Specifying Variables

The first task is accomplished in the LISMOD statement. In the LISMOD statement, you define the lists of variables of interest: YVAR=, XVAR=, ETAVAR=, and XIVAR= lists, respectively for the -variables, -variables, -variables, and the -variables. While you provide the names of variables in these lists, you also define implicitly the numbers of four types of variables: , , , and . The variables in the YVAR= and XVAR= lists are manifest variables and therefore must be present in the analyzed data set. The variables in the ETAVAR= and XIVAR= lists are latent factors, the names of which are assigned by the researcher to represent their roles in the substantive theory. After these lists are defined, the dimensions of the model matrices are also defined by the number of variables on various lists. In addition, the variable orders in the lists are referred to by the row and column variables of the model matrices.

Unlike the LINEQS model, in the LISMOD model you do not need to use the 'F' or 'f' prefix to denote factors in the ETAVAR= or XIVAR= list. You can use any valid SAS variable names for the factors, especially those names that reflect the nature of the factors. To avoid confusion with other names in the model, some general rules are recommended. See the section Naming Variables and Parameters for these general rules about naming variables and parameters.

Specifying Parameters in Model Matrices

The second task is accomplished by the MATRIX statements. In each MATRIX statement, you specify the model matrix by using the matrix names described in the previous table. Then you specify the parameters (free or fixed) in the locations of the model matrix. You can use as many MATRIX statements as needed for defining your model. But each model matrix can be specified only in one MATRIX statement, and each MATRIX statement is used for specifying only one model matrix.

An Example

In the section LISMOD Model, the LISMOD modeling language is used to specify the model described in the section A Structural Equation Example. In the LISMOD statement, you define four lists of variables, as shown in the following statement:

lismod
   yvar  = Anomie67 Powerless67 Anomie71 Powerless71,
   xvar  = Education SEI,
   etav  = Alien67 Alien71,
   xivar = SES;

Endogenous latent factors are specified in the ETAVAR= list. Exogenous latent factors are specified in the XIVAR= list. In this case, Alien67 and Alien71 are the -variables, and SES is the only -variable in the model. Manifest variables that are indicators of endogenous latent factors in are specified in the YVAR= list. In this case, they are the Anomie and Powerless variables, measured at two different time points. Manifest variables that are indicators of exogenous latent factors in are specified in the XVAR= list. In this case, they are the Education and the SEI variables. Implicitly, the dimensions of the model matrices are defined by these lists already; that is, , , , and .

The MATRIX statements are used to specify parameters in the model matrices. For example, in the following statement you define the _LAMBDAX_ () matrix with two nonzero entries:

 matrix _LAMBDAX_ [1,1] = 1.0,
                  [2,1] = lambda;

The first parameter location is for [1,1], which is the effect of SES (the first variable in the XIVAR= list) on Education (the first element in the XVAR= list). A fixed value of is specified there. The second parameter location is for [2,1], which is the effect of SES (the first variable in the XIVAR= list) on SEI (the second variable in the XVAR= list). A parameter named lambda without an initial value is specified there.

Another example is shown as follows:

matrix _THETAY_  [1,1] = theta1,
                 [2,2] = theta2,
                 [3,3] = theta1,
                 [4,4] = theta2,
                 [3,1] = theta5,
                 [4,2] = theta5;

In this MATRIX statement, the error variances and covariances (that is, the matrix) for the -variables are specified. The diagonal elements of the _THETAY_ matrix are specified by parameters theta1, theta2, theta1, and theta2, respectively, for the four -variables Anomie67, Powerless67, Anomie71, and Powerless71. By using the same parameter name theta1, the error variances for Anomie67 and Anomie71 are implicitly constrained. Similarly, the error variances for Powerless67 and Powerless71 are also implicitly constrained. Two more parameter locations are specified. The error covariance between Anomie67 and Anomie71 and the error covariance between Powerless67 and Powerless71 are both represented by the parameter theta5. Again, this is an implicit constraint on the covariances. All other unspecified elements in the _THETAY_ matrix are treated as fixed zeros.

In this example, no parameters are specified for matrices _ALPHA_, _KAPPA_, _NUY_, or _NUX_. Therefore, mean structures are not modeled.

Models 1, 2, 3, and 4—Presence of Both - and -Variables

Submodels 1, 2, 3, and 4 are characterized by the presence of both - and -variables in the model. In fact, Model 1 is the full model with the presence of all four types of variables. All twelve model matrices are parameter matrices in this model.

Depending on the absence of the latent factor lists, manifest variables can replace the role of the latent factors in Models 2–4. For example, the absence of the ETAVAR= list in Model 2 means is equivalent to (). Consequently, you cannot, nor do you need to, use the MATRIX statement to specify parameters in the _LAMBDAY_, _THETAY_, or _NUY_ matrices under this model. Similarly, because is equivalent to () in Model 3, you cannot, nor do you need to, use the MATRIX statement to specify the parameters in the _LAMBDAX_, _THETAX_, or _NUX_ matrices. In Model 4, is equivalent to () and is equivalent to (). None of the six model matrices in the measurement equations are defined in the model. Matrices in which you can specify parameters by using the MATRIX statement are listed in the last column of the table.

Describing Model 4 as a regression model is a simplification. Because can regress on itself in the model equation, the regression description is not totally accurate for Model 4. Nonetheless, if is a null matrix, the equation describes a multivariate regression model with outcome variables and predictor variables . This model is the TYPE 2A model in LISREL VI (Jöreskog and Sörbom; 1985).

You should also be aware of the changes in meaning of the model matrices when there is an equivalence between latent factors and manifest variables. For example, in Model 4 the and are now the covariance matrix and mean vector, respectively, of manifest variables , while in Model 1 (the complete model) these matrices are of the latent factors .

Models 5 and 6 — Presence of -Variables and Absence of -Variables

Models 5 and 6 are characterized by the presence of the -variables and the absence of -variables.

Model 5 is simply a factor model for measured variables , with representing the factor loading matrix, the error covariance matrix, and the factor covariance matrix. If mean structures are modeled, represents the factor means and is the intercept vector. This is the TYPE 1 submodel in LISREL VI (Jöreskog and Sörbom; 1985).

Model 6 is a special case where there is no model equation. You specify the mean and covariance structures (in and , respectively) for the manifest variables directly. The -variables are treated as exogenous variables in this case. Because this submodel uses direct mean and covariance structures for measured variables, it can also be handled more easily by the MSTRUCT modeling language. See the MSTRUCT statement and the section The MSTRUCT Model for more details.

Note that because -variables cannot exist in the absence of -variables (see one of the three aforementioned principles for deriving submodels), adding the ETAVAR= list alone to these two submodels does not generate new submodels that can be analyzed by PROC CALIS.

Models 7, 8 ,9 and 10—Presence of -Variables and Absence of -Variables

Models 7–10 are characterized by the presence of the -variables and the absence of -variables.

Model 7 is a factor model for -variables (TYPE 3B submodel in LISREL VI). It is similar to Model 5, but with regressions among latent factors allowed. When is null, Model 7 functions the same as Model 5. It becomes a factor model for -variables, with representing the factor loading matrix, the error covariance matrix, the factor covariance matrix, the factor means, and the intercept vector.

Model 8 (TYPE 2B submodel in LISREL VI) is a model for studying the mean and covariance structures of -variables, with regression among -variables allowed. When is null, the mean structures of are specified in and the covariance structures are specified in . This is similar to Model 6. However, there is an important distinction. In Model 6, the -variables are treated as exogenous (no model equation at all). But the -variables are treated as endogenous in Model 8 (with or without ). Consequently, the default parameterization would be different for these two submodels. See the section Default Parameters for details about the default parameterization.

Model 9 represents a modified version of the second-order factor model for . It would be a standard second-order factor model when is null. This is the TYPE 3A submodel in LISREL VI. With being null, represents the first-order factors and represents the second-order factors. The first- and second-order factor loading matrices are and , respectively.

Model 10 is another form of factor model when is null, with factors represented by and manifest variables represented by . However, if is indeed a null matrix in applications, you might want to use Model 5, in which the factor model specification is more direct and intuitive.