COSAN Model Statement

The CALIS Procedure

COSAN Model Statement

COSAN matrix_term < + matrix_term... > ;

          where matrix_term represents
               matrix_definition < * matrix_definition ... >
          and matrix_definition represents
               matrix_name (column_number < ,general_form < ,transformation >> )

The COSAN statement constructs the symmetric matrix model for the covariance analysis mentioned earlier (see the section "The Generalized COSAN Model"):

C = F₁P₁F₁' + ... + F_mP_mF_m' ,

F_k = F_k₁ ... F_{k_n(k)}, and P_k = P_k' , k = 1, ... ,m

$F_{k_{j}} = \{\matrix{G_{k_{j}} \cr G^{-1}_{k_{j}} \cr (I-G_{k_{j}})^{-1} \cr} . j=1, ... ,n(k), {and} P_k = \{\matrix{Q_{k} \cr Q^{-1}_k \cr} .$

You can specify only one COSAN statement with each PROC CALIS statement. The COSAN statement contains m matrix_terms corresponding to the generalized COSAN formula. The matrix_terms are separated by plus signs (+) according to the addition of the terms within the model.

Each matrix_term of the COSAN statement contains the definitions of the first n(k)+1 matrices, F_{k_j} and P_k, separated by asterisks (*) according to the multiplication of the matrices within the term. The matrices F_k' of the right-hand-side product are redundant and are not specified within the COSAN statement.

Each matrix_definition consists of the name of the matrix (matrix_name), followed in parentheses by the number of columns of the matrix (column_number) and, optionally, one or two matrix properties, separated by commas, describing the form of the matrix.

The number of rows of the first matrix in each term is defined by the input correlation or covariance matrix. You can reorder and reduce the variables in the input moment matrix using the VAR statement. The number of rows of the other matrices within the term is defined by the number of columns of the preceding matrix.

The first matrix property describes the general form of the matrix in the model. You can choose one of the following specifications of the first matrix property. The default first matrix property is GEN.

Code: Description
IDE: specifies an identity matrix; if the matrix is not square, this specification describes an identity submatrix followed by a rectangular zero submatrix.
ZID: specifies an identity matrix; if the matrix is not square, this specification describes a rectangular zero submatrix followed by an identity submatrix.
DIA: specifies a diagonal matrix; if the matrix is not square, this specification describes a diagonal submatrix followed by a rectangular zero submatrix.
ZDI: specifies a diagonal matrix; if the matrix is not square, this specification describes a rectangular zero submatrix followed by a diagonal submatrix.
LOW: specifies a lower triangular matrix; the matrix can be rectangular.
UPP: specifies an upper triangular matrix; the matrix can be rectangular.
SYM: specifies a symmetric matrix; the matrix cannot be rectangular.
GEN: specifies a general rectangular matrix (default).

The second matrix property describes the kind of inverse matrix transformation. If the second matrix property is omitted, no transformation is applied to the matrix.

Code: Description
INV: uses the inverse of the matrix.
IMI: uses the inverse of the difference between the identity and the matrix.

You cannot specify a nonsquare parameter matrix as an INV or IMI model matrix. Specifying a matrix of type DIA, ZDI, UPP, LOW, or GEN is not necessary if you do not use the unspecified location list in the corresponding MATRIX statements. After PROC CALIS processes the corresponding MATRIX statements, the matrix type DIA, ZDI, UPP, LOW, or GEN is recognized from the pattern of possibly nonzero elements. If you do not specify the first matrix property and you use the unspecified location list in a corresponding MATRIX statement, the matrix is recognized as a GEN matrix. You can also generate an IDE or ZID matrix by specifying a DIA, ZDI, or IMI matrix and by using MATRIX statements that define the pattern structure. However, PROC CALIS would be unable to take advantage of the fast algorithms that are available for IDE and ZID matrices in this case.

For example, to specify a second-order factor analysis model

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

with m₁=3 first-order factors and m₂=2 second-order factors and with n=9 variables, you can use the following COSAN statement:

   cosan F1(3) * F2(2) * P2(2,SYM)+F1(3) * U2(3,DIA) * I1(3,IDE)
         +U1(9,DIA) * I2(9,IDE)

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