PROC CALIS: FACTOR Model Statement :: SAS/STAT(R) 9.2 User's Guide, Second Edition

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 $\text{[math]}$ ,

$\text{[math]}$

where $\text{[math]}$ is a diagonal matrix and $\text{[math]}$ is symmetric. Within this section, $\text{[math]}$ denotes the number of manifest variables corresponding to the rows and columns of matrix $\text{[math]}$ , and $\text{[math]}$ denotes the number of latent variables (factors or components) corresponding to the columns of the loading matrix $\text{[math]}$ .

You can specify only one FACTOR statement with each PROC CALIS statement. You can specify higher-order factor analysis problems by 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 is 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 $\text{[math]}$ is equivalent to the result $\text{[math]}$ ,

$\text{[math]}$

To obtain an identified factor solution, the FACTOR statement imposes zero constraints on the $\text{[math]}$ elements in the upper triangle of $\text{[math]}$ 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 $\text{[math]}$ in the model is set to 0). Note that the rank of $\text{[math]}$ is equal to the number $\text{[math]}$ of components in $\text{[math]}$ . If $\text{[math]}$ is smaller than the number of variables in the moment matrix $\text{[math]}$ , the matrix of predicted model values is singular and maximum likelihood estimates for $\text{[math]}$ cannot be computed. You should compute ULS estimates in this case.

HEYWOOD | HEY

constrains the diagonal elements of $\text{[math]}$ to be nonnegative; in other words, the model is replaced by

$\text{[math]}$

N=m

specifies the number of first-order factors or components. The number $\text{[math]}$ of factors should not exceed the number $\text{[math]}$ of variables in the covariance or correlation matrix analyzed. For the saturated model, $\text{[math]}$ , the COMP option should generally be specified for $\text{[math]}$ ; otherwise, $\text{[math]}$ . For $\text{[math]}$ no factor loadings are estimated, and the model is $\text{[math]}$ , with $\text{[math]}$ . By default, $\text{[math]}$ .

NORM <= KAISER | NONE>

Kaiser-normalizes the rows of the factor pattern for rotation. NORM=KAISER, which is the default, has exactly the same effect as NORM. You can turn off the normalization by NORM=NONE.

RCONVERGE=p

RCONV=p

specifies the convergence criterion for rotation cycles. The option is applicable to rotation by using either the QUARTIMAX, VARIMAX, EQUAMAX, or PARSIMAX method in the ROTATE= option. Rotation stops when the scaled change of the simplicity function value is less than the RCONVERGE= value. The default convergence criterion is

$\text{[math]}$

where $\text{[math]}$ and $\text{[math]}$ are simplicity function values of the current cycle and the previous cycle, respectively, $\text{[math]}$ is a scaling factor, and $\text{[math]}$ is 1E–9 by default and is modified by the RCONVERGE= value.

RITER=n

specifies the maximum number of cycles $\text{[math]}$ for factor rotation by using either the QUARTIMAX, VARIMAX, EQUAMAX, or PARSIMAX method in the ROTATE= option. The default $\text{[math]}$ is the maximum between 10 times the number of variables and 100.

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:

$\text{[math]}$

where the columns of matrix $\text{[math]}$ contain the eigenvectors of $\text{[math]}$ .

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 $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ of the oblique model,

$\text{[math]}$

To specify the structure for matrix $\text{[math]}$ , $\text{[math]}$ , or $\text{[math]}$ , 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 programming 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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