.0, and very large standard errors associated with any estimated coefficient. They must be resolved before evaluating the model results. Each of the

Yvette Reisinger and Felix Movondo

is assessed based on the measurement model only, the GFI of the path model may be unacceptable. Thus, one should pay attention to direct assessment of the path model fit. It is recommended to first fit the measurement model, and then the path model to the latent variable correlation matrix in order to examine the pattern of the discrepancies (McDonald and Ho, 2002).

constructs should also be evaluated separately by examining: t-values, standard errors (SE). and fitted residuals (FR). One should assess squared multiple correlation coefficients (SMC) for the y- and x-variables which indicate how well the dependent y-and independent x-variables measure the latent construct, the total coefficients of determination (TCD) (R2) for all y- and x-variables that show how well the yand x-variables as a group measure the latent constructs, as well as the squared multiple correlations (SMC) and the total coefficient of determination (TCD) (R2) for all structural equations together (see Reisinger and Turner, 1999). Downloaded By: [Monash University] At: 07:37 22 June 2011

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Interpretation of the Fit Indices A good model fit does not indicate the strong relationships among variables; a model can have perfect fit when all variables are uncorrelated. Low correlations among variables indicate the high probability of finding ‘good fit,’ and vice versa. Thus, when correlations are low, one may lack the power to reject the model. The model can also have good fit with small samples (< 200), though RMSEA and CFI are less sensitive to sample size than others (Fan, Thompson and Wang, 1999). Low correlations also indicate low and insignificant path coefficients even when there is a good fit. Consequently, investigators should report not only model fit indices but also the strength of the paths in the model. The model fit can be high but the model itself can be weak. A good fit of the structural model doesn’t mean a good fit of each part of the model. The structural model may have a bad fit because of a poor measurement model. A good fit of the structural model may also conceal a poor fit of the measurement model. As a result, one should present separate fit measures for the major components of the structural model. A misspecified model can also have a good fit. High modification indices may reveal multicollinearity and/or correlated error. Con-

Presenting Fit Measures Since a structural model is a composite of a measurement model and a path model, one should examine and present separate measures of fit corresponding to the major components of the structural model (McDonald and Ho, 2002) Separate chi-square, degrees of freedom, p, d RMSEA and other selected fit measures for the omnibus, measurement and path models should be given (see Table 7). The degrees of freedom in the path model are usually smaller than in the measurement model. The fit of the structural model is usually better if the constraints in the path model are incorrectly specified. If the fit of the structural model is unacceptable or poor, one should learn about the causes of the misfit. Most fit indices are functions of the non-centrality parameter. If the model fit is assessed based on the RMSEA, the GFI of the composite structural model (with its large number of degrees of freedom) may be acceptable as opposed to fit of the path model (with smaller number of degrees of freedom). If the model fit

TABLE 7. Goodness of Fit of the Omnibus, Measurement and Path Models N 530

Model

Chi-square2

df

p

d

RMSEA

Omnibus

127.2

101

.001

.042

.021

measurement

29.1

95

1.000

⫺0.119

.042

path

95.3

9

.000

.171

.149

p= probability; d = non-centrality parameter; RMSEA = root means square of approximation

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sequently, one should report on model modification. Further, good fit indices do not prove good models. A good fit doesn’t mean the exogenous variables cause the endogenous variables (causality). Many equivalent and alternative models may have a good fit, which does not indicate a good fit of the proposed model. Moreover, a model with fewer indicators per factor has usually a better fit than a model with more indicators per factor. In the last case, one should use fit coefficients which reward parsimony to adjust to this tendency.


Model Respecification and Final Model Investigators should identify all specification errors and consider how to improve the model fit and develop a new model, which fits the data better. This new model has to be verified on a second independent sample. Model re-specification must be theoretically justified and based on what the modification indices suggests. Examination of the t-Values and Standardized Residuals Schumacker and Lomax (1996) recommend a re-specification search process by initially examining t-values that help to identify the parameter estimates with little explanatory power. Parameters with insignificant t-values can be deleted from the model. However, where there is theoretical justification for the retention of the parameter then it is better not to remove them. A t-value less than 1.96 indicates that the parameter is not significantly different from zero at the 5% significant level and suggests this parameter can be deleted. The second re-specification to the model may be done through examination of the standardized residuals and modification indices. The standardized residuals (normalized) are provided by the program and represent the differences between the observed correlation/ covariance and the estimated correlation/ covariance matrix. One should identify significant residuals that indicate substantial error for a pair of indicators. Residual values greater than +/⫺ 2.58 are considered statistically significant at 0.05 level and can be adjusted for.

Modification Indices The following MI can be used to modify a starting model: fitted residuals, standardized residuals, modification indices, or Lagrange multipliers for EQS. The modification indices (MI) measure the predicted decrease in the chi-square from modifying a particular relationship in a model. The change in the chi-square results from freeing (relaxing) a single parameter (fixed or constrained) and consequent model reestimation. A better model fit can be obtained by relaxing constrained parameters. Only one parameter is relaxed at a time. The rule of thumb is to add paths associated with parameters showing MI exceeding 100 or add the parameter with the largest MI (even if less than 100). Improvement in fit is measured by a reduction in chi-square. The MI for covariances also create the decrease in chisquare. MI for regression weights can change chi-square if the path between the two variables is restored. MI should only be used if theoretically justified; otherwise the process becomes data driven and capitalizes on chance. AMOS lists the parameter (which arrow to add and delete), the chi-square value for the path labelled “MI,” p of this chi-square and the “parameter change”. Investigators should analyze the MI and parameter change. EQS offers multivariate Lagrange Multiplier and allows coefficients constrained to 0 (no direct paths) to vary instead. Investigators should justify their choice for modifications of their models and report the values of the indexes motivating these changes (Raykov et al., 1991). Model Trimming and Building Model modification can also be achieved by model trimming or building. Model trimming is about deleting paths one at a time until a non-significant chi-square difference indicates the end of the process. As paths are deleted from the model chi-square tends to increase, indicating a worse model fit. The Wald test should be used to determine which arrows to delete (usually those represented by non-significant parameters). On the other hand, model building is about adding paths one at a time. As paths are added chi-square tends to decrease, indicating a better fit. MI can be used to decide



which parameters should be added to the model. In both cases, deleting or adding parameters should be guided by theory. Non-significant t-values for regression weights (below 1.96 at the 0.05 level) give insight as to which parameters should be eliminated. However, if a theory suggests that particular parameters should be included in the model, even non-significant parameters should be retained because the sample size may be too small to detect their real significance (Joreskog and Sorbom, 1989b). The effect of the deletion on the model fit can be assessed by comparing the chi-square values of the two models, particularly, the differences in chi-squares (D2). Consideration should also be given to whether some of the parameters are not necessary to measure the latent constructs. According to Darden (1983), achieving a good fit at all costs is not always recommended; a good fit for a model may be theoretically inappropriate. There are many other models that could fit the data better. A poor fit tells shows the degree to which the model is not supported by the data. Model Parsimony A parsimonious model (with many variables) always fits the data (no parameter is constrained to 0), regardless of whether it makes sense or not. Lack of parsimony (fit) represents a problem for models with few variables. Investigators can increase its fit by adding paths. If the model is too complex one can decrease its complexity by deleting direct effects (straight arrows) from latent variable to another latent variable and from latent variables to the same observed variable. Also, one can delete correlations (curved double-headed arrows) between measurement error terms and the disturbance terms of endogenous variables (residual error terms). These should only be erased from the model if this is theoretically justified. The erasing of directed effects indicates there is no cause-effect relation between two variables. The deletion of non-directed effects indicates the variables have no common cause that has been omitted from the model (McDonald, 1997, 1999; Pearl, 1998, 2000). Investigators should include/delete effects (direct and non-direct) only when they are very confident

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that the causes exist/do not exist. Otherwise they can obtain untestable and underidentified models. Although adding non-directed arcs does not change the causal hypotheses, it may significantly change the model fit. When adding variables investigators have to make change in conclusions (McDonald and Ho, 2002). Alternative Models Alternative or equivalent models exist for almost all models. These are models that fit the analyzed data and the original model equally well. Alternative models (nested with parameter restrictions such as added or freed constraints, or non-nested) should be specified a priori before model modification (MacCallum, Roznowski and Necowitz, 1992). It is important to examine all the alternative models rather than to arbitrary choose one (Spirtes et al., 1998). Alternative models may offer interesting alternative interpretations of the analyzed data. Lee and Hershberger’s (1990) developed ‘replacing rules’ that respecify the original model to construct mathematically equivalent models. TETRAD software program implements an algorithm for searching for covarianceequivalent models. The Problem of Equivalent Models Alternative models are ignored in practice (MacCallum et al., 1993). This is a critical problem because in empirical studies there are alternative models for any given set of multivariate data. These models are represented by different patterns of relations among the variables (different path diagrams) but are similar to and mathematically equivalent the original model in terms of fit to data. A definition of model equivalence does not refer to the equivalent fit. Models are equivalent when their covariance matrixes are equal (MacCallum et al., 1993). Equivalent fit can occur by chance for two models that do not have the same covariance matrix. Thus equivalent models cannot be distinguished from the original model in terms of their fit to the data, rather only in terms of interpretability of parameter estimates and meaningfulness of the model (MacCallum et al., 1993).

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The equivalent models occur in different research designs such as non-correlation research or multidimensional scaling (Tucker, 1972), and covariance structure models (CSM). However, many path analysis models for correlational data do not allow for the construction of equivalent models. Although equivalent models are valid, meaningful and fit to empirical data, for any equivalent model there may be alternative models. Thus, investigators must always check for the possible existence of equivalent models to avoid wrong conclusions. It is unacceptable to pretend that the equivalent models do not exist (MacCallum et al., 1993). Downloaded By: [Monash University] At: 07:37 22 June 2011

Creating Equivalent Models One can generate alternative models by using modification indices (see Stelz, 1986; Verma and Pearl, 1990). Non-directed arcs can be added if a theory suggests so, however, this does not change the causal model (Donald and Ho, 2002). Adding disturbance covariance to improve fit is not desirable (Hoyle and Panter, 1995; Boomsma, 2000). Another way to generate alternative models is to change the pattern of relationships among the latent variables within the structural model, keeping the measurement model fixed (the same relationships between the latent variables and their indicators) (Stelz, 1986; Lee and Hershberger, 1990). This approach can be applied to path models with latent variables, each having a single perfect indicator. Another approach is to replace one parameter with another parameter (one path with another path) to generate equivalent models that do not differ in terms of their parsimony (MacCallum et al., 1993). For the recursive model, Lee and Hershberger (1990) suggest to use a “replacing rule” that does not require the entire model to be recursive (one that contains no reciprocal directional effects or feedback effects and no covariances among disturbance terms). One can use multiple partitioning and divide the model into blocks of latent variables: the preceding, focal and succeeding block. The relationships among latent variables within the focal block and relationships between vari-

ables in different blocks are recursive. Variables in the preceding block may receive no arrows from variables outside of the block. The relationships among variables within the preceding block and within the succeeding block may be non-recursive. Identifying equivalent models represents a significant advance in SEM. The existence of equivalent model(s) presents a serious challenge to investigators. The conclusions and inferences they make to support a new model from a group of equivalent models can be misleading without examination of alternative equivalent models (MacCallum et al., 1993). Thus, investigators should generate as many equivalent models as they can and evaluate the meaningfulness of each equivalent model. Although it becomes difficult to support any specific equivalent model as their number increases (MacCallum et al., 1993) it may also become easier to strengthen the support for the original model, especially if all alternative models are eliminated. Research Design and Alternative Equivalent Models The nature of research design may influence the meaningfulness of alternative models. In the experimental design studies, alternative models can be created by introducing paths from other variables to observed variables. However, these models might not be meaningful. In the longitudinal design studies, alternative models are not meaningful either because relationships among variables may change in time (MacCallum et al., 1993). When the research design is cross-sectional and correlational, or when the model contains a large number of variables one can develop a large number of equivalent models. As a result, it may be difficult to support the original model as being the most meaningful. In addition, the interpretation of equivalent models may be influenced by the nature of the variables examined. For example, introducing other variables to influence demographic variables in equivalent models might make little sense. The original model also gets more support than the equivalent models because it is theoretically justified,


interpretable and fits the data well. However, investigators might not be aware of other past models and theoretical developments (MacCallum et al., 1993). Consequently, investigators must learn the rules for generating equivalent models (Lee and Hershberger, 1990). If equivalent models can be eliminated on theoretical grounds, one can support the original model. However, if equivalent models cannot be eliminated, one should use them as an alternative explanation of the data (MacCallum et al., 1993).


Comparison of Nested Models If one cannot find the best fitting model using the confirmatory strategy, then one should identify such a model by comparing a number of alternative or nested models (with parameter restrictions) to see which comes closest to a theory. Differences between models are indicated by the differences in the chi-square values for the different models. These differences can be tested for statistical significance with the appropriate degrees of freedom. In order to improve the model fit an initial model should go through a series of model respecifications. Final Model Interpretation Evaluation of Model Specification and Evaluation Strategies The presented above model specification and evaluation strategies such as (a) strictly confirmatory (a single a priori model is studied); (b) model generation (an initial model is fit to data and then modified as necessary until an adequate fit is found); and (c) alternative models (multiple a prior models are specified and evaluated) (Joreskog and Sorbom, 1996) should be evaluated. The strictly confirmatory strategy is restrictive; it gives no opportunity for model improvement if the model does not work well. The confirmatory and model generation strategies are commonly used. Since the model generation strategy uses modification indices it requires one to acknowledge that the resulting model is data driven and show that modifications are valid. Model modifications are often not meaningful and arise by

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chance (MacCallum 1986, MacCallum et al., 1992). The modified model must also be evaluated by fitting it to an independent sample. Many investigators do not meet this requirement. Consequently, the model generation and modification strategies are misleading and often abused. The alternative model strategy provides comparative information about alternative explanation of the data and eliminates a confirmation bias, which occurs when one does not consider alternative models to explain data and concludes that original model fit is acceptable, even though values of fit indexes do not always meet even the “acceptability” level (MacCallum and Austin, 2000). Interpretation of Results All models are biased to some degree; there is no perfect or true model (Browne and Cudeck, 1993). A good fit does not imply that a model is correct, but only plausible. There might be other models that fit the data equally well. Investigators are only able to identify parsimonious, meaningful model that fits observed data well. A good fit does not imply either that relationships hypothesized in the model are strong. The relationships among variables can be weak, and a model may have an extremely good fit. These weak relationships are shown in the parameter estimates and large residual variances for endogenous variables. Thus, attention should always be paid to parameters estimates and residual variances which should be examined and reported, regardless of the model fit (MacCallum and Austin, 2000). Generalizability of Findings Even in a good study, conclusions are limited to the particular sample, variables, occasions of measurement and time frame. These limitations must be acknowledged (MacCallum and Austin, 2000). When sample size is small, the expected cross-validation index (ECVI) (Browne and Cudeck, 1989, 1993) should be used to assess how well a solution obtained in one sample fits an independent sample, and to compare alternative models. The sample and population characteristics should be clearly defined.

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This is of critical importance when multisamples are involved and one wants to identify a model fit and estimate parameters across samples from different populations (Linderberger and Baltes, 1997). It is also imperative to describe the indicators that represent latent variables, and explain the effects of occasions of measurement, which can occur over time and may vary with the length of the time interval (Gollob and Reichardt, 1991). The true effect of one variable on another is obtained only when the variables stay the same over any period of time. It is recommended to conduct studies of effects over different time lags (MacCallum and Austin, 2000). Downloaded By: [Monash University] At: 07:37 22 June 2011

The Importance of Time The strength of the relationships in a model may depend on the time lag. In cross-sectional studies (evaluate effects at one point in time only) the causality or directional influences may be questioned. Investigators must ensure that the causal variables do not change between the time this variable is measured and the causal effect occurs. If this is not the case, estimated directional relations are biased (Gollob and Reichardt, 1987) and a model is inappropriate; a longitudinal design should be used (MacCallum and Austin, 2000). However, in the longitudinal designs the causality of the temporal relationships can also be questioned; causality may occur due to an intervening variable or correlations. The true effect of one variable on another is obtained only when the variables stay the same over any period of time (MacCallum and Austin, 2000). MODEL CROSS-VALIDATION Each final model should be cross-validated with a new data set by carrying out a validation test. For this purpose one can do multi-sample analysis. The cross-validation test should be applied (a) when the model did not provide an acceptable fit after modification indices have been used; (b) when the model shows an acceptable fit in the first analysis; (c) to compare competing models; (d) to compare the difference between samples from different populations; and (e) to

assess the impact of moderating variables (Diamantopoulos, 1994; Sharma, Durand and Gur-Arie, 1981). Cross-validation requires large sample to be able to divide it into sub-samples (minimum sample size should be between 300-500 observations). REPORTING RESULTS Difficulties with Reporting Results Many difficulties are associated with presentation of the SEM results due to the differences in SEM computer packages in addressing the same issues and also the lack of agreement in user guides and texts about the style of presentation (Bollen, 1989; Loehlin, 1992; Long 1983a,b). The most frequent problems with reporting results are listed below. Indicators of the latent variables are left undetermined, the type of matrix used is not explained, and the type of data analyzed not explained. The parameter estimates are incomplete (unique variance, non-significant estimates, and residual variances are not provided) and errors and disturbance parameters are not identified. As a result, investigators are urged to learn what details are the most relevant and how much detail is required for an adequate SEM report. For example, researchers should provide a clear and complete specification of models and variables, list the indicators of each latent variable, explain what type of data is analyzed and type of matrix is used (correlation or covariance, with or without deviations to be able to evaluate plausible alternative models) (MacCallum and Austin, 2000). Showing correlations and means and standard deviations for up to 30 variables help to justify the model and better-fitting alternatives. In the set of variables is larger than 30 one should indicate a means to obtain the covariance or correlation matrix (McDonald and Ho, 2002). Investigators are urged to identify the software and method of parameter estimation, present multiple measures of fit with confidence intervals, along with all parameters estimates and associated confidence intervals or standard errors, and criteria for evaluating values of fit indexes (MacCallum and Austin, 2000). According to Raykow et al. (1991), standardized solutions or effect decomposition


should be presented if the study focuses on them (Bentler, 1989; Joreskog and Sorbom, 1988). Detailed guidelines for the reporting of SEM results have been offered by Steiger (1988), Breckler (1990), Raykov, Tomer and Nesselroade (1991), Hoyle and Panter (1995) and Boomsma (2000).


Reporting Parameters and Their Standard Errors It is recommended to report all parameters. The parameter estimates for the measurement model should include factor loadings, specificies and covariances among factors. For the structural model these should include at least path coefficients. One should comment on the sign and size of the estimates. The standard errors of the estimates or, at least which estimates are significant and the level of significance, should be identified to allow the reader to verify that the unique variances are not close to zero that corresponds to an improper (Heywood) solution, as indicated by large standard errors. Standard errors should be included in tables of the parameters in a path diagram or put in parentheses attached to the parameters in path diagram. Since standard errors are not available for the standardized solution the standardized parameters can be obtained by rescaling. It is also recommended to examine and report the standard errors of the unstandardized parameters, and to make statistical inferences based on these before interpreting the standardized coefficients (McDonald and Ho, 2002). The disturbance variances should be identified to show what proportions of variance of the endogenous variables are accounted for by the model. Error variances-covariances of the latent variables may be also reported. It is also advisable to comment on the strength of the squared multiple correlations and coefficients of determination (for LISREL). The former coefficients indicate the amount of error in the model, when the measurement model is contrasted with the structural model. The structural model can reveal whether the measurement model has been estimated at its expense (Anderson and Gerbing, 1988).

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MODEL EQUIVALENCE MEASUREMENT Testing for Measurement Invariance Across Groups (Multiple-Group Comparison) Often one wants to determine if the same measurement model is applicable across different groups (men and women; British and Japanese; or Catholics and Jews) and whether the relationships among the measured variables and latent constructs are the same for each group. First, one should test for measurement invariance between the unconstrained model for all groups together, and then for a model where certain parameters are constrained to be equal between the groups. If the chi-square difference statistic does not reveal a significant difference between the original and the constrained model, one should conclude the model has measurement invariance across groups (the same model applies across groups or the model is equivalent). If the model does not have measurement invariance across groups (the model is different or not equivalent), the meaning of the latent construct across groups is different. Measurement invariance occurs when the factor loadings of measured variables of their respective latent factor do not differ significantly across groups. When testing for multigroup invariance, the investigator should test one sample model for each group separately (e.g., one for a British sample and one for a Japanese sample). This is done to show how consistent the model results are and whether both models fit the data well. However, separate testing does not indicate testing for significant differences in the model’s parameters between groups. If both separate models fit the data well and consistently, the investigator should carry out a multigroup invariance testing. First, one should compute a model fit for the total sample of all groups and examine a baseline chi-square value. Then one should impose equality constraints on various model parameters across groups, fit the model and obtain a chi-square value for the constrained model. In order to impose equality constraints in AMOS one should right click on the regression paths to assign labels to the regression weights. One should enter a label for “Regression weight” when the ob-


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ject properties box is brought. Each path should get a unique label (e.g., “a”). One should check “all groups” so the label for the regression weights applies across groups: this is equality constraint. One should label only the paths connecting latent and measured variables. One should examine a chi-square difference test to see if the difference between a chi-square value of the baseline and constrained model is significant. If it is not significant, a conclusion should be made that the unconstrained baseline multigroup model is the same as the constrained model; the model applies across groups and displays measurement invariance. If the difference between a chi-square value of the baseline and constrained model is significant, one should conclude the model across groups differs and displays measurement non-invariance (http://www2.chass.ncsu.edu/ garson/pa765/structur.htm). See Reisinger and Mavondo, 2004, 2006. If one finds non-invariance across groups (model differs across groups), one must assess the cause of the problem within the model. One should start with the measurement model (see arrows that go from the latent factors to the measured indicators), and then test the structural model and relationships among constructs. Two tests can be run against pairs of samples. In AMOS, in the Manage Groups dialog box, one should delete groups to leave a pair to be tested. One should use the chi-square difference test to see if some pairs of samples are invariant between the two groups in the pair. Chi-square difference should not be significant if the model is invariant (similar) between the two groups. Once the non-invariant pairs are identified, one can remove the equality constraints from the loadings for a given factor and test for invariance between the two groups. One can free one factor at a time to see if the non-invariance is related to a particular factor. One has to free or constrain all indicator loadings, factor covariances, and/or the structural path coefficients to see which models are invariant. In order for the multigroup model to be invariant, the model must be accepted (using fit statistics) when all parameters are constrained to be equal. However, to obtain model acceptance various parameters must be freed. Those free parameters indicate non-invariance (points where the model differs between a pair of

groups)(http://www2.chass.ncsu.edu/garson/pa765/ structur.htm). Testing for Invariance of Path Models Across Groups One may also test for invariance of path models of invariant models across groups and assess whether the relationships among the latent variables are the same for each group. One should follow the same procedure as for measurement invariance testing. One should re-run the model and impose the equality constraints on the structural paths. One should impose equality constraints only on the paths connecting latent variables. A chi-square difference test should be examined. If the chi-square values of the baseline and constrained models are not significantly different, the conclusion should be made that the structural model is invariant between the samples, and thus the model is cross-validated. On the other hand, if the chi-square values of the baseline and constrained models are significantly different, one should conclude there is a moderating effect (causal) on relationships in the model, and their effect differs by group (http://www2.chass. ncsu.edu/garson/pa765/structur.htm). Reporting Measurement Invariance Results When one examines the difference among the groups, the following statistics showing differences between nested models need to be presented (preferably in a table): (a) the values of the chi-square, degrees of freedom and corresponding probability value (p) for each model; (b) the differences in the chi-square value, degrees of freedom, and the probability values; and (c) change in other fit indices and the estimates of interesting parameters. New SEM Model Developments and Research Designs SEM can be used in observational (correlational), cross-sectional and longitudinal, or experimental designs. In the cross-sectional studies multi-sample models and models with structured means are often used. In longitudinal studies the application of SEM to repeated measures designs represents a highly creative



development of models and novel applications in recent years (MacCallum and Austin, 2000). In repeated measures designs one usually uses autoregressive models for measured variables (McArdle and Aber, 1990), called simplex models, and latent variables (MacCallum and Austin, 2000), as well as latent curve models, called growth curve models (Tucker, 1958, Rao, 1958). Curve growth models represent today a rich methodological development (Sayer and Collins, 1999). The latent curve models can be extended into full SEM models. They can be used in multiple-outcome (see MacCallum et al., 1997, Willet and Sayer, 1995) and multisampling designs (Muthen and Curran, 1997). A special case of SEM, confirmatory factor analysis (CFA), can be used to evaluate designs for (a) construct validation and scale refinement, (b) multitraitmulti-method validation, and (c) measurement invariance (MacCallum and Austin, 2000) and to validate the hypothesized relationship (see Floyd and Widaman, 1995). The application of SEM in experimental studies also represents a significant and relatively new potential area of application (MacCallum and Austin, 2000). Experimental studies incorporate categorical variables (that violate the assumption of multivariate normality in ML) but not in exogenous measured variables (Bollen, 1989). When using SEM in these studies one can use coded dummy variables, including covariates, mediators and outcomes and then use the analysis to model the relationships (MacCallum and Austin, 2000). One can also treat experimental variables as representing different populations and conduct multi-sample SEM to fit model to samples that represent different experimental conditions (see Muthen and Curran, 1997). SEM has also been used in meta-analysis (Rounds and Tracey, 1993), test-retest designs (McArdle and Woodcock, 1997), multi-sample models to study moderator effects (e.g., Eisenberg et al., 1997; Harnish, Dodge, Valente, 1995), and the longitudinal study of reliability and validity (Tisak and Tisak, 1996) (cited in MacCallum and Austin, 2000). SEM model developments are continuing. SEM packages are becoming more userfriendly, producing more output and giving researchers a wider choice of issues to analyze. It is believed that by the time this work is pub-

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lished more new SEM applications will become known and available. SEM COMPUTER PROGRAMS There are several computer packages designed for the analysis of SEM such as CALIS (Hartmann, 1992), COSAN (Fraser and McDonald, 1988), MPLUS (Muthen and Muthen, 1998), RAMONA (Browne, Mels and Cowan, 1994), SEPATH (Steiger, 1995), Mx (Neale, 1997), EZPATH, LISCOMP and STREAMS. The most widely available and popular structural equation programs are LISREL (Linear Structural Relations) (Joreskog and Sorbom, 1989b, 1996), EQS (Bentler, 1985, 1995), and AMOS (Analysis of Moment Structures) (Arbuckle, 1997). LISREL is considered as the flagship structural equation modelling (SEM) program. AMOS is a recent package. It has become popular as an easier way of specifying structural models because of its user-friendly graphical interface and the capacity to work through the windows clipboard. EQS is popular but less frequently used. The SEM computer programs address the same issues and supply the same basic information, however, they differ slightly in their solutions, methods of parameter estimation, number and quality of indices reflecting model fit and a considerable amount of information concerning the numeric routine on which these programs rest. Each new version comes with additional features that improves presentation or simplifies functioning by adding useful defaults. For information on the individual SEM packages the reader should consult their user manuals. REFERENCES Aaker, D. and Bagozzi, R. (1979) Unobservable variables in structural equation models with and application in industrial selling. Journal of Marketing Research, 16, 147-158. Allison, S. (1987) Estimation of linear models with incomplete data. In C. Clogg (Ed.) Sociological Methodology (pp. 71-103). San Francisco: Jossey-Bass. Anderson, J. (1987) Structural equation modelling in the social and behavioral sciences: Model building. Child Development, 58, 49-64.


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SUBMITTED June 4, 2006 ACCEPTED November 2, 2006 REFEREED ANONYMOUSLY

doi:10.1300/J073v21n04_05