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 The CALIS Procedure

## FACTOR Model Statement

FACTOR < options > ;
You can use the FACTOR statement to specify an exploratory or confirmatory first-order factor analysis of the given covariance or correlation matrix C,
C = FF' + U,     U = diag
or
C = FPF' + U,     P = P'
where U is a diagonal matrix and P is symmetric. Within this section, n denotes the number of manifest variables corresponding to the rows and columns of matrix C, and m denotes the number of latent variables (factors or components) corresponding to the columns of the loading matrix F.

You can specify only one FACTOR statement with each PROC CALIS statement. You can specify higher-order factor analysis problems using a COSAN model specification. PROC CALIS requires more computing time and memory than PROC FACTOR because it is designed for more general structural estimation problems and is unable to exploit the special properties of the unconstrained factor analysis model.

For default (exploratory) factor analysis, PROC CALIS computes initial estimates for factor loadings and unique variances by an algebraic method of approximate factor analysis. If you use a MATRIX statement together with a FACTOR model specification, initial values are computed by McDonald's (McDonald and Hartmann 1992) method (if possible). For details, see "Using the FACTOR and MATRIX Statements". If neither of the two methods are appropriate, the initial values are set by the START= option.

The unrestricted factor analysis model is not identified because any orthogonal rotated factor loading matrix is equivalent to the result F,

To obtain an identified factor solution, the FACTOR statement imposes zero constraints on the m(m - 1)/2 elements in the upper triangle of F by default.

The following options are available in the FACTOR statement.

COMPONENT | COMP
computes a component analysis instead of a factor analysis (the diagonal matrix U in the model is set to 0). Note that the rank of FF' is equal to the number m of components in F. If m is smaller than the number of variables in the moment matrix C, the matrix of predicted model values is singular and maximum likelihood estimates for F cannot be computed. You should compute ULS estimates in this case.

HEYWOOD | HEY
constrains the diagonal elements of U to be nonnegative; in other words, the model is replaced by
C = FF' + U2 ,     U = diag

N = m
specifies the number of first-order factors or components. The number m of factors should not exceed the number n of variables in the covariance or correlation matrix analyzed. For the saturated model, m=n, the COMP option should generally be specified for U = 0; otherwise, df < 0. For m = 0 no factor loadings are estimated, and the model is C = U, with U = diag. By default, m=1.

NORM
normalizes the rows of the factor pattern for rotation using Kaiser's normalization.

ROTATE | R = name
specifies an orthogonal rotation. By default, ROTATE=NONE. The possible values for name are as follows:

PRINCIPAL | PC
specifies a principal axis rotation. If ROTATE=PRINCIPAL is used with a factor rather than a component model, the following rotation is performed:

where the columns of matrix T contain the eigenvectors of Fold' Fold.
QUARTIMAX | Q
specifies quartimax rotation.
VARIMAX | V
specifies varimax rotation.
EQUAMAX | E
specifies equamax rotation.
PARSIMAX | P
specifies parsimax rotation.
NONE
performs no rotation (default).

### Using the FACTOR and MATRIX Statements

You can specify the MATRIX statement and the FACTOR statement to compute a confirmatory first-order factor or component analysis. You can define the elements of the matrices F, P, and U of the oblique model,
C = FPF' + U2 ,     P = P' ,     U = diag

To specify the structure for matrix F, P, or U, you have to refer to the matrix _F_ , _P_ , or _U_ in the MATRIX statement. Matrix names automatically set by PROC CALIS always start with an underscore. As you name your own matrices or variables, you should avoid leading underscores.

The default matrix forms are as follows.
_F_
lower triangular matrix (0 upper triangle for problem identification, removing rotational invariance)
_P_
identity matrix (constant)
_U_
diagonal matrix

For details about specifying the elements in matrices, see the section "MATRIX Statement". If you are using at least one MATRIX statement in connection with a FACTOR model statement, you can also use the BOUNDS or PARAMETERS statement and program statements to constrain the parameters named in the MATRIX statement. Initial estimates are computed by McDonald's (McDonald and Hartmann 1992) method. McDonald's method of computing initial values works better if you scale the factors by setting the factor variances to 1 rather than by setting the loadings of the reference variables equal to 1.

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