The structural equation models considered in this book are special cases of the so-calledgeneral LInear Structural RELationships(LISREL)model. To define it formally, denote by Y the vector ofpobserved variables in a considered study (p> 1), byhthat ofqlatent factors assumed in it (q> 0), and byethe vector ofppertinent residuals (error terms). Designating byL
thep×qmatrix of factor loadings, bynthe p × 1 vector of observed vari- able mean intercepts, byatheq× 1 vector of latent variable intercepts, and byztheq× 1 vector of corresponding latent disturbance terms, the general LISREL model is represented by the following pair of equations:
Y =n+Lh+eand (A1.1)
h=a+ Bh+z, (A1.2)
where in addition the matrix I – B is assumed invertible. In this introductory text, for simplicity the additional assumption of normality of the variables in Y,h,e, andzis made, which are continuous, as well as that of uncorre- latedness ofeandz,eandh, andzandh. Components of each of the vectors
e,z, andhmay be correlated among themselves, however, assuming overall model identification. Further, the model classes considered in this book also assumen=a= 0, which does not pose any special restriction of gener- ality in many social and behavioral studies when the units and origins of measurement are not meaningful (or the analyzed variables may be consid- ered mean-centered without loss of relevant information for a particular in- vestigation.) The validity of Equations (A1.1) and (A1.2) is assumed for each individual in a sample, but for simplicity of notation we suppress the sub- ject subindex in this Appendix. We stress that the LISREL model defined by Equations (A1.1) and (A1.2) also includes the case where manifest variables influence observed and/or latent variables, which is seen by noting that ob- served predictors can be formally represented by error-free latent variables (i.e.,h’s with correspondinge’s being 0 and unitary pertinent elements of
L) in the right-hand sides of each of these two equations.
Under these assumptions, via simple algebra Equations (A1.1) and (A1.2) entail that the implied, observed variable covariance matrixShas the following form:
S=L(I–B)–1Y(I–B')–1L' +Q, (A1.3)
where priming denotes transposition,Y= Cov(x) is the latent disturbance terms covariance matrix, andQ= Cov(e) is that of the residuals.
A simple inspection of the right-hand side of Equation (A1.3) demon- strates that the parameters of the general LISREL model are among: (a) the
factor loadings (elements ofL); (b) the latent regression coefficients (ele- ments of B); and (c) the variances and covariances of the latent disturbance terms and residuals (the elements ofYandQ) that collectively represent the independent variables of the model. (In case a considered model has no latent dependent variable, as in confirmatory factor analysis models, seth=
zandB= 0 in (A1.2) and use the immediately preceding sentence in this paragraph to obtain its model parameters.) This observation presents the rationale behind the rules for determining model parameters discussed in this chapter. Also, denoting bygthe vector of all model parameters (i.e., all variances of and covariances between independent variables as well as all regression coefficients and factor loadings), Equation (A1.3) states that S =
S(g). That is, an implication of any considered structural equation model is the structuring of all elements of the pertinent population covariance ma- trix in terms of fewer in number, more fundamental parameters, viz. those ing. This is the essence of the model parameterization that is invoked as a consequence of adopting a particular structural equation model.
The fit function for the model fitting and estimation approach used throughout this book, that of the maximum likelihood method, is de- fined as
FML= “distance”(S,S(g)) = –1n|SS(g)
–1| + tr(SS(g)–1) –p, (A1.4)
where |.| denotes matrix determinant and tr(.) trace (sum of main diagonal elements). Using special numerical optimization algorithms, this fit func- tion is minimized across the parameter space, i.e., the set of all admissible values of all model parameters (in particular, typically positive variances). When the model is correct and fitted to the covariance matrix for a large sample,T=n FML, minfollows a central chi-square distribution with degrees of freedom being those of the fitted model, whereFML, minis the minimum of (A1.4),n = N– 1 andNis sample size. This fit function is derivable in the context of the likelihood ratio test theory, when comparing a given model to a saturated model fitted to the same set of observed variables (e.g., Bollen, 1989). The minimizer of (A1.4),g$, consists of the point estimates of all model parameters; their standard errors are obtained as the correspond- ing elements on the main diagonal of the inverted observed information matrix.
The unweighted least squares (ULS) method is based on minimizing, across the set of all possible values forg, the fit function
FULS= .5 tr[(S –S(g))
2
] , (A1.5)
while the generalized least squares (GLS) approach minimizes the fit function
FGLS= .5 tr[(I – S
The asymptotically distribution free (ADF) method (also called weighted least squares method) minimizes the fit function
FADF=(s–s(g))'W
–1(s–s(g)) , (A1.7)
wheresdenotes the strung-out vector of nonredundant elements ofS,s(g)
the similar vector of their counterparts inS(g), and W is a weight matrix that represents a consistent estimate of the large sample covariance matrix of the elements ofS(considered as random variables themselves). As shown in the literature (e.g., Bollen, 1989), with large samples the ULS, ML, and GLS estimation methods can be considered special cases of the ADF method, ob- tained with appropriate choices of the matrix W. Corrections of the chi- square value and parameter standard errors, which are functions of the ma- trix W, yield from the ML test statistic and standard errors robust ML test sta- tistics and robust standard errors, respectively (e.g., Bentler, 2004).
The matrix of covariance residuals,S–S(g$), which is also often referred to as matrix of model residuals, contains information about local goodness of fit of the model. These residuals, i.e., the elements ofS–S(g$), are ex- pressed in the original metrics of manifest variables that may be quite dis- similar across variables, and thus are in general hard to interpret. Their standardized counterparts, called standardized residuals, are expressed in a uniform metric across all variables and can therefore be used to locate pairs of variables whose interrelationship indices are markedly misfit. Like the individual case residuals in regression analysis, in SEM the model resid- uals are not unrelated to one another. However, those of them whose stan- dardized versions are in absolute value higher than 2 generally indicate parts of the model that are considerably inconsistent with the data. A posi- tive residual means underprediction by the model of the covariance for the two variables involved, and may be made smaller by introducing a parame- ter additionally contributing to their interrelationship index (as reflected in its counterpart element ofS(g$)). Conversely, a negative residual means overprediction by the model of that covariance, and may be rendered smaller in magnitude (i.e., absolute value) by deleting a parameter contrib- uting to this interrelationship index (as reflected in its counterpart inS(g$)). An examination of model residuals, in particular standardized residuals, is therefore recommendable as an essential step in the process of model fit evaluation.