Within-List Covariances

Using k variable names in the variables list on the left-hand side of an equal sign in a COV statement means that the parameter list (parameter-spec) on the right-hand side refers to all distinct variable pairs. Order is very important. The order relation between the left-hand-side variable pairs and the right-hand-side parameter list is illustrated by the following within-list covariances specification in the COV statement:

   cov E1-E4 = PHI1-PHI6 ;

This specification is equivalent to the following specification:

   cov E2 E1 = phi1,
       E3 E1 = phi2, E3 E2 = phi3,
       E4 E1 = phi4, E4 E2 = phi5, E4 E3 = phi6;

Because cov(a,b) is the same as cov(b,a), you need to specify only one of them for a covariance parameter of any two variables.

When you use prefix names on the right-hand sides, you do not have to count the exact number of parameters. For example, the following statement generates distinct parameter names by appending distinct numbers to the prefix phi:

   cov E1-E4 = phi__ ;

Between-List Covariances

Using and variable names in the two lists (separated by an asterisk) on the left-hand side of an equal sign in a COV statement means that the covariance parameter list on the right-hand side refers to all distinct variable pairs. Again, order is very important. The order relation between the left-hand-side variable pairs and the right-hand-side parameter list is illustrated by the following between-list covariances specification:

   cov E1 E2 * E3 E4 = phi1-phi4;

This is equivalent to the following specification:

   cov  E1 E3 = phi1, E1 E4 = phi2,
        E2 E3 = phi3, E2 E4 = phi4;

Using prefix names on the right-hand sides lets you achieve the same purpose without counting the number of parameters. That is, you can specify the between-list covariances equivalently by the following statement:

   cov  E1 E2 * E3 E4 = phi__ ;

Figure 88.3 Within-List and Between-List Covariances

Automatic Covariance Parameters among Exogenous Manifest Variables

In LINEQS models, all covariances among exogenous manifest variables are free parameters to estimate unless you specify otherwise in the COV statement. The parameter names for these covariances are generated by PROC TCALIS with the prefix _Add and appended with integer suffixes. If you want any of these covariances be fixed values, you must specify them explicitly in the COV statement.

Emphasis must be put on the manifest variables for automatic covariance parameters generation. Covariaces among exogenous latent variables by default are fixed zeros unless they are specified otherwise in the COV statement. The same applies to any covariance between an exogenous latent variable and an exogenous manifest variable. Only among the set of exogenous mainfest variables are the covariance parameters automatically generated if they are not specified explicitly in the COV statement. This is different than the case of variance parameters. Variances of all exogenous variables are automatically generated if they are not explicitly specified in the STD statement.

Modifying a Parameter Specification from a Reference Model

If you define a new LINEQS model by using a reference (or base) model in the REFMODEL statement, in some situations you might want to modify some parameter specifications from the COV statement of the reference model before transferring the specifications to the new model. If you want to change a particular COV parameter specification from the reference model, you can simply put the corresponding parameter location with a new parameter specification in the new model. If you want to delete a particular parameter location from the reference model, you can put that parameter location with a missing parameter value in the new model. For example, if the covariance between variables V1 and V2 are defined in the reference model and you do not want this parameter location in your new model, you can use the following statement to delete the parameter location in the new model:

   cov  V1 V2 = .;

Note that the missing value syntax is valid only when you use with the REFMODEL statement. See the section Modifying a LINEQS Model from a Reference Model for a more detailed example of LINEQS model respecification.

In LINEQS models, all covariances among exogenous manifest variables are free parameters to estimate unless you specify otherwise in the COV statement. This is also true for LINEQS models specified under the REFMODEL statement. In the reference model, covariance parameters generated by PROC TCALIS are named with the _Add prefix and appended with integer suffixes. These parameter specifications in the reference model do not transfer to the new model. Only after the new model is resolved from the reference model, the REFMODEL statement options, the RENAMEPARM statement, and model respecifications do the remaining unspecified covariances among exogenous manifest variables generate additional covariance parameters with the _Add prefix and integer suffixes. In this way, the covariance parameters in the new model are not constrained to be the same as the corresponding parameters in the reference model. If you want any of these covariance parameters to be constrained across the models, you must specify them explicitly in the COV statement of the reference model so that the covariance specification is transferred to the new model.