Introduction to Structural Equation Modeling: Issues and ...

1 An NCME Instructional Module onIntroduction to StructuralEquation modeling : Issuesand practical ConsiderationsPui-Wa Lei and Qiong Wu,The Pennsylvania State UniversityStructural Equation modeling (SEM) is a versatile statistical modeling tool. Its estimationtechniques, modeling capacities, and breadth of applications are expanding rapidly. This moduleintroduces some common terminologies. General steps of SEM are discussed along with importantconsiderations in each step. Simple examples are provided to illustrate some of the ideas forbeginners. In addition, several popular specialized SEM software programs are briefly discussedwith regard to their features and availability. The intent of this module is to focus on foundationalissues to inform readers of the potentials as well as the limitations of SEM. Interested readers areencouraged to consult additional references for advanced model types and more : Structural Equation modeling , path model, measurement modelStructural Equation modeling (SEM) has gained popular-ity across many disciplines in the past two decades dueperhaps to its generality and flexibility.

2 As a statistical mod-eling tool, its development and expansion are rapid andPui-Wa Lei is an assistant professor, Department of Educationaland School Psychology and Special Education, The PennsylvaniaState University, 230 CEDAR Building, University Park, PA Her primary research interests include structuralequation modeling and item response theory. Qiong Wu is a doctoralstudent in the Department of Educational and School Psychology andSpecial Education, The Pennsylvania State University, 230 CEDARB uilding, University Park, PA 16802; Her interests aremeasurement, statistical modeling , and high-stakes InformationITEMS is a series of units designed to facilitate instruction in ed-ucational measurement. These units are published by the NationalCouncil on Measurement in Education. This module may be photo-copied without permission if reproduced in its entirety and used forinstructional purposes.

3 Information regarding the development of newITEMS modules should be addressed to: Dr. Mark Gierl, Canada Re-search Chair in Educational Measurement and Director, Centre forResearch in Applied Measurement and Evaluation, Department of Ed-ucational Psychology, 6-110 Education North, University of Alberta,Edmonton, Alberta, Canada T6G With advances in estimation techniques, basic mod-els, such as measurement models, path models, and theirintegration into a general covariance structure SEM anal-ysis framework have been expanded to include, but are byno means limited to, the modeling of mean structures, in-teraction or nonlinear relations, and multilevel purpose of this module is to introduce the foundationsof SEM modeling with the basic covariance structure mod-els to new SEM researchers. Readers are assumed to havebasic statistical knowledge in multiple regression and anal-ysis of variance (ANOVA).

4 References and other resourceson current developments of more sophisticated models areprovided for interested is Structural Equation modeling ? Structural Equation modeling is a general term that hasbeen used to describe a large number of statistical modelsused to evaluate the validity of substantive theories withempirical data. Statistically, it represents an extension ofgeneral linear modeling (GLM) procedures, such as theANOVA and multiple regression analysis. One of the pri-mary advantages of SEM (vs. other applications of GLM)is that it can be used to study the relationships among la-tent constructs that are indicated by multiple measures. It isalso applicable to both experimental and non-experimentaldata, as well as cross-sectional and longitudinal data. SEMtakes a confirmatory (hypothesis testing) approach to theFall 200733multivariate analysis of a Structural theory, one that stipu-lates causal relations among multiple variables.

5 Thecausalpattern of intervariable relations within the theory is spec-ified a priori. The goal is to determine whether a hypothe-sized theoretical model is consistent with the data collectedto reflect this theory. The consistency is evaluated throughmodel-data fit, which indicates the extent to which the pos-tulated network of relations among variables is is a large sample technique (usuallyN>200; ,Kline, 2005, pp. 111, 178) and the sample size required issomewhat dependent on model complexity, the estimationmethod used, and the distributional characteristics of ob-served variables (Kline, pp. 14 15). SEM has a number ofsynonyms and special cases in the literature including pathanalysis, causal modeling , and covariance structure simple terms, SEM involves the evaluation of two models:ameasurementmodel and apathmodel.

6 They are ModelPath analysis is an extension of multiple regression in that itinvolves various multiple regression models or equations thatare estimated simultaneously. This provides a more effectiveand direct way of modeling mediation, indirect effects, andother complex relationship among variables. Path analysiscan be considered a special case of SEM in which structuralrelations among observed (vs. latent) variables are relations are hypotheses about directional influ-ences or causal relations of multiple variables ( , howindependent variables affect dependent variables). Hence,path analysis (or the more generalized SEM) is sometimesreferred to as causal modeling . Because analyzing interrela-tions among variables is a major part of SEM and these in-terrelations are hypothesized to generate specific observedcovariance (or correlation) patterns among the variables,SEM is also sometimes called covariance structure SEM, a variable can serve both as a source variable(called anexogenousvariable, which is analogous to an in-dependent variable) and a result variable (called anendoge-nousvariable, which is analogous to a dependent variable)in a chain of causal hypotheses.

7 This kind of variable isoften called amediator. As an example, suppose that fam-ily environment has a direct impact on learning motivationwhich, in turn, is hypothesized to affect achievement. In thiscase motivation is a mediator between family environmentand achievement; it is the source variable for achievementand the result variable for family environment. Furthermore,feedback loops among variables ( , achievement can inturn affect family environment in the example) are per-missible in SEM, as are reciprocal effects ( , learningmotivation and achievement affect each other).1In path analyses, observed variables are treated as if theyare measured without error, which is an assumption thatdoes not likely hold in most social and behavioral observed variables contain error, estimates of path co-efficients may be biased in unpredictable ways, especially forcomplex models ( , Bollen, 1989, p.)

8 151 178). Estimatesof reliability for the measured variables, if available, can beincorporated into the model to fix their error variances ( ,squared standard error of measurement via classical testtheory). Alternatively, if multiple observed variables thatare supposed to measure the same latent constructs areavailable, then a measurement model can be used to sepa-rate the common variances of the observed variables fromtheir error variances thus correcting the coefficients in themodel for ModelThe measurement of latent variables originated from psy-chometric theories. Unobserved latent variables cannot bemeasured directly but are indicated or inferred by responsesto a number of observable variables (indicators). Latentconstructs such as intelligence or reading ability are oftengauged by responses to a battery of items that are designedto tap those constructs.

9 Responses of a study participant tothose items are supposed to reflect where the participantstands on the latent variable. Statistical techniques, suchas factor analysis, exploratory or confirmatory, have beenwidely used to examine the number of latent constructs un-derlying the observed responses and to evaluate the adequacyof individual items or variables as indicators for the latentconstructs they are supposed to measurement model in SEM is evaluated through con-firmatory factor analysis (CFA). CFA differs from exploratoryfactor analysis (EFA) in that factor structures are hypoth-esized a priori and verified empirically rather than derivedfrom the data. EFA often allows all indicators to load on allfactors and does not permit correlated residuals. Solutionsfor different number of factors are often examined in EFAand the most sensible solution is interpreted.

10 In contrast,the number of factors in CFA is assumed to be known. InSEM, these factors correspond to the latent constructs rep-resented in the model. CFA allows an indicator to load onmultiple factors (if it is believed to measure multiple latentconstructs). It also allows residuals or errors to correlate (ifthese indicators are believed to have common causes otherthan the latent factors included in the model). Once the mea-surement model has been specified, Structural relations ofthe latent factors are then modeled essentially the same wayas they are in path models. The combination of CFA modelswith Structural path models on the latent constructs repre-sents the general SEM framework in analyzing ModelsCurrent developments in SEM include the modeling of meanstructures in addition to covariance structures, the modelingof changes over time (growth models) and latent classes orprofiles, the modeling of data having nesting structures ( ,students are nested within classes which, in turn, are nestedwith schools; multilevel models), as well as the modeling ofnonlinear effects ( , interaction).