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<H1>What is Structural Equation Modeling?</H1>
<P><IMG src="http://www2.gsu.edu/~mkteer/fitfunc.gif" align=bottom>
<P>Structural equation modeling, or SEM, is a very general, chiefly linear, chiefly cross-sectional statistical modeling technique. Factor analysis, path analysis and regression all represent special cases of SEM.
<P>SEM is a largely confirmatory, rather than exploratory, technique. That is, a researcher are more likely to use SEM to determine whether a certain model is valid., rather than using SEM to "find" a suitable model--although SEM analyses often involve a certain exploratory element.
<P>In SEM, interest usually focuses on <a href="http://www.gsu.edu/~mkteer/sem2.html#latentvar" target="_blank" >latent constructs</A>--abstract psychological variables like "intelligence" or "attitude toward the brand"--rather than on the manifest variables used to measure these constructs. Measurement is recognized as difficult and error-prone. By explicitly modeling measurement error, SEM users seek to derive unbiased estimates for the relations between latent constructs. To this end, SEM allows multiple measures to be associated with a single latent construct.
<P>A structural equation model implies a structure of the covariance matrix of the measures (hence an alternative name for this field, "analysis of covariance structures"). Once the model's parameters have been estimated, the resulting model-implied covariance matrix can then be compared to an empirical or data-based covariance matrix. If the two matrices are consistent with one another, then the structural equation model can be considered a plausible explanation for relations between the measures.
<P>Compared to regression and factor analysis, SEM is a relatively young field, having its roots in papers that appeared only in the late 1960s. As such, the methodology is still developing, and even fundamental concepts are subject to challenge and revision. This rapid change is a source of excitement for some researchers and a source of frustration for others.</P>
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A structural equation model may include two types of latent constructs--exogenous and endogenous. In the most traditional system, exogenous constructs are indicated by the Greek character "ksi" (at left) 
and endogenous constructs are indicated by the Greek character "eta" (at right). These two types of constructs are distinguished on the basis of whether or not they are dependent variables in any equation in the system of equations represented by the model. Exogenous constructs are independent variables in all equations in which they appear, while endogenous constructs are dependent variables in at least one equation--although they may be independent variables in other equations in the system. In graphical terms, each endogenous construct is the target of at least one one-headed arrow, while exogenous constructs are only targeted by two-headed arrows. 
Parameters representing regression relations between latent constructs are typically labeled with the Greek character "gamma" (at left) for the regression of an endogenous construct on an exogenous construct, or with the Greek character "beta" (at right) for the regression of one endogenous construct on another endogenous construct. 
Typically in SEM, exogenous constructs are allowed to covary freely. Parameters labeled with the Greek character "phi" (at left) represent these covariances. This covariance comes from common predictors of the exogenous constructs which lie outside the model under consideration. 
Few SEM researchers expect to perfectly predict their dependent constructs, so model typically include a structural error term, labeled with the Greek character "zeta" (at right). To achieve consistent parameter estimation, these error terms are assumed to be uncorrelated with the model's exogenous constructs. (Violations of this assumption come about as a result of the 
SEM researchers use manifest variables--that is, actual measures and scores--to ground their latent construct models with real data. Manifest variables associated with exogenous constructs are labeled X, while those associated with endogenous constructs are labeled Y. Otherwise, there is no fundamental distinction between these measures, and a measure that is labeled X in one model may be labeled Y in another. 
In SEM, each latent construct is usually associated with multiple measures. SEM researchers most commonly link the latent constructs to their measures through a factor analytic measurement model. That is, each latent construct is modeled as a common factor underlying the associated measures. These "loadings" linking constructs to measures are labeled with the Greek character "lambda" (at left). Structural equation models can include two separate "lambda matrices, one on the X side and one on the Y side. In SEM applications, the most common measurement model is the congeneric measurement model, where each measure is associated with only one latent construct, and all covariation between measures is a consequence of the relations between measures and constructs. 
SEM users typically recognize that their measures are imperfect, and they attempt to model this imperfection. Thus, structural equation models include terms representing measurement error. In the context of the factor analytic measurement model, these measurement error terms are uniquenesses or unique factors associated with each measure. Measurement error terms associated with X measures are labeled with the Greek character "delta" (at left) while terms associated with Y measures are labeled with "epsilon" (at right). Conceptually, almost every measure has an associated error term. In other words, almost every measure is acknowledged to include some error. 
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