LINEQS Model Statement

The CALIS Procedure

LINEQS Model Statement

LINEQS equation < , equation ... > ;

          where equation represents dependent = term < + term ... >
          and where term represents one of the following:
                $\bullet$ coefficient-name < (number) > variable-name
                $\bullet$ prefix-name < (number) > variable-name
                $\bullet$ < number > variable-name

The LINEQS statement defines the LINEQS model

$eta &=& {\beta}eta + {\gamma}{\xi}\ C&=& J(I- B)^{-1} {{\Gamma}}{{\Phi}}{{\Gamma}}^' ((I- B)^{-1})^' J^'$

You can specify only one LINEQS statement with each PROC CALIS statement. There are some differences from Bentler's notation in choosing the variable names. The length of each variable name is restricted to eight characters. The names of the manifest variables are defined in the DATA= input data set. The VAR statement can be used to select a subset of manifest variables in the DATA= input data set to analyze. You do not need to use a V prefix for manifest variables in the LINEQS statement nor do you need to use a numerical suffix in any variable name. The names of the latent variables must start with the prefix letter F (for Factor); the names of the residuals must start with the prefix letters E (for Error) or D (for Disturbance). The trailing part of the variable name can contain letters or digits. The prefix letter E is used for the errors of the manifest variables, and the prefix letter D is used for the disturbances of the latent variables. The names of the manifest variables in the DATA= input data set can start with F, E, or D, but these names should not coincide with the names of latent or error variables used in the model. The left-hand side (that is, endogenous dependent variable) of each equation should be either a manifest variable of the data set or a latent variable with prefix letter F. The left-hand-side variable should not appear on the right-hand side of the same equation; this means that matrix ${\beta}$ should not have a nonzero diagonal element. Each equation should contain, at most, one E or D variable.

The equations must be separated by a comma. The order of the equations is arbitrary. The displayed output generally contains equations and terms in an order different from the input.

Coefficients to estimate are indicated in the equations by a name preceding the independent variable's name. The coefficient's name can be followed by a number inside parentheses indicating the initial value for this coefficient. A number preceding the independent variable's name indicates a constant coefficient. If neither a coefficient name nor a number precedes the independent variable's name, a constant coefficient of 1 is assumed.

If the initial value of a parameter is not specified in the equation, the initial value is chosen in one of the following ways:

If you specify the RANDOM= option in the PROC CALIS statement, the variable obtains a randomly generated initial value r, such that $0 \le r \le 1$ . The uninitialized parameters in the diagonals of the central model matrices are given the nonnegative random values r multiplied by 10, 100, or the value specified in the DEMPHAS= option.
If the RANDOM= option is not used, PROC CALIS tries to estimate the initial values.
If the initial values cannot be estimated, the value of the START= option is used as an initial value.

In Bentler's notation, estimated coefficients are indicated by asterisks. Referring to a parameter in Bentler's notation requires the specification of two variable names that correspond to the row and column of the position of the parameter in the matrix. Specifying the estimated coefficients by parameter names makes it easier to impose additional constraints with code. You do not need any additional statements to express equality constraints. Simply specify the same name for parameters that should have equal values.

If your model contains many unconstrained parameters and it is too cumbersome to find different parameter names, you can specify all those parameters by the same prefix name. A prefix is a short name followed by a colon. The CALIS procedure then generates a parameter name by appending an integer suffix to this prefix name. The prefix name should have no more than five or six characters so that the generated parameter name is not longer than eight characters. To avoid unintentional equality constraints, the prefix names should not coincide with explicitly defined parameter names.

For example, you can specify confirmatory second-order factor analysis model

S = F₁ F₂ P₂ F₂' F₁' + F₁ U₂² F₁' + U₁²

by using the LINEQS and STD statements:

   LINEQS
          V1 = X1 F1 + E1,
          V2 = X2 F1 + E2,
          V3 = X3 F1 + E3,
          V4 = X4 F2 + E4,
          V5 = X5 F2 + E5,
          V6 = X6 F2 + E6,
          V7 = X7 F3 + E7,
          V8 = X8 F3 + E8,
          V9 = X9 F3 + E9,
          F1 = Y1 F4 + D1,
          F2 = Y1 F4 + Y2 F5 + D2,
          F3 = Y2 F5 + D3;
   STD
          E1-E9 = 9 * U:,
          D1-D3 = 3 * V:,
          F4 F5 = 2 * P;
   RUN;

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