Introduction to Structural Equation Modeling with Latent Variables

As discussed in the preceding section, if you try to fit the errors-in-variables model with measurement errors in both X and Y without applying certain constraints, the model is not identified and you cannot obtain unique estimates of the parameters. For example, the errors-in-variables model with measurement errors in both X and Y has five parameters (one coefficient and four variances). The covariance matrix of the observed variables Y and X has only three elements that are free to vary, since Cov(Y,X)=Cov(X,Y). Therefore, the covariance structure can be expressed as three equations in five unknown parameters. Since there are fewer equations than unknowns, there are many different sets of values for the parameters that provide a solution for the equations. Such a model is said to be underidentified.

If the number of parameters equals the number of free elements in the covariance matrix, then there might exist a unique set of parameter estimates that exactly reproduce the observed covariance matrix. In this case, the model is said to be just-identified or saturated.

If the number of parameters is less than the number of free elements in the covariance matrix, there might exist no set of parameter estimates that reproduces the observed covariance matrix exactly. In this case, the model is said to be overidentified. Various statistical criteria, such as maximum likelihood, can be used to choose parameter estimates that approximately reproduce the observed covariance matrix. If you use ML, FIML, GLS, or WLS estimation, PROC CALIS can perform a statistical test of the goodness of fit of the model under the certain statistical assumptions.

If the model is just-identified or overidentified, it is said to be identified. If you use ML, FIML, GLS, or WLS estimation for an identified model, PROC CALIS can compute approximate standard errors for the parameter estimates. For underidentified models, PROC CALIS obtains approximate standard errors by imposing additional constraints resulting from the use of a generalized inverse of the Hessian matrix.

You cannot guarantee that a model is identified simply by counting the parameters. For example, for any latent variable, you must specify a numeric value for the variance, or for some covariance involving the variable, or for a coefficient of an indicator variable. Otherwise, the scale of the latent variable is indeterminate, and the model is underidentified regardless of the number of parameters and the size of the covariance matrix. As another example, an exploratory factor analysis with two or more common factors is always underidentified because you can rotate the common factors without affecting the fit of the model.

PROC CALIS can usually detect an underidentified model by computing the approximate covariance matrix of the parameter estimates and checking whether any estimate is linearly related to other estimates (Bollen 1989, pp. 248–250), in which case PROC CALIS displays equations showing the linear relationships among the estimates. Another way to obtain empirical evidence regarding the identification of a model is to run the analysis several times with different initial estimates to see whether the same final estimates are obtained. Bollen (1989) provides detailed discussions of conditions for identification in a variety of models.